text:
\begin{document}

\title{Entire solutions for a class of fourth order semilinear elliptic equations with weights}

\author{}

\date{}

{\Large
\centerline{Entire solutions for a class of fourth order}
\centerline{semilinear elliptic equations with weights}
}
\bigskip

\centerline{\scshape Paolo Caldiroli\footnote{email: paolo.caldiroli@unito.it} and Gabriele Cora\footnote{email: 330933@studenti.unito.it}}
\medskip

{\footnotesize

 \centerline{Dipartimento di Matematica, Universit\`a di Torino}
 \centerline{via Carlo Alberto, 10 -- 10123 Torino, Italy}

}
\bigskip

\begin{abstract}
We investigate the problem of entire solutions for a class of fourth order, dilation invariant, semilinear elliptic equations with power-type weights and with subcritical or critical growth in the nonlinear term. These equations define non compact variational problems and are characterized by the presence of a term containing lower order derivatives, whose strength is ruled by a parameter $\lambda$. We can prove existence of entire solutions found as extremal functions for some Rellich-Sobolev type inequalities. Moreover, when the nonlinearity is suitably close to the critical one and the parameter $\lambda$ is large, symmetry breaking phenomena occur and in some cases the asymptotic behavior of radial and non radial ground states can be somehow described.
\vspace{6pt}\\
\textbf{Keywords:} Weighted biharmonic operator, extremal functions, Rellich-So\-bo\-lev inequality, breaking symmetry.\vspace{6pt}\\
\textbf{2010 Mathematics Subject Classification:} 26D10, 47F05.
\end{abstract}

\section{Introduction}

In recent years much interest has been addressed to a class of equations shaped on
\begin{equation}
\label{eq:NLRellich}
\Delta(|x|^{\alpha}\Delta u)=|x|^{-\beta}|u|^{q-2}u\quad\text{in }\R^{n}
\end{equation}
where the dimension $n$ and the parameters $\alpha$, $\beta$ and $q>2$ are asked to satisfy suitable restrictions. In particular, in the case of the pure biharmonic operator, we quote, e.g., \cite{AOM}, \cite{BerFarFerGaz}, \cite{B}, \cite{BM}, \cite{C14}, \cite{Lions}, \cite{Mit93}, \cite{Mit96}, \cite{Mus13}, \cite{NousSwanYang}.

Equations like (\ref{eq:NLRellich}) arise in a natural way from variational inequalities of the form
$$
S_{\alpha,q}\left(\int_{\R^{n}}|x|^{-\beta}|u|^{q}~\!dx\right)^{2/q}\le\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx\quad\forall u\in C^{\infty}_{c}(\R^{n}\setminus\{0\})
$$
which, for $q>2$, can be considered as nonlinear versions of the weighted Rellich inequality (this case occurs taking $q=2$ and $-\beta=\alpha-4$).

In addition, (\ref{eq:NLRellich}) can be viewed as a higher order version of equations like
\begin{equation}
\label{eq:NLHardy}
-\mathrm{div}(|x|^{a}\nabla u)=|x|^{-b}|u|^{q-2}u\quad\text{in }\R^{n}
\end{equation}
where again $n\in\N$, $a,b\in\R$ and $q>2$ are subject to some constraints. Even (\ref{eq:NLHardy}) comes from a class of variational inequalities which are often known in the literature as Caffarelli-Kohn-Nirenberg inequalities (\cite{CKN}) and can be read as interpolation of the linear weighted Hardy inequality with the weighted Sobolev inequality.

As one can expect, both for (\ref{eq:NLRellich}) and for (\ref{eq:NLHardy}), the restrictions on the parameters are needed in order to guarantee that the supporting variational inequalities hold true. In particular, these restrictions necessarily involve some dilation invariance which is a typical feature of any problem displaying scaling processes.

In this work we study a class of equations which are built as linear combination of (\ref{eq:NLRellich}) and (\ref{eq:NLHardy}). In particular we are interested in the existence of nontrivial solutions to
\begin{equation}
\label{eq:pb}
\left\{\begin{array}{l}
\Delta(|x|^{\alpha}\Delta u)-\lambda~\!\mathrm{div}(|x|^{\alpha-2}\nabla u)=|x|^{-\beta}|u|^{q-2}u\quad\text{in }\R^{n}\\
\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx<\infty
\end{array}\right.
\end{equation}
where $n\ge 5$, $q>2$,
\begin{equation}
\label{eq:alpha-beta}
4-n<\alpha<n~\!,~~
\beta=n-\frac{q(n-4+\alpha)}{2}~\!,
\end{equation}
and $\lambda$ is a real parameter subject to some limitation.
In particular we look for ground states of (\ref{eq:pb}), i.e., solutions to (\ref{eq:pb}) characterized as minimizers for the following problems:
\begin{equation}
\label{eq:Sq}
S_{\alpha,q}(\lambda):=\inf_{\scriptstyle u\in D^{2,2}(\R^{n};|x|^{\alpha})\atop\scriptstyle u\ne 0}\frac{\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx+\lambda\int_{\R^{n}}|x|^{\alpha-2}|\nabla u|^{2}~\!dx}{\left(\int_{\R^{n}}|x|^{-\beta}|u|^{q}~\!dx\right)^{{2}/{q}}}~\!.
\end{equation}
Here $D^{2,2}(\R^{n};|x|^{\alpha})$ is the space defined as the completion of $C^{\infty}_{c}(\R^{n})$ with respect to the norm
\begin{equation}
\label{eq:norm}
\|u\|_{2,\alpha}^{2}:=\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx~\!.
\end{equation}
We point out that the role of entire solutions and especially of ground states of (\ref{eq:pb}) is rather meaningful since this kind of solutions naturally appear as limiting profiles in the blowup analysis of related classes of nonlinear problems.

As discussed later, thanks to already known variational inequalities (see \cite{CM-Milan11}, \cite{GM13}), one has that $S_{\alpha,q}(\lambda)>0$ and the minimization problem (\ref{eq:Sq}) makes sense whenever
\begin{equation}
\label{eq:q}
2<q\le 2^{**}:=\frac{2n}{n-4}
\end{equation}
and
\begin{equation}
\label{eq:delta-nabla}
\lambda>-\gamma_{\alpha}\text{~~where~~}\gamma_{\alpha}:=\inf_{\scriptstyle u\in D^{2,2}(\R^{n};|x|^{\alpha})\atop\scriptstyle u\ne 0}\frac{\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx}{\int_{\R^{n}}|x|^{\alpha-2}|\nabla u|^{2}~\!dx}~\!.
\end{equation}
In particular it is known that if $n\ge 5$ and $\alpha\in(4-n,n)$ then $\gamma_{\alpha}>0$. Moreover for $\alpha\in[0,n)$ then $\gamma_{\alpha}=(n-\alpha)^{2}/4$. These facts are discussed in \cite{CM-Bangalore}, \cite{Mor}, \cite{GM11}.

Taking $\beta$ as in (\ref{eq:alpha-beta}) makes problem (\ref{eq:pb})  invariant with respect to the action of the weighted dilation group
\begin{equation}
\label{eq:dilation}
\rho\mapsto (\rho*u)(x)=\rho^{\frac{n-4+\alpha}{2}}u(\rho x)\quad(\rho>0).
\end{equation}
This invariance is responsible of a lack of compactness in the study of the minimization problem (\ref{eq:Sq}). Adapting some techniques already used for different problems (\cite{BM}, \cite{CM-Milan11}), we develop a suitable argument allowing us to recover some local compactness, and we get a first existence result, stated as follows.

\begin{Theorem}
\label{T:ground-state}
Let $n\ge 5$ and assume (\ref{eq:alpha-beta}). Then:
\begin{itemize}
\item[(i)]
For $q\in(2,2^{**})$ and $\lambda\in(-\gamma_{\alpha},\infty)$ problem (\ref{eq:pb}) admits a ground state.
\item[(ii)]
For $q=2^{**}$, for every $\alpha\in(4-n,n)$ there exists $\lambda_{\alpha}>-\gamma_{\alpha}$ such that problem (\ref{eq:pb}) admits a ground state if $\lambda\in(-\gamma_{\alpha},\lambda_{\alpha})$ (the value $\lambda_{\alpha}$ is given by (\ref{eq:lambda-n-alpha})).
\end{itemize}
\end{Theorem}

By exploiting the rotational symmetry of the domain and of the weights in (\ref{eq:pb}), we can drop the upper bound on $q$ in (\ref{eq:q}) by looking for radial ground states for problems (\ref{eq:pb}), namely, non trivial, radial weak solutions of (\ref{eq:pb}) characterized as minimum points for
\begin{equation}
\label{eq:Sqrad}
S_{\alpha,q}^{\mathrm{rad}}(\lambda):=\inf_{\scriptstyle u\in D^{2,2}_{\text{rad}}(\R^{n};|x|^{\alpha})\atop\scriptstyle u\ne 0}\frac{\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx+\lambda\int_{\R^{n}}|x|^{\alpha-2}|\nabla u|^{2}~\!dx}{\left(\int_{\R^{n}}|x|^{-\beta}|u|^{q}~\!dx\right)^{{2}/{q}}}
\end{equation}
where $D^{2,2}_{\text{rad}}(\R^{n};|x|^{\alpha})$ is the space of radial functions belonging to $D^{2,2}(\R^{n};|x|^{\alpha})$. We have that:

\begin{Theorem}
\label{T:radial-ground-state}
Let $n\ge 5$ and assume (\ref{eq:alpha-beta}). Then for every $q\in(2,\infty)$ and $\lambda>-(n-\alpha)^{2}/{4}$, problem (\ref{eq:pb}) admits a radial ground state. Moreover such a ground state has constant sign and is unique up to the weighted dilation (\ref{eq:dilation}).
\end{Theorem}

The second part of our work consists in the study of global ground states of (\ref{eq:pb}) given by Theorem \ref{T:ground-state}. In particular we are interested in investigating radial symmetry or not of these solutions. We find symmetry breaking in different situations.
A first result in this direction is the following.

\begin{Theorem}
\label{T:SB1}
Let $n\ge 5$ and $\lambda>0$. There exist $\overline\alpha>0$ and $\overline{q}\in(2,2^{**})$, both depending on $\lambda$, such that if $q\in(\overline{q},2^{**}]$ and $|\alpha|<\overline\alpha$ then $S_{\alpha,q}(\lambda)<S_{\alpha,q}^{\mathrm{rad}}(\lambda)$. In particular, if $q\in(\overline{q},2^{**})$ and $|\alpha|<\overline\alpha$ then global ground states of (\ref{eq:pb}) are not radially symmetric.
\end{Theorem}

The previous result is obtained by noticing that $S_{0,2^{**}}(\lambda)<S_{0,2^{**}}^{\mathrm{rad}}(\lambda)$ and using some continuity of the mappings $(\alpha,q)\mapsto S_{\alpha,q}(\lambda)$ and $(\alpha,q)\mapsto S_{\alpha,q}^{\mathrm{rad}}(\lambda)$. We have no sharp information on the region of values $(\alpha,q)$ for which global ground states of (\ref{eq:pb}) are non radial. On the other hand, for fixed $q$ and $\alpha$, again symmetry breaking is displayed for $\lambda$ large, as stated in the next result.

\begin{Theorem}
\label{T:non-radial-ground-state}
Let $n\ge 5$ and assume (\ref{eq:alpha-beta}). Let $2^{*}=\frac{2n}{n-2}$ be the critical exponent for the first order Sobolev embedding.
\begin{itemize}
\item[(i)] If $q\in(2^{*},2^{**})$ then for $\lambda$ large enough (depending on $q$) any global ground state of (\ref{eq:pb}) is not radially symmetric.
\item[(ii)] If $q\in(2,2^{*}]$ and
\begin{equation}
\label{eq:FelliSchneider}
\frac{1}{n-1}\left(\frac{n-4+\alpha}{2}\right)^{2}>\frac{1}{q-2}-\frac{1}{q+2}
\end{equation}
then for $\lambda$ large enough (depending on $q$) any global ground state of (\ref{eq:pb}) is not radially symmetric.
\end{itemize}
\end{Theorem}

When $q\in(2,2^{*})$ we can better describe the limit profile of ground states of (\ref{eq:pb}) as $\lambda\to\infty$. To this aim, we need to introduce the lower order problem
\begin{equation}
\label{eq:pb1}
\left\{\begin{array}{l}
-\mathrm{div}(|x|^{\alpha-2}\nabla u)=|x|^{-\beta}|u|^{q-2}u\quad\text{in }\R^{n}\\
\int_{\R^{n}}|x|^{\alpha-2}|\nabla u|^{2}~\!dx<\infty
\end{array}\right.
\end{equation}
The natural variational space for problem (\ref{eq:pb1}) is $D^{1,2}(\R^{n};|x|^{\alpha-2})$ defined as the completion of $C^{\infty}_{c}(\R^{n}\setminus\{0\})$ with respect to the norm
$$
\|u\|_{1,\alpha-2}^{2}=\int_{\R^{n}}|x|^{\alpha-2}|\nabla u|^{2}~\!dx~\!.
$$
Ground states of problem (\ref{eq:pb1}) are defined as weak solutions of (\ref{eq:pb1}) minimizing
$$
\widetilde{S}_{\alpha,q}=\inf_{\scriptstyle u\in D^{1,2}(\R^{n};|x|^{\alpha-2})\atop\scriptstyle u\ne 0}\frac{\int_{\R^{n}}|x|^{\alpha-2}|\nabla u|^{2}~\!dx}{\left(\int_{\R^{n}}|x|^{-\beta}|u|^{q}~\!dx\right)^{2/q}}.
$$
We finally can show:
\begin{Theorem}
\label{T:limiting-profile}
Let $n\ge 5$, $q\in(2,2^{*})$ and assume (\ref{eq:alpha-beta}). If $\lambda_{k}\to\infty$ and $u_{k}\in D^{2,2}(\R^{n};|x|^{\alpha})$ is a ground state of (\ref{eq:pb}) with $\lambda=\lambda_{k}$, then there exists a sequence $(\rho_{k})\subset(0,\infty)$ such that, for a subsequence, $\lambda_{k}^{-\frac{1}{q-2}}\rho_{k}*u_{k}$ converges strongly in $D^{1,2}(\R^{n};|x|^{\alpha-2})$ to a ground state of (\ref{eq:pb1}). The same holds for radial ground states.
\end{Theorem}

We point out that condition (\ref{eq:FelliSchneider}) has been found in \cite{FS} for having symmetry breaking of ground states of the lower order problem (\ref{eq:pb1}).

We finally observe that most of the results contained in this work are presented in \cite{Ctesi}, generalize in a nontrivial way and complete with new contributions some previous results discussed in \cite{C14} limited to the case $\alpha=0$.

\section{Preliminaries}

Here we introduce the space $D^{2,2}(\R^{n};|x|^{\alpha})$ for $n\ge 5$ and $\alpha\in(4-n,n)$ and we discuss its main embedding properties. The starting point can be given by the weighted Rellich inequality stating that
\begin{equation}
\label{eq:Rellich}
\delta_{\alpha}\int_{\R^{n}}|x|^{\alpha-4}|u|^{2}~\!dx\le\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx\quad\forall u\in C^{\infty}_{c}(\R^{n}\setminus\{0\})
\end{equation}
with optimal constant
\begin{equation}
\label{eq:deltaalpha}
\delta_{\alpha}=\left[\left(\frac{n-2}{2}\right)^{2}-\left(\frac{\alpha-2}{2}\right)^{2}\right]^2.
\end{equation}
We refer to the paper \cite{CM-CalVar12} and to its bibliography for a deeper discussion on (\ref{eq:Rellich}) and some generalizations. Inequality (\ref{eq:Rellich}) allows us to define the space $D^{2,2}(\R^{n};|x|^{\alpha})$ as the completion of $C^{\infty}_{c}(\R^{n}\setminus\{0\})$ with respect to the Hilbertian norm given by (\ref{eq:norm}).
Let $2^{**}=\frac{2n}{n-4}$ be the critical exponent for the second order Sobolev embedding. It is known (see, e.g., \cite{CM-Milan11}) that if $q\in[2,2^{**}]$ and $\beta$ is given as in (\ref{eq:alpha-beta}) then $D^{2,2}(\R^{n};|x|^{\alpha})$ is continuously embedded into $L^{q}(\R^{n};|x|^{\beta})$, that is the space of mappings in $L^{q}$ with respect to the measure $|x|^{-\beta}dx$.
\medskip

For future convenience let us introduce also the space $D^{1,2}(\R^{n};|x|^{\widetilde\alpha})$ which can be defined as the completion of $C^{\infty}_{c}(\R^{n}\setminus\{0\})$ with respect to the Hilbertian norm
$$
\|u\|^2_{1,\widetilde\alpha}=\int_{\R^{n}}|x|^{\widetilde\alpha}|\nabla u|^{2}~\!dx~\!.
$$
This definition of $D^{1,2}(\R^{n};|x|^{\widetilde\alpha})$ is well posed when $\widetilde\alpha>2-n$, thanks to the weighted Hardy inequality
\begin{equation}
\label{eq:Hardy}
h_{\widetilde\alpha}\int_{\R^{n}}|x|^{\widetilde\alpha-2}|u|^{2}~\!dx\le\int_{\R^{n}}|x|^{\widetilde\alpha}|\nabla u|^{2}~\!dx\quad\forall u\in C^{\infty}_{c}(\R^{n}\setminus\{0\})
\end{equation}
which holds with optimal constant
$$
h_{\widetilde\alpha}=\left(\frac{n-2+\widetilde\alpha}{2}\right)^{2}.
$$
In fact here we consider the case $\widetilde\alpha=\alpha-2$. Let $2^{*}=\frac{2n}{n-2}$ be the critical exponent for the first order Sobolev embedding. As a direct consequence of the Caffarelli-Kohn-Nirenberg inequalities \cite{CKN}, if $q\in[2,2^{*}]$ and $\beta$ is given as in (\ref{eq:alpha-beta}) then $D^{1,2}(\R^{n};|x|^{\alpha-2})$ is continuously embedded into $L^{q}(\R^{n};|x|^{\beta})$.
\medskip

As noticed in the Introduction, for $n\ge 5$ and $\alpha\in(4-n,n)$, the space $D^{2,2}(\R^{n};|x|^{\alpha})$ turns out to be continuously embedded into $D^{1,2}(\R^{n};|x|^{\alpha-2})$. We denote by $\gamma_{\alpha}$ the embedding constant, defined as in (\ref{eq:delta-nabla}). The value of $\gamma_{\alpha}$ is explicitly known only for $\alpha$ in a sub-interval of the admissible one $(4-n,n)$. More precisely, there exists $\alpha^{*}\in(4-n,0)$ such that $\gamma_{\alpha}=(n-\alpha)^{2}/4$ for $\alpha\in(\alpha^{*},n)$, whereas this expression is not valid for all $\alpha\in(4-n,n)$ (see \cite{CM-Bangalore}).
\medskip

Hence, for $n\ge 5$, $q\in(2,2^{**}]$, $\alpha$ and $\beta$ as in (\ref{eq:alpha-beta}), and $\lambda>-\gamma_{\alpha}$, the minimization problem (\ref{eq:Sq}) is meaningful since the corresponding infimum value $S_{\alpha,q}(\lambda)$ is positive. Let us recall the following well known property linking the minimization problem (\ref{eq:Sq}) with (\ref{eq:pb}).

\begin{Lemma}
\label{L:inf=equation}
If $S_{\alpha,q}(\lambda)>0$ and $u\in D^{2,2}(\R^{n};|x|^{\alpha})$ is a minimum point for (\ref{eq:Sq}), then $U=\tau^{\frac{1}{q-2}}u$ is a solution of (\ref{eq:pb}), being $\tau=S_{\alpha,q}(\lambda)\left(\int_{\R^{n}}|x|^{-\beta}|u|^{q}~\!dx\right)^{\frac{2-q}{q}}$.
\end{Lemma}

Since we are interested also in radial ground states of (\ref{eq:pb}), we introduce also spaces of radial functions. More precisely, with obvious notation, we set
\begin{gather*}
D^{2,2}_{rad}(\R^{n};|x|^{\alpha}):=\{u\in D^{2,2}(\R^{n};|x|^{\alpha})~|~u=u(|x|)\},\\
D^{1,2}_{rad}(\R^{n};|x|^{\widetilde\alpha}):=\{u\in D^{1,2}(\R^{n};|x|^{\widetilde\alpha})~|~u=u(|x|)\},
\end{gather*}
with the same restrictions on $n$, $\alpha$ and $\widetilde\alpha$ as before. For $n\ge 5$ and $\alpha\in(4-n,n)$ the value of the best constant embedding of $D^{2,2}_{rad}(\R^{n};|x|^{\alpha})$ into $D^{1,2}_{rad}(\R^{n};|x|^{\alpha-2})$ is sharp:
\begin{equation}
\label{eq:nabla-delta-rad}
\inf_{\scriptstyle u\in D^{2,2}_{rad}(\R^{n};|x|^{\alpha})\atop\scriptstyle u\ne 0}\frac{\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx}{\int_{\R^{n}}|x|^{\alpha-2}|\nabla u|^{2}~\!dx}=\frac{(n-\alpha)^{2}}{4}
\end{equation}
(see \cite{CM-Bangalore} and \cite{GM11}).

The Emden-Fowler transform, defined by
\begin{equation}
\label{eq:EmdenFowler}
u(x)=|x|^{\frac{4-n-\alpha}{2}}w(-\log|x|)
\end{equation}
provides a nice isomorphism between the space $D^{2,2}_{rad}(\R^{n};|x|^{\alpha})$ and the standard Sobolev space $H^{2}(\R)$, and between $D^{1,2}_{rad}(\R^{n};|x|^{\alpha-2})$ and $H^{1}(\R)$. Indeed:

\begin{Lemma}
\label{L:EmdenFowler}
Let $n\ge 5$ and $\alpha\in(4-n,n)$. For any radial mapping $u\colon\R^{n}\setminus\{0\}\to\R$ let $w\colon\R\to\R$ be defined by (\ref{eq:EmdenFowler}), and viceversa.
\begin{itemize}
\item[(i)]
$u\in D^{2,2}_{rad}(\R^{n};|x|^{\alpha})$ if and only if $w\in H^{2}(\R)$. Moreover
$$
\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx=\omega_{n}\int_{\R}\left(|w''|^{2}+2\widetilde{\delta}_{\alpha}|w'|^{2}+\delta_{\alpha}|w|^{2}\right)~\!dt
$$
where $\omega_{n}=|\mathbb{S}^{n-1}|$ and $\widetilde{\delta}_{\alpha}=\left(\frac{n-2}{2}\right)^{2}+\left(\frac{\alpha-2}{2}\right)^{2}$.
\item[(ii)]
$u\in D^{1,2}_{rad}(\R^{n};|x|^{\alpha-2})$ if and only if $w\in H^{1}(\R)$. Moreover
$$
\int_{\R^{n}}|x|^{\alpha-2}|\nabla u|^{2}~\!dx=\omega_{n}\int_{\R}\left(|w'|^{2}+\widetilde{h}_{\alpha}|w|^{2}\right)~\!dt
$$
where $\widetilde{h}_{\alpha}={h}_{\alpha-2}=\left(\frac{n-4+\alpha}{2}\right)^{2}$.
\item[(iii)]
For $q\ge 2$ and $\beta$ as in (\ref{eq:alpha-beta}), $u\in L^{q}(\R^{n};|x|^{-\beta})$ if and only if $w\in L^{q}(\R)$. In this case
$$
\int_{\R^{n}}|x|^{-\beta}|u|^{q}~\!dx=\omega_{n}\int_{\R}|w|^{q}~\!dt~\!.
$$
\end{itemize}
\end{Lemma}
For a proof we refer to \cite{CM-Milan11}.

Thanks to Lemma \ref{L:EmdenFowler} the spaces $D^{2,2}_{rad}(\R^{n};|x|^{\alpha})$ and $D^{1,2}_{rad}(\R^{n};|x|^{\alpha-2})$ are embedded into $L^{q}(\R^{n};|x|^{-\beta})$ for \emph{every} $q\ge 2$. In particular, taking account also of (\ref{eq:nabla-delta-rad}), the minimization problem (\ref{eq:Sqrad}) is meaningful for all $\lambda>-\frac{(n-\alpha)^{2}}{4}$ and $q>2$. Moreover, an analogue of Lemma \ref{L:inf=equation} holds true, namely:

\begin{Lemma}
\label{L:radial-inf=equation}
If $S_{\alpha,q}^{rad}(\lambda)>0$ and $u\in D^{2,2}_{rad}(\R^{n};|x|^{\alpha})$ is a minimum point for (\ref{eq:Sqrad}), then $U=\tau^{\frac{1}{q-2}}u$ is a solution of (\ref{eq:pb}), being $\tau=S_{\alpha,q}^{rad}(\lambda)\left(\int_{\R^{n}}|x|^{-\beta}|u|^{q}~\!dx\right)^{\frac{2-q}{q}}$.
\end{Lemma}

\section{Global ground states}
In this section we prove Theorem \ref{T:ground-state}.

\begin{Lemma}
\label{L:Ekeland}
There exists a sequence $(u_{k})$ in $D^{2,2}(\R^{n};|x|^{\alpha})$ satisfying
\begin{gather}
\label{eq:lambda-norm}
\int_{\R^{n}}\left(|x|^{\alpha}|\Delta u_{k}|^2+\lambda|x|^{\alpha-2}|\nabla u_{k}|^{2}\right)~\!dx=S_{\alpha,q}(\lambda)^{{q}/{q-2}}+o(1)\\
\label{eq:q-norm}
\int_{\R^{n}}|x|^{-\beta}|u_{k}|^{q}~\!dx=S_{\alpha,q}(\lambda)^{{q}/{q-2}}\\
\label{eq:B2}
\int_{B_{2}}|x|^{-\beta}|u_{k}|^{q}~\!dx=\frac{1}{2}S_{\alpha,q}(\lambda)^{{q}/{q-2}}\\
\label{eq:almost-solution}
\Delta(|x|^{\alpha}\Delta u_{k})-\lambda~\!\mathrm{div}(|x|^{\alpha-2}\nabla u_{k})-|x|^{-\beta}|u_{k}|^{q-2}u_{k}\to 0\text{ in $(D^{2,2}(\R^{n};|x|^{\alpha}))'$.}
\end{gather}
\end{Lemma}

\proof
Set
$$
F(u)=\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx+\lambda\int_{\R^{n}}|x|^{\alpha-2}|\nabla u|^{2}~\!dx\quad\text{and}\quad G(u)=\int_{\R^{n}}|x|^{-\beta}|u|^{q}~\!dx
$$
so that
$$
S_{\alpha,q}(\lambda)=\inf\{F(u)~|~G(u)=1\}.
$$
Since the constraint $G(u)=1$ defines a smooth manifold, by the Ekeland variational principle, one can find a sequence $(v_{k})\subset D^{2,2}(\R^{n};|x|^{\alpha})$ such that
$$
F'(v_{k})-\mu_{k}G'(v_{k})\to 0\text{~~in~~}(D^{2,2}(\R^{n};|x|^{\alpha}))'\text{~~where~~}\mu_{k}=\frac{\langle\nabla F(v_{k}),\nabla G(v_{k})\rangle}{\|\nabla G(v_{k})\|_{2,\alpha}},
$$
$\nabla F(v_{k})$, $\nabla G(v_{k})$ denote the Riesz representative in $D^{2,2}(\R^{n};|x|^{\alpha})$ of the functionals $F'(v_{k})$, $G'(v_{k})$, respectively, and $\langle~\!,\rangle$ stands for the inner product in $D^{2,2}(\R^{n};|x|^{\alpha})$ corresponding to the norm (\ref{eq:norm}). One can easily check that the sequence $(v_{k})$ is bounded in $D^{2,2}(\R^{n};|x|^{\alpha})$, $\mu_{k}\to\frac{2}{q}S_{\alpha,q}(\lambda)$ and $\sup_{k}\|G'(v_{k})\|<\infty$. Hence
$$
F'(v_{k})-\mu G'(v_{k})\to 0\text{~~in~~}(D^{2,2}(\R^{n};|x|^{\alpha}))'\text{~~where~~}\mu=\frac{2}{q}S_{\alpha,q}(\lambda)~\!.
$$
Now the sequence
$$
\tilde{u}_{k}=S_{\alpha,q}(\lambda)^{1/(q-2)}v_{k}
$$
turns out to satisfy (\ref{eq:lambda-norm}), (\ref{eq:q-norm}) and (\ref{eq:almost-solution}). Finally, for every $k$ one can find $\rho_{k}>0$ such that
$$
u_{k}(x)=\rho_{k}^{\frac{n-4+\alpha}{2}}\tilde{u}_{k}(\rho_{k} x)
$$
satisfies (\ref{eq:B2}). The sequence $(u_{k})$ always verifies (\ref{eq:lambda-norm}), (\ref{eq:q-norm}) and (\ref{eq:almost-solution}) because the functionals $F$ and $G$ are invariant with respect to (\ref{eq:dilation}).
\QED

A key tool in our argument is the following compactness lemma. This result is an adaptation of a tool already used in previous works, like \cite{BM} or \cite{CM-Milan11}.

\begin{Lemma}
\label{L:compact}
Let $(u_{k})$ be a sequence in $D^{2,2}(\R^{n};|x|^{\alpha})$ satisfying (\ref{eq:almost-solution}). If
\begin{gather}
\label{eq:weak-conv}
u_{k}\to 0\text{ weakly in }D^{2,2}(\R^{n};|x|^{\alpha})\\
\label{eq:small}
\limsup\int_{B_{R}}|x|^{-\beta}|u_{k}|^{q}~\!dx<S_{\alpha,q}(\lambda)^{q/(q-2)}\text{ for some $R>0$,}
\end{gather}
then $|x|^{-\beta}|u_{k}|^{q}\to 0$ strongly in $L^{1}_{\mathrm{loc}}(B_{R})$.
\end{Lemma}

\proof

Fix $R'\in(0,R)$ and take a cut-off function $\varphi\in C^{\infty}_{c}(B_{R})$ such that $\varphi=1$ on $B_{R'}$. We point out that the sequence $(\varphi^{2} u_{k})$ is bounded in $D^{2,2}(\R^{n};|x|^{\alpha})$. Using $\varphi^{2} u_{k}$ as a test function in (\ref{eq:almost-solution}) we obtain
\begin{equation}
\label{eq:almost-sol-test}
\begin{split}
\int_{\R^{n}}\varphi^{2} u_{k}\Delta(|x|^{\alpha}\Delta u_{k})~\!dx-\lambda&\int_{\R^{n}}\varphi^{2}u_{k}~\!\mathrm{div}(|x|^{\alpha-2}\nabla u_{k})~\!dx\\
&=\int_{\R^{n}}|x|^{-\beta}\varphi^{2}|u_{k}|^{q}~\!dx+o(1).
\end{split}
\end{equation}
By (\ref{eq:weak-conv}) $u_{k}\to 0$ weakly in $H^{2}_{\mathrm{loc}}(\R^{n}\setminus\{0\})$ and then, by compactness, $u_{k}\to 0$ strongly in $H^{1}_{\mathrm{loc}}(\R^{n}\setminus\{0\})$. Hence we have that
\begin{gather*}
\int_{\R^{n}}|x|^{\alpha}|\Delta(\varphi u_{k})|^{2}~\!dx=\int_{\R^{n}}|x|^{\alpha}\varphi^{2}|\Delta u_{k}|^{2}~\!dx+o(1)\\
\int_{\R^{n}}|x|^{\alpha}(\Delta u_{k})\Delta(\varphi^{2} u_{k})~\!dx=\int_{\R^{n}}|x|^{\alpha}\varphi^{2}|\Delta u_{k}|^{2}~\!dx+o(1)\\
\int_{\R^{n}}|x|^{\alpha-2}\nabla u_{k}\cdot\nabla(\varphi^{2}u_{k})~\!dx=
\int_{\R^{n}}|x|^{\alpha-2}|\nabla(\varphi u_{k})|^{2}~\!dx+o(1).
\end{gather*}
Then, after integration by parts,
\begin{gather*}
\label{eq:Delta2}
\int_{\R^{n}}\varphi^{2} u_{k}\Delta(|x|^{\alpha}\Delta u_{k})~\!dx=\int_{\R^{n}}|x|^{\alpha}|\Delta(\varphi u_{k})|^{2}~\!dx+o(1)\\
\label{eq:nabla}
\int_{\R^{n}}\varphi^{2}u_{k}~\!\mathrm{div}(|x|^{\alpha-2}\nabla u_{k})~\!dx =-\int_{\R^{n}}|x|^{\alpha-2}|\nabla(\varphi u_{k})|^{2}~\!dx+o(1).
\end{gather*}
Consequently (\ref{eq:almost-sol-test}) reduces to
\begin{equation}
\label{eq:almost-sol-test2}
\int_{\R^{n}}|x|^{\alpha}|\Delta(\varphi u_{k})|^{2}~\!dx+\lambda\int_{\R^{n}}|x|^{\alpha-2}|\nabla(\varphi u_{k})|^{2}~\!dx=\int_{\R^{n}}|x|^{-\beta}\varphi^{2}|u_{k}|^{q}~\!dx+o(1).
\end{equation}
By (\ref{eq:small}) there exists $\varepsilon_{0}>0$ such that
\begin{equation}
\label{eq:epsilon}
\int_{B_{R}}|x|^{-\beta}|u_{k}|^{q}~\!dx\le\varepsilon_{0}<S_{\alpha,q}(\lambda)^{q/(q-2)}\quad\forall k\text{ large.}
\end{equation}
Therefore, using the H\"{o}lder inequality and (\ref{eq:epsilon}), we estimate
\begin{equation}
\label{eq:almost-sol-test3}
\int_{\R^{n}}|x|^{-\beta}\varphi^{2}|u_{k}|^{q}~\!dx\le \varepsilon_{0}^{(q-2)/q}\left(\int_{\R^{n}}|x|^{-\beta}|\varphi u_{k}|^{q}~\!dx\right)^{2/q}.
\end{equation}
On the other side, by definition of $S_{\alpha,q}(\lambda)$,
\begin{equation}
\label{eq:almost-sol-test4}
\int_{\R^{n}}|x|^{\alpha}|\Delta(\varphi u_{k})|^{2}~\!dx+\lambda\int_{\R^{n}}|x|^{\alpha-2}|\nabla(\varphi u_{k})|^{2}~\!dx\ge S_{\alpha,q}(\lambda)\left(\int_{\R^{n}}|x|^{-\beta}|\varphi u_{k}|^{q}~\!dx\right)^{\frac{2}{q}}.
\end{equation}
Therefore from (\ref{eq:almost-sol-test2})--(\ref{eq:almost-sol-test4}) it follows that
$$
S_{\alpha,q}(\lambda)\left(\int_{\R^{n}}|x|^{-\beta}|\varphi u_{k}|^{q}~\!dx\right)^{2/q}\le\varepsilon_{0}^{(q-2)/q}\left(\int_{\R^{n}}|x|^{-\beta}|\varphi u_{k}|^{q}~\!dx\right)^{2/q}+o(1).
$$
As $\varepsilon_{0}<S_{\alpha,q}(\lambda)^{q/(q-2)}$ we infer that
$$
\int_{\R^{n}}|x|^{-\beta}|\varphi u_{k}|^{q}~\!dx\to 0
$$
and then, since $\varphi=1$ on $B_{R'}$ and $R'$ is arbitrary in $(0,R)$, $|x|^{-\beta}|u_{k}|^{q}\to 0$ strongly in $L^{1}_{\mathrm{loc}}(B_{R})$.
\QED
\medskip

\noindent
\textbf{Proof of Theorem \ref{T:ground-state}, part (i).}
Let $(u_{k})$ be a sequence in $D^{2,2}(\R^{n};|x|^{\alpha})$ satisfying (\ref{eq:lambda-norm})--(\ref{eq:almost-solution}), as given by Lemma \ref{L:Ekeland}. Since $\lambda>-\gamma_{\alpha}$, by (\ref{eq:lambda-norm}), the sequence $(u_{k})$ is bounded in $D^{2,2}(\R^{n};|x|^{\alpha})$ and then it admits a subsequence, still denoted $(u_{k})$, weakly converging to some $u\in D^{2,2}(\R^{n};|x|^{\alpha})$. If $u\ne 0$, then $u$ is a minimizer for $S_{\alpha,q}(\lambda)$ and $u_{k}\to u$ strongly in $D^{2,2}(\R^{n};|x|^{\alpha})$. The proof of this fact is definitely standard: one can adapt to our situation a well known argument (see, e.g., \cite{Str}, Chapt.~1, Sect.~4). Hence we have to exclude that $u=0$. We argue by contradiction, assuming that $u=0$. In this case, by Lemma \ref{L:compact},
\begin{equation}
\label{eq:B1-zero}
\int_{B_{1}}|x|^{-\beta}|u_{k}|^{q}\!~dx\to 0.
\end{equation}
Therefore, by (\ref{eq:B2}),
\begin{equation}
\label{eq:annulus}
\int_{B_{2}\setminus B_{1}}|x|^{-\beta}|u_{k}|^{q}\!~dx\to\frac{1}{2}S_{q}(\lambda)^{{q}/{q-2}}.
\end{equation}
Since $q\in(2,2^{**})$ and $u_{k}\to 0$ weakly in $H^{2}_{\mathrm{loc}}(\R^{n}\setminus\{0\})$, the Rellich compactness Theorem implies that $u_{k}\to 0$ strongly in $L^{q}(B_{2}\setminus B_{1})$, in contradiction with (\ref{eq:annulus}). Therefore $u$ cannot be zero and the proof is complete.
\QED
\medskip

Now we focus on the case of critical exponent $q=2^{**}$. To this purpose let us denote by $S^{**}$ the best constant for the second order standard Sobolev embedding, defined by
$$
S^{**}:=\inf_{\scriptstyle u\in C^{\infty}_{c}(\R^{n})\atop\scriptstyle u\ne 0}\frac{\int_{\R^{n}}|\Delta u|^{2}~\!dx}{\left(\int_{\R^{n}}|u|^{2^{**}}~\!dx\right)^{2/2^{**}}}.
$$
As $S^{**}>0$ one can introduce the space $D^{2,2}(\R^{n})$ as the completion of $C^{\infty}_{c}(\R^{n})$ with respect to the norm $\|\Delta u\|_{L^{2}}$. One has that $C^{\infty}_{c}(\R^{n}\setminus\{0\})$ is dense in $D^{2,2}(\R^{n})$ and then $D^{2,2}(\R^{n})$ coincides with the space $D^{2,2}(\R^{n};|x|^{\alpha})$ with $\alpha=0$ and $S^{**}=S_{0,2^{**}}(0)$. Let us recall the following result.

\begin{Lemma}\label{L:EFJ}(\cite{EFJ})
The function $U(x)={\left(1+|x|^{2}\right)^{-\frac{n-4}{2}}}$ is a minimizer  for $S^{**}$ in $D^{2,2}(\R^{n})$.
\end{Lemma}

A condition for existence of a ground state for problem (\ref{eq:pb}) in case of critical exponent is stated by the following result.
\begin{Lemma}
\label{L:<}
If $S_{\alpha,2^{**}}(\lambda)<S^{**}$ then $S_{\alpha,2^{**}}(\lambda)$ is attained in $D^{2,2}(\R^{n};|x|^{\alpha})$.
\end{Lemma}

\proof
As in the proof of Theorem \ref{T:ground-state}, part (i), there exists a sequence $(u_{k})$ in $D^{2,2}(\R^{n};|x|^{\alpha})$ satisfying (\ref{eq:lambda-norm})--(\ref{eq:almost-solution}) and there exists $u\in D^{2,2}(\R^{n};|x|^{\alpha})$ such that $u_{k}\to u$ weakly in $D^{2,2}(\R^{n};|x|^{\alpha})$ and strongly in $H^{1}_{\mathrm{loc}}(\R^{n}\setminus\{0\})$. If $u\ne 0$ then, with a stardard argument, $u$ turns out to be a minimizer. Assume by contradiction that $u=0$. Then (\ref{eq:B1-zero}) and (\ref{eq:annulus}) hold. Let us fix a cut-off function $\varphi\in C^{\infty}_{c}(\R^{n}\setminus\{0\})$ such that $0\le\varphi\le 1$ and $\varphi(x)=1$ for $1\le|x|\le 2$. Arguing as in the first part of the proof of Lemma \ref{L:compact} we obtain (\ref{eq:almost-sol-test2}). By (\ref{eq:lambda-norm}) and (\ref{eq:q-norm}), we also have that
\begin{equation}
\label{eq:almost-sol-crit11}
\begin{split}
\int_{\R^{n}}|x|^{\alpha}|\Delta(\varphi u_{k})|^{2}~\!dx&+\lambda\int_{\R^{n}}|x|^{\alpha-2}|\nabla(\varphi u_{k})|^{2}~\!dx\\
&\le S_{\alpha,2^{**}}(\lambda)\left(\int_{\R^{n}}|x|^{-\beta}|\varphi u_{k}|^{2^{**}}~\!dx\right)^{\frac{2}{2^{**}}}+o(1).
\end{split}
\end{equation}
Since
$$
\int_{\R^{n}}|x|^{\alpha-2}|\nabla(\varphi u_{k})|^{2}~\!dx\le C\int_{B_{2}\setminus B_{1}}\left(|\nabla u_{k}|^{2}+u_{k}^{2}\right)~\!dx
$$
for some constant $C>0$, and $u_{k}\to 0$ strongly in $H^{1}_{\mathrm{loc}}(\R^{n}\setminus\{0\})$, (\ref{eq:almost-sol-test2}) reduces to
\begin{equation}
\label{eq:almost-sol-crit2}
\int_{\R^{n}}|x|^{\alpha}|\Delta(\varphi u_{k})|^{2}~\!dx=\int_{\R^{n}}|x|^{-\beta}\varphi^{2}|u_{k}|^{2^{**}}~\!dx+o(1).
\end{equation}
Now we apply the identity
\begin{equation*}
\begin{split}
|x|^{-\alpha}|\Delta(|x|^{\frac{\alpha}{2}}w)|^{2}&=|\Delta w|^{2}+\alpha^{2}|x|^{-4}|x \cdot \nabla w|^{2}+\frac{\alpha^{2}}{4}\left(n-2+\frac{\alpha}{2}\right)^{2}|x|^{-4}w^{2}\\
&\quad+2\alpha|x|^{-2}(x\cdot\nabla w)\Delta w+\alpha^{2}\left(n-2+\frac{\alpha}{2}\right)|x|^{-4}w(x\cdot\nabla w)\\
&\quad+\alpha\left(n-2+\frac{\alpha}{2}\right)|x|^{-2}w\Delta w
\end{split}
\end{equation*}
with $w=\varphi u_{k}$ and, using again the fact that $u_{k}\to 0$ strongly in $H^{1}_{\mathrm{loc}}(\R^{n}\setminus\{0\})$, we infer that
$$
\int_{\R^{n}}|x|^{\alpha}|\Delta(\varphi u_{k})|^{2}~\!dx=\int_{\R^{n}}\left|\Delta(|x|^{\alpha/2}\varphi u_{k})\right|^{2}~\!dx+o(1).
$$
Hence from (\ref{eq:almost-sol-test2}), (\ref{eq:almost-sol-crit11}) and (\ref{eq:almost-sol-crit2}) it follows that
\begin{equation*}
\begin{split}
S_{\alpha,2^{**}}(\lambda)\left(\int_{\R^{n}}|x|^{-\beta}|\varphi u_{k}|^{2^{**}}~\!dx\right)^{\frac{2}{2^{**}}}&\ge\int_{\R^{n}}\left|\Delta(|x|^{\alpha/2}\varphi u_{k})\right|^{2}~\!dx+o(1)\\
&\ge S^{**}\left(\int_{\R^{n}}||x|^{\alpha/2}\varphi u_{k}|^{2^{**}}~\!dx\right)^{\frac{2}{2^{**}}}+o(1).
\end{split}
\end{equation*}
Since
$$
\int_{\R^{n}}||x|^{\alpha/2}\varphi u_{k}|^{2^{**}}~\!dx=\int_{\R^{n}}|x|^{-\beta}|\varphi u_{k}|^{2^{**}}~\!dx
$$
and, by hypothesis, $S_{\alpha,2^{**}}(\lambda)<S^{**}$, we deduce that
$$
\int_{\R^{n}}|x|^{-\beta}|\varphi u_{k}|^{2^{**}}~\!dx\to 0
$$
in contradiction with (\ref{eq:annulus}), as $\varphi=1$ on $B_{2}\setminus B_{1}$.
\QED

\begin{Lemma}
\label{L:lambda}
If
\begin{equation}
\label{eq:condition-lambda}
-\gamma_{\alpha}<\lambda<\left(\alpha-\frac{\alpha^{2}}{4}\right)\frac{(\alpha^{2}-4\alpha)(n-3)-2(n-2)^{2}(n-4)}{(n-4+\alpha)^{2}(n-3)+(n-4)^{2}}
\end{equation}
then $S_{\alpha,2^{**}}(\lambda)<S^{**}$.
\end{Lemma}

\proof
Set $u(x)=|x|^{-\frac{\alpha}{2}}U(x)$ where $U$ is as in Lemma \ref{L:EFJ}. One can check that
\begin{gather*}
\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx=\int_{\R^{n}}|\Delta U|^{2}~\!dx+A_{\alpha}\int_{\R^{n}}|x|^{-4}U^{2}~\!dx
\\
\int_{\R^{n}}|x|^{\alpha-2}|\nabla u|^{2}~\!dx=B_{\alpha}\int_{\R^{n}}|x|^{-4}U^{2}~\!dx
\\
\int_{\R^{n}}|x|^{-\beta}|u|^{2^{**}}~\!dx=\int_{\R^{n}}U^{2^{**}}~\!dx
\end{gather*}
where
\begin{gather*}
A_{\alpha}=\left(\frac{\alpha^{2}}{4}-\alpha\right)\left(\frac{\alpha^{2}}{4}-\alpha-\frac{(n-2)^{2}(n-4)}{2(n-3)}\right)\\
B_{\alpha}=\left(\frac{n-4+\alpha}{2}\right)^{2}+\frac{(n-4)^{2}}{4(n-3)}.
\end{gather*}
For detailed computation see \cite{CM-Milan11} or \cite{Ctesi}.
Then
$$
S_{\alpha,q}(\lambda)\le\frac{\int_{\R^{n}}|\Delta U|^{2}~\!dx}{\left(\int_{\R^{n}}U^{2^{**}}~\!dx\right)^{2/2^{**}}}+\left(A_{\alpha}+\lambda~\!B_{\alpha}\right)\frac{\int_{\R^{n}}U^{2}~\!dx}{\left(\int_{\R^{n}}U^{2^{**}}~\!dx\right)^{2/2^{**}}}~\!.
$$
Hence the strict inequality $S_{\alpha,q}(\lambda)<S^{**}$ holds true when
\begin{equation}
\label{eq:lambda-n-alpha}
\lambda<\lambda_{\alpha}:=-\frac{A_{\alpha}}{B_{\alpha}}=\left(\alpha-\frac{\alpha^{2}}{4}\right)\frac{(\alpha^{2}-4\alpha)(n-3)-2(n-2)^{2}(n-4)}{(n-4+\alpha)^{2}(n-3)+(n-4)^{2}}.
\end{equation}
We point out that $\lambda_{\alpha}=0$ when $\alpha=0,4$, $\lambda_{\alpha}>0>-\gamma_{\alpha}$ when $\alpha\in(4-n,0)\cup(4,n)$, whereas $\lambda_{\alpha}<0$ when $\alpha\in(0,4)$. In fact, with some calculation one can check that in this last case $\lambda_{\alpha}>-\frac{(n-\alpha)^{2}}{4}=-\gamma_{\alpha}$. Hence (\ref{eq:condition-lambda}) is completely proved.
\QED

Clearly the proof of Theorem \ref{T:ground-state}, part (ii) is a direct consequence of Lemmas \ref{L:<} and \ref{L:lambda}.

When $\alpha=0$, condition (\ref{eq:condition-lambda}) is optimal for the validity of the strict inequality $S_{\alpha,2^{**}}(\lambda)<S^{**}$. Indeed one has:

\begin{Proposition}
\label{P}
If $\lambda\ge 0$ then $S_{0,2^{**}}(\lambda)=S^{**}$. Moreover, for $\lambda>0$ the infimum $S_{0,2^{**}}(\lambda)$ is not achieved in $D^{2,2}(\R^{n})$.
\end{Proposition}
For a proof, see \cite{C14} or \cite{Ctesi}.

\section{Radial ground states}
Here we prove Theorem \ref{T:radial-ground-state}. In view of Lemma \ref{L:EmdenFowler} we have that
\begin{equation}
\label{eq:radial-minimization}
S_{\alpha,q}^{\mathrm{rad}}(\lambda)=\omega_{n}^{\frac{q-2}{q}}\inf_{\scriptstyle w\in H^{2}(\R)\atop\scriptstyle w\ne 0}\frac{\int_{\R}\left(|w''|^{2}+2a_{\lambda}|w'|^{2}+b_{\lambda}|w|^{2}\right)~\!dt}{\left(\int_{\R}|w|^{q}~\!dt\right)^{{2}/{q}}}
\end{equation}
where
\begin{equation}
\label{eq:alambda-blambda}
a_{\lambda}=\frac{(n-2)^{2}}{4}+\frac{(\alpha-2)^{2}}{4}+\frac{\lambda}{2}\quad\text{and}\quad b_{\lambda}=\left(\frac{(n-\alpha)^{2}}{4}+\lambda\right)\left(\frac{n-4+\alpha}{2}\right)^{2}.
\end{equation}
We point out that, thanks to the assumption $\lambda>-(n-\alpha)^{2}/{4}$, the values $a_{\lambda}$ and $b_{\lambda}$ are positive. Now we use the following key result, proved in \cite{BM}:

\begin{Theorem}
\label{T:BM}
For every $a,b>0$ and $q>2$ the minimization problem
$$
\inf_{\scriptstyle w\in H^{2}(\R)\atop\scriptstyle w\ne 0}\frac{\int_{\R}\left(|w''|^{2}+2a|w'|^{2}+b|w|^{2}\right)~\!dt}{\left(\int_{\R}|w|^{q}~\!dt\right)^{{2}/{q}}}
$$
admits a minimum point. In addition, if $a^{2}\ge b$ then the minimum point is positive and unique, up to the natural invariances of the problem (i.e., translation, inversion, multiplication by a non zero constant).
\end{Theorem}

In the case in consideration
$$
a_{\lambda}^{2}-b_{\lambda}=\left(\frac{(n-2)(\alpha-2)}{2}-\frac{\lambda}{2}\right)^{2}.
$$
Hence by Theorem \ref{T:BM} there exists a positive function $w\in H^{2}(\R)$ which is a minimizer for the problem defined by the right hand side of (\ref{eq:radial-minimization}). Such a minimizer is unique up to translation, inversion, and multiplication by a non zero constant. Then, using Lemma \ref{L:EmdenFowler}, we infer that the mapping $u$ defined by (\ref{eq:EmdenFowler}) belongs to $D^{2,2}_{rad}(\R^{n};|x|^{\alpha})$, is a positive minimizer for $S_{\alpha,q}^{rad}(\lambda)$ and is the unique minimizer up to the weighted dilation (\ref{eq:dilation}) and to a multiplicative constant. Then one applies Lemma \ref{L:radial-inf=equation} to get a radial ground state for problem (\ref{eq:pb}).
\QED

\section{Symmetry breaking and limiting profiles}
This section contains the proof of the symmetry breaking results stated in Theorems \ref{T:SB1} and \ref{T:non-radial-ground-state}, and the description of the limit profile of ground states for $q\in(2,2^{*})$, when $\lambda\to\infty$ (Theorem \ref{T:limiting-profile}).

Let us start with the discussion of Theorem \ref{T:SB1}, whose proof lies on the following semicontinuity inequalities.

\begin{Lemma}
\label{L:semicontinuity-inequalities}
For $\lambda\ge 0$ one has
\begin{equation}
\label{eq:sc-ineq}
\limsup_{(\alpha,q)\to(0,2^{**}_{-})}S_{\alpha,q}(\lambda)\le S_{0,2^{**}}(\lambda)\quad\text{and}\quad S^{\mathrm{rad}}_{0,2^{**}}(\lambda)\le\liminf_{(\alpha,q)\to(0,2^{**}_{-})}S^{\mathrm{rad}}_{\alpha,q}(\lambda)~\!.
\end{equation}
\end{Lemma}

\proof
Fix $\lambda\ge 0$. For every $u\in C^{\infty}_{c}(\R^{n}\setminus\{0\})$, $u\ne 0$, set
$$
Q_{\alpha,q}(u)=\frac{\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx+\lambda\int_{\R^{n}}|x|^{\alpha-2}|\nabla u|^{2}~\!dx}{\left(\int_{\R^{n}}|x|^{-\beta}|u|^{q}~\!dx\right)^{{2}/{q}}}.
$$
Since $Q_{\alpha,q}(u)\to Q_{0,2^{**}}(u)$ as $(\alpha,q)\to(0,2^{**}_{-})$ and $C^{\infty}_{c}(\R^{n}\setminus\{0\})$ is dense in $D^{2,2}(\R^{n};|x|^{\alpha})$, the first inequality in (\ref{eq:sc-ineq}) immediately follows. In order to check the second inequality, we proceed in this way. For every $q\in(2,2^{**})$ let $\theta_{q}=\frac{q-2}{2^{**}-2}$. By the H\"{o}lder inequality, one has that
$$
\int_{\R^{n}}|x|^{-\beta}|u|^{q}~\!dx\le\left(\int_{\R^{n}}|x|^{\alpha-4}|u|^{2}~\!dx\right)^{1-\theta_{q}}\left(\int_{\R^{n}}|x|^{\frac{n\alpha}{n-4}}|u|^{2^{**}}~\!dx\right)^{\theta_{q}}
$$
and

consequently
\begin{equation}
\label{eq:sn1}
S_{\alpha,q}^{\mathrm{rad}}(\lambda)^{\frac{q}{\theta_{q}2^{**}}}\ge S_{\alpha,2}^{\mathrm{rad}}(\lambda)^{\frac{(1-\theta_{q})2}{\theta_{q}2^{**}}}S_{\alpha,2^{**}}^{\mathrm{rad}}(\lambda)\ge \delta_{\alpha}^{\frac{(1-\theta_{q})2}{\theta_{q}2^{**}}}S_{\alpha,2^{**}}^{\mathrm{rad}}(\lambda)
\end{equation}
(see (\ref{eq:Rellich})--(\ref{eq:deltaalpha})).
Now we estimate $S_{\alpha,2^{**}}^{\mathrm{rad}}(\lambda)$ in terms of $S_{0,2^{**}}^{\mathrm{rad}}(\lambda)$.
For every $\alpha$ let $\tau_{\alpha}:=1+\frac{\alpha}{n-4}$ and for every radial $u\in C^{\infty}_{c}(\R^{n}\setminus\{0\})$, set $\widetilde{u}(x)=u(|x|^{1/\tau_{\alpha}})$. One can check that
\begin{gather*}
\int_{\R^{n}}|\widetilde{u}|^{2^{**}}~\!dx=\tau_{\alpha}\int_{\R^{n}}\!|x|^{\frac{n\alpha}{n-4}}|u|^{2^{**}}~\!dx~\!,~~\int_{\R^{n}}\!|x|^{-2}|\nabla\widetilde{u}|^{2}~\!dx=\tau_{\alpha}^{-1}\!\int_{\R^{n}}\!|x|^{\alpha-2}|\nabla u|^{2}~\!dx,\\
\int_{\R^{n}}\!|\Delta\widetilde{u}|^{2}~\!dx=\tau_{\alpha}^{-3}\!\int_{\R^{n}}\!|x|^{\alpha}|\Delta u+R_{\alpha}u|^{2}~\!dx~~\text{where}~ R_{\alpha}u=(\tau_{\alpha}-1)(n-2)\nabla u\cdot\frac{x}{|x|^{2}}.
\end{gather*}
Setting $\varepsilon_{\alpha}=|\tau_{\alpha}-1|(n-2)\gamma_{\alpha}^{-1/2}$ and using (\ref{eq:delta-nabla}), as $\alpha\in(4-n,n)$ we can estimate
$$
\int_{\R^{n}}|x|^{\alpha}|R_{\alpha}u|^{2}~\!dx\le\varepsilon_{\alpha}^{2}\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx
$$
and then, by the Cauchy-Schwarz inequality,
$$
\int_{\R^{n}}|x|^{\alpha}|\Delta u+R_{\alpha}u|^{2}~\!dx\le (1+\varepsilon_{\alpha})^{2}\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx~\!.
$$
Therefore
\begin{equation*}
\begin{split}
S_{0,2^{**}}^{\mathrm{rad}}(\lambda)&\le\frac{\int_{\R^{n}}(|\Delta\widetilde{u}|^{2}+\lambda|x|^{-2}|\nabla\widetilde{u}|^{2})~\!dx}{\left(\int_{\R^{n}}|\widetilde{u}|^{2^{**}}~\!dx\right)^{\frac{2}{2^{**}}}}\\
&\le\frac{(1+\varepsilon_{\alpha})^{2}}{\tau_{\alpha}^{3+2/2^{**}}}~\!
\frac{\int_{\R^{n}}\left(|x|^{\alpha}|\Delta u|^{2}+\lambda|x|^{\alpha-2}|\nabla u|^{2}\right)~\!dx}{\left(\int_{\R^{n}}|x|^{\frac{n\alpha}{n-4}}|u|^{2^{**}}~\!dx\right)^{\frac{2}{2^{**}}}}\\
&\quad+\frac{\lambda}{\tau_{\alpha}^{1+2/2^{**}}}\left(1-\frac{(1+\varepsilon_{\alpha})^{2}}{\tau_{\alpha}^{2}}\right)\frac{\int_{\R^{n}}|x|^{\alpha-2}|\nabla u|^{2}~\!dx}{\left(\int_{\R^{n}}|x|^{\frac{n\alpha}{n-4}}|u|^{2^{**}}~\!dx\right)^{\frac{2}{2^{**}}}}\\
&\le K_{\alpha}\frac{\int_{\R^{n}}\left(|x|^{\alpha}|\Delta u|^{2}+\lambda|x|^{\alpha-2}|\nabla u|^{2}\right)~\!dx}{\left(\int_{\R^{n}}|x|^{\frac{n\alpha}{n-4}}|u|^{2^{**}}~\!dx\right)^{\frac{2}{2^{**}}}}
\end{split}
\end{equation*}
where
$$
K_{\alpha}={\left(\frac{(1+\varepsilon_{\alpha})^{2}}{\tau_{\alpha}^{3+2/2^{**}}}+\frac{\lambda\gamma_{\alpha}^{-1}}{\tau_{\alpha}^{1+2/2^{**}}}\left|1-\frac{(1+\varepsilon_{\alpha})^{2}}{\tau_{\alpha}^{2}}\right|\right)}.
$$
Hence
\begin{equation}
\label{eq:sn2}
S_{\alpha,2^{**}}^{\mathrm{rad}}(\lambda)\ge K_{\alpha}^{-1}S_{0,2^{**}}^{\mathrm{rad}}(\lambda)~\!.
\end{equation}
For $|\alpha|$ small, one has that $C^{-1}\le\gamma_{\alpha}\le C$ and $C^{-1}\le\delta_{\alpha}\le C$ for some constant $C>0$. Consequently, if $(\alpha,q)\to(0,2^{**}_{-})$, then $\tau_{\alpha}\to 1$, $\theta_{q}\to 1$, $\varepsilon_{\alpha}\to 0$, and $K_{\alpha}\to 1$. Thus the second inequality in (\ref{eq:sc-ineq}) follows from (\ref{eq:sn1})--(\ref{eq:sn2}).
\QED
\medskip

\noindent
\textbf{Proof of Theorem \ref{T:SB1}.} Fix $\lambda>0$. By Theorem \ref{T:radial-ground-state} $S_{0,2^{**}}^{\mathrm{rad}}(\lambda)$ is achieved in $D^{2,2}_{rad}(\R^{n})$. Instead, by Proposition \ref{P}, $S_{0,2^{**}}(\lambda)$ is not attained. Hence $S_{0,2^{**}}^{\mathrm{rad}}(\lambda)>S_{0,2^{**}}(\lambda)$. Therefore the conclusion follows by applying Lemma \ref{L:semicontinuity-inequalities}.
\QED

Now let us address to Theorem \ref{T:non-radial-ground-state}. It is convenient to normalize the minimization problems defined by (\ref{eq:Sq}) and (\ref{eq:Sqrad}) as follows. For every $\varepsilon\ge 0$ set
\begin{gather*}
\widetilde{S}_{\alpha,q}(\varepsilon):=\inf_{\scriptstyle u\in C^{\infty}_{c}(\R^{n}\setminus\{0\})\atop\scriptstyle u\ne 0}\frac{\varepsilon\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx+\int_{\R^{n}}|x|^{\alpha-2}|\nabla u|^{2}~\!dx}{\left(\int_{\R^{n}}|x|^{-\beta}|u|^{q}~\!dx\right)^{{2}/{q}}}\\
\widetilde{S}_{\alpha,q}^{rad}(\varepsilon):=\inf_{\scriptstyle u\in C^{\infty}_{c}(\R^{n}\setminus\{0\})\atop\scriptstyle u=u(|x|),~u\ne 0}\frac{\varepsilon\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx+\int_{\R^{n}}|x|^{\alpha-2}|\nabla u|^{2}~\!dx}{\left(\int_{\R^{n}}|x|^{-\beta}|u|^{q}~\!dx\right)^{{2}/{q}}}
\end{gather*}
We remark that if $\lambda>0$ then
\begin{equation}
\label{eq:lambda-eps}
\widetilde{S}_{\alpha,q}(\lambda^{-1})=\lambda^{-1}{S}_{\alpha,q}(\lambda)\quad\text{and}\quad\widetilde{S}_{\alpha,q}^{rad}(\lambda^{-1})=\lambda^{-1}{S}_{\alpha,q}^{rad}(\lambda).
\end{equation}

\begin{Lemma}\label{L:Seps}
Let $n\ge 5$ and $\alpha\in(4-n,n)$.
\begin{itemize}
\item[(i)]
For every $q\in(2,2^{**}]$ one has that $\widetilde{S}_{\alpha,q}(\varepsilon)\to\widetilde{S}_{\alpha,q}(0)$  as $\varepsilon\to 0$.
\item[(ii)]
For every $q>2$ one has that $\widetilde{S}_{\alpha,q}^{rad}(\varepsilon)\to\widetilde{S}_{\alpha,q}^{rad}(0)$ as $\varepsilon\to 0$.
\end{itemize}
\end{Lemma}

\proof (i) Fix $\alpha\in(4-n,n)$ and $q\in(2,2^{**}]$ and set
$$
\widetilde{Q}_{\varepsilon}(u):=\frac{\varepsilon\int_{\R^{n}}|x|^{\alpha}|\Delta u|^{2}~\!dx+\int_{\R^{n}}|x|^{\alpha-2}|\nabla u|^{2}~\!dx}{\left(\int_{\R^{n}}|x|^{-\beta}|u|^{q}~\!dx\right)^{{2}/{q}}}\quad\forall u\in C^{\infty}_{c}(\R^{n}\setminus\{0\}),~u\ne 0.
$$
Since $\widetilde{S}_{\alpha,q}(\varepsilon)\le\widetilde{Q}_{\varepsilon}(u)$, when $\varepsilon\to 0$ one has that
$$
\limsup \widetilde{S}_{\alpha,q}(\varepsilon)\le\widetilde{Q}_{0}(u)\quad\forall u\in C^{\infty}_{c}(\R^{n}\setminus\{0\}),~u\ne 0
$$
namely $\limsup\widetilde{S}_{\alpha,q}(\varepsilon)\le\widetilde{S}_{\alpha,q}(0)$. On the other hand $\widetilde{Q}_{\varepsilon}(u)\ge \widetilde{Q}_{0}(u)$ and then $\widetilde{S}_{\alpha,q}(\varepsilon)\ge\widetilde{S}_{\alpha,q}(0)$. Hence the conclusion immediately follows. Clearly (ii) is proved in the same way.
\QED
\medskip

\noindent
\textbf{Proof of Theorem \ref{T:non-radial-ground-state}.}
If $q\in(2^{*},2^{**}]$ then $\widetilde{S}_{\alpha,q}^{rad}(0)>0$ whereas $\widetilde{S}_{\alpha,q}(0)=0$. Indeed, by Lemma \ref{L:EmdenFowler},
$$
\widetilde{S}_{\alpha,q}^{rad}(0)=\omega_{n}^{1-\frac{2}{q}}\inf_{\scriptstyle w\in C^{\infty}_{c}(\R)\atop\scriptstyle w\ne 0}\frac{\int_{\R}(|w'|^{2}+\widetilde{h}_{\alpha}|w|^{2})~\!dt}{\left(\int_{\R}|w|^{q}~\!dt\right)^{2/q}}
$$
which is positive because $H^{1}(\R)$ is embedded into $L^{q}$. Instead, taking $u\in C^{\infty}_{c}(\R^{n})$, $u\ne 0$, with support in the unit ball, fixing $x_{0}\in\R^{n}$ with $|x_{0}|=1$ and setting
$$
u_{\delta}(x)=\delta^{-\frac{n-2}{2}}u\left(\frac{x-x_{0}}{\delta}\right),
$$
one can check that $\widetilde{Q}_{0}(u_{\delta})\to 0$ as $\delta\to 0$, because $q>2^{*}$. Hence, by Lemma \ref{L:Seps}, $\widetilde{S}_{\alpha,q}(\varepsilon)<\widetilde{S}_{\alpha,q}^{rad}(\varepsilon)$ for $\varepsilon>0$ small, and then ${S}_{\alpha,q}(\lambda)<{S}_{\alpha,q}^{rad}(\lambda)$ for $\lambda$ large, by (\ref{eq:lambda-eps}). If $q\in(2,2^{*}]$ and (\ref{eq:FelliSchneider}) holds, then $\widetilde{S}_{\alpha,q}(0)<\widetilde{S}_{\alpha,q}^{rad}(0)$, as proved in \cite{FS} and one concludes as before that ${S}_{\alpha,q}(\lambda)<{S}_{\alpha,q}^{rad}(\lambda)$ for $\lambda$ large.
\QED

In the following we study the behavior of ground states of problems (\ref{eq:pb}) for fixed $q\in(2,2^{*})$, in the limit $\lambda\to\infty$.

\begin{Lemma}
\label{L:limit-profile}
Let $n\ge 5$, $q\in(2,2^{*})$ and assume (\ref{eq:alpha-beta}). Let $\varepsilon_{k}\to 0^{+}$ and for every $k$ let $v_{k}\in D^{2,2}(\R^{n};|x|^{\alpha})$ be a minimizer for $\widetilde{S}_{\alpha,q}(\varepsilon_{k})$ with
\begin{equation}
\label{eq:assum-k}
\int_{\R^{n}}|x|^{-\beta}|v_{k}|^{q}~\!dx=1\quad\text{and}\quad\int_{B_{2}}|x|^{-\beta}|v_{k}|^{q}~\!dx=\frac{1}{2}~\!.
\end{equation}
If $v_{k}\to v$ weakly in $D^{1,2}(\R^{n};|x|^{\alpha-2})$ then $v_{k}\to v$ strongly in $D^{1,2}(\R^{n};|x|^{\alpha-2})$ and $v$ is a minimizer for $\widetilde{S}_{\alpha,q}(0)$.
\end{Lemma}

\proof
Let us write, briefly, $\widetilde{S}_{k}=\widetilde{S}_{\alpha,q}(\varepsilon_{k})$ and $\widetilde{S}_{0}=\widetilde{S}_{\alpha,q}(0)$. Since $v_{k}$ is a minimizer for $\widetilde{S}_{k}$ and (\ref{eq:assum-k}) holds, we have that
\begin{equation}
\label{eq:tilde-bounded}
\widetilde{S}_{0}\le\int_{\R^{n}}|x|^{\alpha-2}|\nabla v_{k}|^{2}~\!dx\le\widetilde{S}_{k}
\end{equation}
and then,
\begin{equation}
\label{eq:zero-k}
\varepsilon_{k}\int_{\R^{n}}|x|^{\alpha}|\Delta v_{k}|^{2}~\!dx\to 0
\end{equation}
for $\widetilde{S}_{k}\to\widetilde{S}_{0}$ by Lemma \ref{L:Seps}. Now we want to exclude that $v=0$. To do this, we argue by contradiction, assuming that $v\to 0$ weakly in $D^{1,2}(\R^{n};|x|^{\alpha-2})$. Since $v_{k}$ is a minimizer for $\widetilde{S}_{k}$, it is so for $S_{\alpha,q}(\varepsilon_{k}^{-1})$ and, by Lemma \ref{L:inf=equation},
\begin{equation}
\label{eq:pb-epsk}
\varepsilon_{k}\Delta(|x|^{\alpha}\Delta v_{k})-\mathrm{div}(|x|^{\alpha-2}\nabla v_{k})=\widetilde{S}_{k}|x|^{-\beta}|v_{k}|^{q-2}v_{k}\quad\text{on }\R^{n}.
\end{equation}
Taking a cut-off function $\varphi\in C^{\infty}_{c}(\R^{n})$ with $\mathrm{supp}(\varphi)\subset B_{2}$ and $\varphi\equiv 1$ in $B_{1}$, we can use $\varphi^{2}v_{k}$ as a test function in (\ref{eq:pb-epsk}) getting that
\begin{equation}
\label{eq:start-k}
\varepsilon_{k}\int_{\R^{n}}|x|^{\alpha}\Delta v_{k}\Delta(\varphi^{2}v_{k})~\!dx+\int_{\R^{n}}|x|^{\alpha-2}\nabla v_{k}\cdot\nabla(\varphi^{2}v_{k})~\!dx=\widetilde{S}_{k}\int_{\R^{n}}|x|^{-\beta}\varphi^{2}|v_{k}|^{q}~\!dx~\!.
\end{equation}
We estimate each term of (\ref{eq:start-k}) as follows. Firstly
\begin{equation*}
\begin{split}
&\left|\int_{\R^{n}}|x|^{\alpha}\Delta v_{k}\Delta(\varphi^{2}v_{k})~\!dx\right|\le\int_{\R^{n}}|x|^{\alpha}|\Delta v_{k}|\big|\Delta(\varphi^{2})v_{k}+2\nabla(\varphi^{2})\cdot\nabla v_{k}\big|~\!dx\\
&\hspace{7.8cm}+\int_{\R^{n}}|x|^{\alpha}|\varphi\Delta v_{k}|^{2}~\!dx\\
&\le\left(\int_{\R^{n}}|x|^{\alpha}|\Delta v_{k}|^{2}~\!dx\right)^{1/2}\left(C\int_{B_{2}\setminus B_{1}}(|v_{k}|^{2}+|\nabla v_{k}|^{2})~\!dx\right)^{1/2}+\int_{\R^{n}}|x|^{\alpha}|\Delta v_{k}|^{2}~\!dx
\end{split}
\end{equation*}
because $\varphi$ is constant in $B_{1}$ and out of $B_{2}$. Then
\begin{equation}
\label{eq:Delta-k}
\varepsilon_{k}\int_{\R^{n}}|x|^{\alpha}\Delta v_{k}\Delta(\varphi^{2}v_{k})~\!dx\to 0
\end{equation}
thanks to (\ref{eq:zero-k}) and because the sequence $(v_{k})$ is bounded in $D^{1,2}(\R^{n};|x|^{\alpha-2})$ and then also in $H^{1}(B_{2}\setminus B_{1})$.
Secondly, as $v_{k}\to 0$ weakly in $D^{1,2}(\R^{n};|x|^{\alpha-2})$ we also obtain that
\begin{equation}
\label{eq:grad-k}
\int_{\R^{n}}|x|^{\alpha-2}\nabla v_{k}\cdot\nabla(\varphi^{2}v_{k})~\!dx=\int_{\R^{n}}|x|^{\alpha-2}|\nabla(\varphi v_{k})|^{2}~\!dx+o(1)~\!.
\end{equation}
In addition, using H\"older inequality and (\ref{eq:assum-k})
\begin{equation}
\label{eq:nonlinear-k}
\int_{\R^{n}}|x|^{-\beta}|v_{k}|^{q}\varphi^{2}~\!dx\le 2^{-\frac{q-2}{q}}\left(\int_{\R^{n}}|x|^{-\beta}|\varphi v_{k}|^{q}~\!dx\right)^{2/q}.
\end{equation}
By definition of $\widetilde{S}_{0}$ we also have that
\begin{equation}
\label{eq:CKN-k}
\int_{\R^{n}}|x|^{\alpha-2}|\nabla(\varphi v_{k})|^{2}~\!dx\ge\widetilde{S}_{0}\left(\int_{\R^{n}}|x|^{-\beta}|\varphi v_{k}|^{q}~\!dx\right)^{2/q}.
\end{equation}
Plugging (\ref{eq:Delta-k})--(\ref{eq:CKN-k}) into (\ref{eq:start-k}) and taking into account that $\widetilde{S}_{k}\to\widetilde{S}_{0}$, we obtain
$$
\widetilde{S}_{0}\left(\int_{\R^{n}}|x|^{-\beta}|\varphi v_{k}|^{q}~\!dx\right)^{2/q}\le 2^{-\frac{q-2}{q}}\widetilde{S}_{0}\left(\int_{\R^{n}}|x|^{-\beta}|\varphi v_{k}|^{q}~\!dx\right)^{2/q}+o(1)~\!.
$$
Being $q>2$, we infer that $\int_{B_{1}}|x|^{-\beta}|v_{k}|^{q}~\!dx\to 0$ and, by (\ref{eq:assum-k}),
\begin{equation}
\label{eq:k-Lq-annulus}
\int_{B_{2}\setminus B_{1}}|x|^{-\beta}|v_{k}|^{q}~\!dx\to\frac{1}{2}~\!.
\end{equation}
On the other hand, if $v_{k}\to 0$ weakly in $D^{1,2}(\R^{n};|x|^{\alpha-2})$, in particular $v_{k}\to 0$ weakly in $H^{1}_{loc}(\R^{n}\setminus\{0\})$ and, by the Rellich compactness theorem, $v_{k}\to  0$ strongly in $L^{q}(B_{2}\setminus B_{1})$ because $q<2^{*}$. This is in contradiction with (\ref{eq:k-Lq-annulus}). Therefore $v\ne 0$. Since, by (\ref{eq:zero-k}), $(v_{k})$ is a minimizing sequence for $\widetilde{S}_{0}$ with a nonzero weak limit, it is standard to conclude that $v$ is a minimizer for $\widetilde{S}_{0}$ and $v_{k}\to v$ strongly in $D^{1,2}(\R^{n};|x|^{\alpha-2})$.
\QED
\medskip

\noindent
\textbf{Proof of Theorem \ref{T:limiting-profile}.}
Let $u_{k}\in D^{2,2}(\R^{n};|x|^{\alpha})$ be a ground state of (\ref{eq:pb}) with $\lambda=\lambda_{k}$ and set $S_{k}=S_{\alpha,q}(\lambda_{k})$ and $\widetilde{u}_{k}=S_{k}^{-\frac{1}{q-2}}u_{k}$. Then $\widetilde{u}_{k}$ turns out to be a minimizer for $\widetilde{S}_{\alpha,q}(\lambda_{k}^{-1})$ with
$$
\int_{\R^{n}}|x|^{-\beta}|\widetilde{u}_{k}|^{q}~\!dx=1~\!.
$$
Moreover there exists $\rho_{k}>0$ such that
$$
\int_{B_{2}}|x|^{-\beta}|\rho_{k}*\widetilde{u}_{k}|^{q}~\!dx=\frac{1}{2}~\!.
$$
Hence $v_{k}=\rho_{k}*\widetilde{u}_{k}$ is again a minimizer for $S_{\alpha,q}(\lambda_{k}^{-1})$ and satisfies (\ref{eq:assum-k}). Since (\ref{eq:tilde-bounded}) holds, there exists $v\in D^{1,2}(\R^{n};|x|^{\alpha})$ such that, for a subsequence, $v_{k}\to v$ weakly in $D^{1,2}(\R^{n};|x|^{\alpha})$. Since $\lambda_{k}\to\infty$, by Lemma \ref{L:limit-profile}, $v_{k}\to v$ strongly in $D^{1,2}(\R^{n};|x|^{\alpha})$ and $v$ is a minimizer for $\widetilde{S}_{\alpha,q}(0)$. Then $u=\widetilde{S}_{\alpha,q}(0)^{\frac{1}{q-2}}v$ turns out to be a ground state for problem (\ref{eq:pb1}) and $\lim_{k}\lambda_{k}^{-\frac{1}{q-2}}\rho_{k}*u_{k}=\widetilde{S}_{\alpha,q}(0)^{\frac{1}{q-2}}\lim_{k} S_{k}^{-\frac{1}{q-2}}\rho_{k}*u_{k}=u$ strongly in $D^{1,2}(\R^{n};|x|^{\alpha})$.
\QED
\medskip

\noindent
\textbf{Acknowledgments.} The first author is partially supported by the PRIN2012 grant ``Variational and perturbative aspects of nonlinear differential problems''. He is also member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

\label{References}


text:
\begin{document}

\title{Moduli space of metrics of nonnegative sectional or positive Ricci curvature on Brieskorn quotients in dimension 5}

\author{Jonathan Wermelinger}

\date{\today}

\maketitle
\begin{abstract}\noindent We show that the moduli space of Riemannian metrics of nonnegative sectional curvature on certain quotients of Brieskorn varieties in dimension 5 has infinitely many path components. The same is true for the corresponding moduli space of positive Ricci curvature.\end{abstract}

\section{Introduction}
The subject of moduli spaces of Riemannian metrics with lower curvature bounds has been an active field of study over the last years. The first results on the moduli spaces of 5-manifolds have been presented in \cite{TWi17}. Recently, the first examples of 5-manifolds whose moduli spaces have infinitely many path-components for nonnegative sectional curvature \cite{DGA19} and positive Ricci curvature \cite{G20} have been identified.

In this note, we continue the study of moduli spaces of metrics of nonnegative sectional curvature of quotients of Brieskorn varieties in dimension 5 which was started by Dessai and Gonz\'{a}lez-\'{A}lvaro. One of their results is about the moduli spaces of manifolds homotopy equivalent to $\mathbb{R}\text{P}^5$ whose universal cover is $S^5$, while we will consider quotients of $S^2\times S^3$ under certain involutions. This really only changes the diffeomorphism classification of or quotients and we may apply a very similar procedure as presented in \cite{DGA19}. We also want to point out that our manifolds are also different from the examples presented by Goodman \cite{G20}, since his manifold's second homotopy group is trivial as a $\mathbb{Z}[Z_2]$-module and our's are non-trivial as we will discuss below.

The main result we are going to prove is the following.

\begin{theorem*}
Let $N^5$ be an orientable, closed, smooth $5$-dimensional manifold whose universal cover is $S^2\times S^3$, satisfying $\pi_1(N)=\Z_2$, $w_2(N)\neq 0$ and $\pi_2(N)$ is non-trivial as a $\Z[\Z_2]$-module. Then the moduli space $\mathcal{M}_{sec\geq 0}(N)$ of nonnegative sectional curvature has infinitely many path components. The same is true for the moduli space $\mathcal{M}_{Ric> 0}(N)$ of metrics of positive Ricci curvature.
\end{theorem*}

As we will see, there are five pairwise non-diffeomorphic manifolds satisfying the conditions of this theorem. These are all represented by what we will call a Brieskorn quotient. Furthermore, these manifolds are not all pairwise homotopy equivalent. Unless otherwise stated, all manifolds will be oriented if orientable and all diffeomorphisms will be orientation-preserving.

This note is organized as follows. In section 2, we define Brieskorn manifolds and a cohomogeneity one action on them. From this action, we select an involution and define Brieskorn quotients. We discuss the diffeomorphism classification of manifolds satisfying the assumptions of the main theorem and relate them to the Brieskorn quotients. In section 3, we equip the quotients and other constructions with Riemannian metrics of nonnegative sectional curvature. Section 4 then introduces $Spin^c$-structures and eta-invariants. Finally, the corresponding moduli space is investigated and the proof of our main theorem is given in section 5.

\section{Brieskorn varieties and manifolds}

\subsection{Definition}\label{subsec:def}

Let $z=(z_0,z_1,z_2,z_3)\in\C^4$ and consider the polynomial $$f:\C^{4}\rightarrow \C: z\mapsto z_0^d+z_1^2+z_2^2+z_3^2.$$
For $\epsilon \in \C$ we get the hypersurface $V_\epsilon^3(d):=f^{-1}(\epsilon)$, which is an algebraic manifold with no singular points for $\epsilon\neq 0$, but only a variety if $\epsilon=0$ with an isolated singular point $z=0$ if $d\geq 2$. Let $S^{7}:=\{z\in \C^{4}|\text{ }|z_0|^2+...+|z_3|^2=1\}$ and $D^{8}:=\{z\in \C^{4}|\text{ }|z_0|^2+...+|z_3|^2\leq 1\}$ denote the unit sphere and the closed disk in $\C^{4}=\R^{8}$ respectively. We may then define the link $$\Sigma^{5}_\epsilon(d):=V_\epsilon^3(d)\cap S^{7}$$ and $$W^{6}_\epsilon(d):=V_\epsilon^3(d)\cap D^{8},$$ both equipped with the natural orientation. The manifolds $\Sigma^5_\epsilon(d)$ are examples of \emph{Brieskorn varieties} (see \cite{HM68} and \cite{M68} for more details).

An important fact is that for $\epsilon$ small enough, $\Sigma^{5}_\epsilon(d)$ is diffeomorphic to $\Sigma^{5}_0(d)$ (see \cite[Satz 14.3]{HM68}). Also, $W^{6}_\epsilon(d)$ is homotopy equivalent to a bouquet $S^3\vee ... \vee S^3$ of $3$-spheres (see \cite[p.83]{HM68}).

If $d$ is odd, then $\Sigma^5_0(d)$ is diffeomorphic to $S^5$ (see \cite[14.8 (zweiter) Satz]{HM68}).

Throughout this note, we will consider the case when $d$ is even. In this case, it can be shown that $\Sigma^5_0(d)$ is diffeomorphic to $S^2\times S^3$ (see \cite[Proposition 7]{GT98}).

\subsection{Group action and quotient}\label{subsec:grpac}

Recall that an action of a compact Lie group $G$ on a smooth manifold $M$ is said to be of \emph{cohomogeneity one} if the orbit space $M/G$ is one-dimensional. Consider the following group actions. $$a:S^1\times \C^4 \rightarrow \C^4: (w,(z_0,z_1,z_2,z_3))\mapsto (w^2z_0,w^dz_1,w^dz_2,w^dz_3),$$ $$b:O(3)\times \C^4 \rightarrow \C^4: (A,(z_0,z_1,z_2,z_3))\mapsto (z_0,(A(z_1,z_2,z_3)^T)^T).$$

Suppose $d$ is even. In case $\epsilon\neq 0$, we get an action of $\Z_{2d}\times O(3)$ with ineffective kernel $\{(\pm 1, Id)\}$, whereas for $\epsilon = 0$ we get an action of $S^1\times O(3)$ with the same ineffective kernel as before. Both of these actions are orientation-preserving on $\Sigma^{5}_\epsilon(d)$.

The action of $S^1\times O(3)$ on $\Sigma^{5}_0(d)$ is of cohomogeneity one with orbit space an interval. This can be seen by noticing that the norm of the zero component is not affected by this action.

The principal isotropy is $\Z_2\times O(1)$, corresponding to interior points of the quotient interval. The singular isotropies are $S^1\times O(1)$ and $\Z_2\times O(2)$, corresponding to the vanishing and maximal norm of $z_0$ respectively. The corresponding singular orbits are both of codimension two. For more details on the above cohomogeneity one action, see \cite[\textsection 4.2]{DGA19} and \cite[\textsection 1]{GVWZ06}.

Consider $\tau:=(1,-Id)\in S^1\times O(3)$, i.e. $\tau(z)=(z_0,-z_1,-z_2,-z_3)$ for $z=(z_0,z_1,z_2,z_3)\in\C^4$. This gives an orientation-preserving involution which acts freely on $\Sigma^{5}_\epsilon(d)$ for $\epsilon<1$. Since the polynomial $f(z)=z^d_0+z^2_1+z^2_2+z^2_3$ is $\tau$-invariant, we get an induced map $\tilde{f}:S^{7}/\tau\rightarrow \C$.

We can now define $N^5_\epsilon(d):=\Sigma^{5}_\epsilon(d)/\tau$, which we will call a \emph{Brieskorn quotient}. For $d$ odd, these are homotopy equivalent to $\R\text{P}^5$. For $d$ even, these quotients are not all pairwise homotopy equivalent (see \cite[p.1180 Remark (2)]{GT98}). In both cases, $N^5_\epsilon(d)$ is diffeomorphic to $N^5_0(d)$ for $\epsilon$ sufficiently small (see \cite[Proposition 4.1]{DGA19}).

The above action of $S^1\times O(3)$ on $\Sigma^{5}_0(d)$ descends to a cohomogeneity one action on $N^5_\epsilon(d)$ which also has codimension two singular orbits.

For $0<\epsilon<1$, the fixed points of the action of $\tau$ on $W^{6}_\epsilon(d)$ are $p_i=(\lambda_i,0,0,0)$ , $1\leq i\leq d$, where $\lambda_i$ is a complex d-root of $\epsilon$ for all $i$. These fixed points are isolated and all lie in the interior of $W^{6}_\epsilon(d)$. The subaction $\Z_{2d}\times \{Id\}$ of $\Z_{2d}\times O(2)$ on $W^{6}_\epsilon(d)$ permutes these fixed points.

\subsection{Diffeomorphism classification}\label{subsec:diffclass}

In order to give a diffeomorphism classification of the above quotients, we need to introduce $Pin$ structures. Define the groups $Pin^\pm(n)$ to be the two distinct double covers of $O(n)$, corresponding to the central extensions of $O(n)$ by $\Z_2$: $$1\rightarrow \Z_2\rightarrow Pin^\pm(n)\rightarrow O(n) \rightarrow 1.$$
They are topologically the same space but have different group structures. A $Pin^\pm$-structure on a smooth manifold $M$ is a reduction of the structure group $O(n)$ of its tangent bundle to $Pin^\pm(n)$.

A manifold $M$ admits a $Pin^+$-structure if and only if $w_2(M)=0$ and a $Pin^-$-structure if and only if $w_2(M)=w_1(M)^2$, for $w_1,w_2$ the first and second Stiefel-Whitney class respectively. Furthermore, $Pin^\pm$-structures are in bijection with $H^1(M;\Z_2)$, if any exist (see \cite{KT90}).

If we ask for $Pin^\pm$-structures on manifolds with boundaries to restrict to $Pin^\pm$-structures on the boundary, we get $Pin^\pm$-bordism groups $\Omega^{Pin^\pm}_n$. Kirby and Taylor determined $\Omega^{Pin^+}_4\cong \Z_{16}$ (see \cite{KT90}), and if we identify an element with its inverse, we get $\Omega^{Pin^+}_4/\pm\cong \Z_{8}$.

Recall that a \emph{characteristic submanifold} of a manifold $M^n$ with $\pi_1(M)=\Z_2$ is a submanifold $P^{n-1}\subset M^n$ such that there exists a submanifold $A\subset \tilde{M}$ with $\tilde{M}=A\cup T(A)$, $\partial A = \partial T(A)=\tilde{P}$ and $P=\tilde{P}/T$. Here $\tilde{P}$ is the universal covering and $T$ the corresponding deck-transformation.

Su now proved the following classification result (see \cite[Theorem 1.2.]{S12}).

\begin{theorem}\label{thm:classification5mfds}
Let $M^5$ be an orientable, smooth 5-manifold with $\pi_1(M)\cong \Z_2$ and universal cover $\tilde{M}\cong S^2\times S^3$. If $\pi_2(M)\cong \Z$ is non-trivial as a $\Z[\Z_2]$-module and $w_2(M)\neq 0$, then $M$ is diffeomorphic to $N^5_0(d)$ for some $d=0,2,4,6,8$. The quotients $N^5_0(d)$ are classified by the $Pin^+$-bordism class of their characteristic submanifold $P_d$ with $[P_d]=d\in\Omega^{Pin^+}_4/\pm=\{0,1,...,8\}$.
\end{theorem}

The module structure on $\pi_2(M)$ comes from the $\pi_1(M)$-action on it.

Geiges and Thomas showed that for $d$ even, $N^5_0(d)$ satisfies all of the above conditions (see \cite[Proposition 6, 7 and Lemma 11]{GT98}). In particular, they show that the induced isomorphism $\tau_*:H_2(\Sigma^5_0(d);\mathbb{Z})\rightarrow H_2(\Sigma^5_0(d);\mathbb{Z})$ is non-trivial (see \cite[Lemma 5]{GT98}) and therefore it follows by the use of Hurewicz' map that $\pi_2(N^5_0(d))$ is non-trivial as a $\mathbb{Z}[\mathbb{Z}_2]$-module. Note furthermore that the $Pin^\pm$-bordism class does not depend on the choice of a characteristic submanifold (see \cite[Lemma 9]{GT98}).

From the Proof of \cite[Lemma 11]{GT98} and using the above theorem, one can also deduce the following.

\begin{lemma}\label{lemma:diffquo}
For $d,d'\in\N$ even, $N^5_\epsilon(d)$ and $N^5_\epsilon(d')$ are diffeomorphic if and only if $d\equiv \pm d' \text{ mod }16$.
\end{lemma}

We thus get five different oriented diffeomorphism types for the quotients. For a fixed $d$ even, and hence a fixed diffeomorphism type, there are infinitely many $d'\in \N$ such that $N^5_\epsilon(d)$ is diffeomorphic to $N^5_\epsilon(d')$.

\section{Metrics of nonnegative sectional curvature}\label{sec:metrics}

By the work of Grove and Ziller, one can equip cohomogeneity one manifolds with an invariant metric of nonnegative sectional curvature, provided the singular orbits are of codimension two (see \cite[Theorem E]{GZ00}).

As we have seen above, the quotients $N^5_0(d)$ admit a cohomogeneity one action by $S^1\times O(3)$ with codimension two singular orbits. Hence, $N^5_0(d)$ can be equipped with an invariant metric of $sec\geq 0$, which we will denote by $g^d_{GZ}$ and call its Grove-Ziller metric.

In order to get a metric of positive scalar curvature (and $sec\geq 0$), we will apply Cheeger deformation to the Grove-Ziller metric. See \cite[\textsection 6.1]{AB15} for a general treatment of Cheeger deformation.

Let $G$ be a compact Lie group acting by isometries on a Riemannian manifold $(M,g)$ and $Q$ a biinvariant metric on $G$. The action $g_1\cdot(p,g_2)=(g_1\cdot p,g_1\cdot g_2)$ of $G$ on $M\times G$ is free and isometric with respect to $g+\frac{1}{t}Q$, where $t\in\R^{>0}$. The quotient $(M\times G)/G$ is diffeomorphic to $M$. The \emph{Cheeger deformation} of $g$ is the metric $g_t$ which turns the projection $$\pi_t:(M\times G,g+\frac{1}{t}Q)\rightarrow (M,g_t):(p,u)\mapsto u^{-1}p$$ into a Riemannian submersion. The Lie group $G$ acts by isometries on $(M,g_t)$ for all $t>0$. It can be shown that $g_t$ varies smoothly in $t$ and that it extends smoothly to $t=0$ by setting $g_0=g$.

Let $p\in M$, $G_p$ the isotropy group and $\mathfrak{g}_p$ the corresponding Lie algebra. Denote by $\mathfrak{m}_p$ the $Q$-orthogonal complement of $\mathfrak{g}_p$ in $\mathfrak{g}$, the Lie algebra of $G$. For $X\in \mathfrak{g}$ and $p\in M$, let $X^*_p=\frac{d}{dt}(exp(tX)\cdot p)|_{t=0}$. One can then identify the orthogonal complement with the tangent space of the $G$-orbit of $p$, $$\mathfrak{m}_p\xrightarrow{\cong}T_p(Gp):X\mapsto X^*.$$ Consider the automorphism $P_p:\mathfrak{m}_p\rightarrow \mathfrak{m}_p$ defined by the relation $Q(P_p(X),Y)=g(X^*_p,Y^*_p)$ for all $Y\in \mathfrak{m}_p$.

Cheeger deformations preserve nonnegative sectional curvature and under certain circumstances, also produce positive scalar curvature. For $X,Y\in\mathfrak{g}$, let $[X,Y]$ denote the Lie bracket of $X$ and $Y$. Then one gets the following (see \cite[Proposition 5.2]{DGA19}).

\begin{proposition}\label{Prop:cheegposscal}
Let $(M,g)$ be a Riemannian manifold and $G$ a compact Lie group acting on it by isometries. If $sec_g\geq 0$, then $sec_{g_t}\geq 0$ for all $t\geq 0$. If in addition there exist $X,Y\in \mathfrak{m}_p$ such that $[P_p(X),P_p(Y)]\neq 0$ for all $p\in M$, then $scal_{g_t}>0$ for all $t>0$.
\end{proposition}

Let us apply this to $(N^5_0(d),g^d_{GZ})$. The action is by $S^1\times O(3)=SO(2)\times O(3)$ with principal isotropy group $\Z_2\times O(1)$ and singular isotropy groups $S^1\times O(1)$ and $\Z_2\times O(2)$. Since $S^1\times O(3)$ is non-abelian and the Lie algebra of $\Z_2\times O(1)$ is $0$-dimensional, there exist non-commuting elements in this case. Now consider the singular isotropy groups, which are both 1-dimensional. It follows that the dimension of $\mathfrak{m}_p$ is 3 in both cases. By the classification of maximal tori of orthogonal groups, we know that $G=SO(2)\times O(3)$ has a 2-dimensional abelian maximal torus. Its Lie algebra $\mathfrak{g}$ is 4-dimensional and therefore splits into a 2-dimensional abelian and a 2-dimensional non-abelian part. $P_p$ being an automorphism, this means that there must be non-commuting elements in its image. Thus the conditions of Proposition \ref{Prop:cheegposscal} are satisfied.

Let $t'>0$ be fixed and $g^d:=(g^d_{GZ})_{t'}$ the Cheeger deformation of the Grove-Ziller metric at time $t'$. The metric $g^d$ will be called the \emph{modified Grove-Ziller metric} on $N^5_0(d)$, which is of $sec\geq 0$, $scal>0$ (by Proposition \ref{Prop:cheegposscal}) and invariant under the action of $S^1\times O(3)$.

We can now extend this metric to

$$\tilde{Z}(d):=\tilde{f}^{-1}([0,\epsilon_0])=\bigcup_{0\leq \epsilon \leq \epsilon_0}N^5_\epsilon(d),$$
where $\tilde{f}:S^7/\tau\rightarrow \C$ is the map induced from the polynomial $f$ discussed in \textsection \ref{subsec:grpac}. This extended metric can further be averaged over $O(3)$, yielding an $O(3)$-invariant metric $h$ on $\tilde{Z}(d)$.

If we restrict $h$ to $N^5_{\epsilon_0}(d)$, we can lift it to $\Sigma^5_{\epsilon_0}(d)$ via pull-back and extend it to a metric on $W^6_{\epsilon_0}(d)$ in such a way that it is of product form near the boundary. Furthermore, the latter metric can be assumed to be $O(3)$-invariant and of positive sectional curvature when restricted to a sufficiently small invariant neighborhood of each fixed point of the action of $\tau$ on $W^6_{\epsilon_0}(d)$ (see \cite{DGA19} Corollary 5.5 and discussion thereafter). After averaging, we finally get an $O(3)$-invariant metric $k$ on $W^6_{\epsilon_0}(d)$.

Now, using $O(3)$-actions, we can produce Cheeger deformations $g^d_t$, $h_t$ and $k_t$ of the above metrics.  These metrics satisfy the following properties (see \cite[\textsection 5 and 6]{DGA19}).

\begin{lemma}\label{lemma:metrics}
Let $g^d_t$, $g_t$ and $k_t$ be the Cheeger deformations of the metrics $g^d$, $h$ and $k$ from above. Then,

\begin{itemize}
\item for every $t> 0$, the metric $g^d_t$ is of $sec\geq 0$ and $scal>0$,
\item there exists a $t_0\geq 0$ such that the restriction of $h_t$ to $N^5_{\epsilon}(d)$ has $scal>0$ for all $t\geq t_0$ and $0\leq \epsilon\leq \epsilon_0$, and
\item $k_t$ on $W^6_{\epsilon_0}(d)$ also has $scal>0$ and is of product form near the boundary for all $t\geq t_0$.
\end{itemize}
\end{lemma}

\section{$Spin^c$-structures and reduced eta-invariants}

The eta-invariant we want to use to distinguish path components in the space of metrics requires some additional structure on the considered manifolds. In this section, we give a short introduction on these $Spin^c$-structures (see \cite[Appendix D]{LM89} for more details).

Let $Spin^c(n):=Spin(n)\times_{\Z_2}U(1)$, where the action is diagonal by $\{\pm(1,1)\}$. Then there is a short exact sequence
$$1 \rightarrow \Z_2\rightarrow Spin^c(n)\xrightarrow{\xi} SO(n)\times U(1) \rightarrow 1$$ where $\xi$ is the map induced by $\rho\times (-)^2$ with $\rho:Spin(n)\rightarrow SO(n)$ the standard double-covering. A $Spin^c$-structure on an n-dimensional, oriented Riemannian manifold $(X,g)$ consists of a principal $Spin^c(n)$-bundle $P_{Spin^c(n)}$ over $X$, a principal $U(1)$-bundle $P_{U(1)}$ over $X$ and a bundle map $\xi:P_{Spin^c(n)}\rightarrow P_{SO(n)}\times P_{U(1)}$ such that $\xi(pg)=\xi(p)\rho(g)$ for all $p\in P_{Spin^c(n)}$ and $g\in Spin^c(n)$. Here $P_{SO(n)}$ denotes the principal bundle of oriented orthonormal frames on $X$. If $X$ is equipped with a $Spin^c$-structure, it is called a $Spin^c$-manifold.

A Riemannian manifold $X$ can be equipped with a $Spin^c$-structure if and only if there exists an element $\omega\in H^2(X;\Z)$ such that $w_2(X)\equiv \omega \text{ mod }2$. The $Spin^c$-structures on $X$, if any exist, are parametrized by $2H^2(X;\Z)\oplus H^1(X;\Z_2)$. If $X$ is a complex manifold, it carries a canonical $Spin^c$-structure. In this case, the principal $U(1)$-bundle corresponds to the determinant line bundle of the complex tangent bundle of $X$.

If $X$ is a $Spin^c$-manifold, one can construct a complex spinor bundle $S(X)$ over it. The Levi-Civita connection on $X$ together with a connection $\nabla^c_X$ on $P_{U(1)}$ determine a connection on $S(X)$, which turns $S(X)$ into a Dirac bundle. There is an associated $Spin^c$-Dirac operator $D^c_X:\Gamma(S(X))\rightarrow \Gamma(S(X))$, where $ \Gamma(S(X))$ is the set of sections on $S(X)$. Given a Hermitian complex vector bundle $E$ over $X$, one also gets a twisted $Spin^c$-Dirac operator $D^c_{X,E}:\Gamma(S(X)\otimes E)\rightarrow \Gamma(S(X)\otimes E)$.

\subsection{The Atiyah-Patodi-Singer index theorem}

Eta-invariants were introduced in the context of the Atiyah-Patodi-Singer (APS) index theorems, where they appear as metric dependent boundary correction terms (see \cite{APSI75} and \cite{APSII75}).

Let $(W^{2n},g_W)$ be a compact $Spin^c$-manifold with boundary $\partial W=M$ such that the metric $g_W$ is of product form near the boundary and $g_M$ is the induced metric on $M$. Fix a connection $\nabla^c_W$ on the principal $U(1)$-bundle $P_{U(1)}\rightarrow W$ corresponding to the $Spin^c$-structure, which is constant in normal direction near the boundary and with curvature form $\Omega^c$. As $W$ is of even dimension, there is a direct sum splitting $S(W)=S^+(W)\oplus S^-(W)$. If we subject the restricted $Spin^c$-Dirac operator to APS-boundary conditions, we obtain an operator $D^{c,+}_W:\Gamma(S^+(W))\rightarrow \Gamma(S^-(W))$ and an adjoint operator $(D^{c,+}_W)^*$, both of which have finite dimensional kernel (see \cite[Appendix D]{LM89}).

Then the index $\text{ind} (D^{c,+}_W):=\text{dim(ker(}D^{c,+}_W))-\text{dim(ker(}(D^{c,+}_W)^*))\in \Z$ is well-defined, and by the APS index theorem (\cite[Theorem 4.2]{APSI75}), it is given by $$\text{ind} (D^{c,+}_W)=\int_We^{\frac{1}{2}c}\hat{\mathcal{A}}(W,g_W)-\frac{1}{2}\Big(h(M,g_M)+\eta(M,g_M)\Big).$$ Here $ \hat{\mathcal{A}}(W,g_W)$ is the $\hat{\mathcal{A}}$-series in Pontryagin forms, $c:=\frac{1}{2\pi}\Omega^c$ denotes the first Chern form in terms of the curvature form $\Omega^c$ of $\nabla^c$, $h(M,g_M):=\text{dim}(\text{ker}(D^c_M))$ and $\eta(M,g_M)$ is the eta-invariant of $D^c_M$. To introduce it, we need to consider $$\eta(z):=\sum_\lambda \frac{\text{sign}(\lambda)}{|\lambda|^z}, \qquad z\in\C, \,\text{Re}(z)\gg 0,$$ where the sum is running over all non-zero eigenvalues $\lambda$ of $D^c_M$. This function extends in a unique way to a meromorphic function (also denoted by $\eta$) which is holomorphic at $z=0$ (see \cite[\textsection 2]{APSI75}) and we may define $\eta(M,g_M):=\eta(0)$.

The twisted case is completely analogous. Let $E$ be a Hermitian complex vector bundle over $W$ and $\nabla^E$ a Hermitian connection on it, which is constant in normal direction near the boundary. The twisted APS index formula (\cite[(4.3)]{APSI75}) is then given by $$\text{ind}(D^{c,+}_{W,E})=\int_Wch(E,\nabla^E)e^{\frac{1}{2}c}\hat{\mathcal{A}}(W,g_W)-\frac{1}{2}\Big(h_E(M,g_M)+\eta_E(M,g_M)\Big).$$ Here $h_E(M,g_M):=\text{dim}(\text{ker}(D^c_{M,E}))$ is the kernel, $ch(E,\nabla^E):=\text{tr}(\text{exp}(\frac{i\Omega^c_E}{2\pi}))$ is the Chern character in terms of the curvature form $\Omega^c_E$ of $\nabla^E$ and $\eta_E(M,g_M)$ is defined as above with $D^{c}_{M,E}$.

We have the following well known vanishing theorem (see \cite{LM89} and \cite{APSII75}).

\begin{theorem}\label{thm:vanish}
Let $(W^{2n},g_W)$, $(M,g_M)$ and $E$ be as above, with $\Omega^c_W$ and $\Omega^c_M$ the curvature forms corresponding to the principal $U(1)$-bundle of the $Spin^c$-structure respectively. Suppose $E$ is flat. Then if $scal_{g_M}>0$ and $\Omega^c_M=0$, we have $h(M,g_M)=0$ and $h_E(M,g_M)=0$. If $scal_{g_W}>0$ and $\Omega^c_W=0$, then $\emph{ind}(D^{c,+}_{W})=0$ and $\emph{ind}(D^{c,+}_{W,E})=0$.
\end{theorem}

Now suppose $M$ is connected. Let $\alpha:\pi_1(M)\rightarrow U(k)$ be a unitary representation, so that it determines a flat vector bundle $E_\alpha:=\tilde{M}\times_{\pi_1(M)}\C^k$ over $M$, where $\tilde{M}$ is the universal cover of $M$ (see \cite{K87} (1.2.4)). We can then define the reduced eta-invariant $$\tilde{\eta}_{E_\alpha}(M,g_M):=\eta_{E_\alpha}(M,g_M)-k\cdot\eta(M,g_M).$$

This invariant will allow us to tell path components apart in the space of Riemannian metrics of positive scalar curvature. Indeed, if $\mathcal{R}_{scal>0}(M)$ denotes the set of Riemannian metrics on $M$ with positive scalar curvature, we have the following result (see \cite[p. 417]{APSII75}, \cite[Proposition 3.3]{DGA19}).

\begin{proposition}\label{prop:etainvsame}
Let $M^{2n-1}$ be a closed connected $Spin^c$-manifold, $\nabla^c_M$ a flat connection on $P_{U(1)}\rightarrow M$ and $\alpha:\pi_1(M)\rightarrow U(k)$ a unitary representation. Let $g_0,g_1\in\mathcal{R}_{scal>0}(M)$ be in the same path component. Then $\tilde{\eta}_{\alpha}(M,g_0)=\tilde{\eta}_{\alpha}(M,g_1)$.
\end{proposition}

The reduced eta-invariant is in general difficult to compute. In an equivariant setting however, there are explicit formulas which allow us to determine them. Let $(M,g_M)$ be a closed Riemannian manifold and suppose a compact Lie group $G$ acts by isometries on it. Then one can define $G$-equivariant $Spin^c$-structures by requiring all of the corresponding bundles to be $G$-equivariant. If furthermore the unitary connection on the associated principal $U(1)$-bundle is $G$-equivariant, then one gets a $G$-equivariant spinor bundle $S_G(M)$ and a corresponding $G$-equivariant $Spin^c$-Dirac operator $D^{c,G}_M$. Fix $g\in G$ and let $\lambda$ be an eigenvalue of $D^{c,G}_M$. The action of $g$ on the Eigenspace $E_\lambda$ will be denoted by $g^\#_\lambda$. One can then consider $$\eta_g(z):=\sum_{0\neq\lambda \in E_\lambda}\frac{\text{sign}(\lambda)\cdot \text{tr}(g^\#_\lambda)}{|\lambda|^z}$$ which converges absolutely for $\text{Re}(z)\gg 0$ and define $\eta(M,g_M)_g:=\eta_g(0)$, the equivariant eta-invariant at $g$ (after meromorphic extension, see \cite{D78}).

Suppose now that $M$ is connected and that $G=\pi_1(M)$ is finite. The metric $g_M$ lifts to a metric $\tilde{g}_{\tilde{M}}$ on the universal cover $\tilde{M}$, as does the $Spin^c$-structre on $M$. Recall that $\alpha$ is a unitary representation and $E_\alpha$ the corresponding flat vector bundle, defined above. In this situation, we get the following formula (see \cite[Theorem 3.4.]{D78}). $$\eta_{E_\alpha}(M,g_M)=\frac{1}{|G|}\sum_{g\in G}\eta(\tilde{M},\tilde{g}_{\tilde{M}})_g\cdot \chi_\alpha(g),$$ where $\chi_\alpha:G\rightarrow \C:g\mapsto tr(\alpha(g))$ is the character of $\alpha$ and $|G|$ the order of $G$.

\subsection{Donnelly's Fixed Point Formula}

Let $(W^{2n},g_W)$ and $(M^{2n-1},g_M)$ be as in the previous paragraph and suppose a compact Lie group $G$ acts on $W$ by isometries and preserves the $Spin^c$-structure. Assume also that $g_W$ is of product form near the boundary $\partial W=M$. Let $D^{c,+}_W$ be the $G$-equivariant $Spin^c$-Dirac operator on $(W,g_W)$ subject to the APS-boundary condition and $D_M$ the $G$-equivariant $Spin^c$-Dirac operator on $(M,g_M)$ (with corresponding adjoint operators $(D^{c,+}_W)^*$ and $D^*_M$). In this case, the kernels of these operators are $G$-representations and the equivariant index $\text{ind}_G(D^{c,+}_W):=\text{ker}(D^{c,+}_W)-\text{ker}((D^{c,+}_W)^*)$ is a virtual complex $G$-representation in the ring of representations $R(G)$. For $g\in G$, let $h_g$ be the character of the $G$-representation $\text{ker}(D_M)$ at $g$ and $\eta(M,g_M)_g$ the equivariant eta-invariant of $D_M$ at $g$. Then we have the following fixed point formula (see \cite[Theorem 1.2.]{D78}).
$$ \text{ind}_G(D^{c,+}_W)_g + \frac{1}{2}\Big(h_g+\eta(M,g_M)_g\Big)=\sum_{N\subset W^g}a_N,$$ where $N$ denotes a connected component of the fixed point set $W^g$ and $a_N$ is a so-called local contribution at $N$. In our specific case, this local contribution can be computed explicitely (see \cite[Proposition 3.5]{DGA19}).

\begin{proposition}\label{prop:localcontr}
Let $(W^{2n},g_W)$ and $(M^{2n-1},g_M)$ be as above, and suppose $\tau\in G$ induces an involution on $W$ with isolated fixed points which lie in the interior of $W$. Then the local contribution at any fixed point $p$ is given by $a_{\{p\}}=2^{-n}$.
\end{proposition}

We can now compute the reduced eta-invariant in the case of the Brieskorn quotients. Let $d\in\N$ be even and $\epsilon\neq 0$.

Let $W^6_{\epsilon}(d)$ be equipped with a $G$-invariant metric $g_W$, where $G$ is a closed subgroup of $\Z_{2d}\times O(3)$ containing the involution $\tau$ defined above. Fix a Hermitian metric on its tangent bundle which is $G$-invariant. Consider the $G$-equivariant $Spin^c$-structure induced by its complex manifold structure and the Hermitian metric. The induced $G$-equivariant $Spin^c$-structure on the boundary $\Sigma^5_{\epsilon}(d)=\partial W^6_{\epsilon}(d)$ descends to a $Spin^c$-structure on $N^5_{\epsilon}(d)=\Sigma^5_{\epsilon}(d)/\tau$, which we will call its preferred $Spin^c$-structure.

Now fix a flat connection on the principal $U(1)$-bundles over $W^6_{\epsilon}(d)$, $\Sigma^5_{\epsilon}(d)$ and $N^5_{\epsilon}(d)$ corresponding to the $Spin^c$-structure (which we can do since the second cohomology groups with real coefficients are trivial and therefore the first Chern classes of the bundles vanish). If $\tau$ acts as $-Id$ on $\C$, then $E_\alpha:=\Sigma^5_{\epsilon}(d)\times_{<\tau>}\C$ is a non-trivial flat complex line bundle.

\begin{proposition}\label{prop:etainveps}
Let $0<\epsilon_0 < 1$ and $g_N$ be a metric of $scal>0$ on $N^5_\epsilon(d)$. Suppose that the lift $g_\Sigma$ of $g_N$ to $\Sigma^5_\epsilon(d)$ extends to a metric $g_W$ on $W^6_\epsilon(d)$ which is $\tau$-invariant, of product form near the boundary and of $scal>0$. Then $$\tilde{\eta}_{E_\alpha}(N^5_\epsilon(d),g_N)=-\frac{d}{4}.$$

\begin{proof}
Using the formulas from the previous section, we get for $G=\Z_2=\{1,\tau\}$:

$$\eta_{E_\alpha}(N^5_\epsilon(d),g_N)=\frac{1}{2}\Big(\eta(\Sigma^5_\epsilon(d),g_\Sigma)_1\cdot \chi_\alpha(1)+\eta(\Sigma^5_\epsilon(d),g_\Sigma)_\tau\cdot \chi_\alpha(\tau)\Big)$$
$$=\frac{1}{2}\Big(\eta(\Sigma^5_\epsilon(d),g_\Sigma)-\eta(\Sigma^5_\epsilon(d),g_\Sigma)_\tau\Big)$$
and using the trivial representation (whose character is always 1)
$$\eta(N^5_\epsilon(d),g_N)=\frac{1}{2}\Big(\eta(\Sigma^5_\epsilon(d),g_\Sigma)+\eta(\Sigma^5_\epsilon(d),g_\Sigma)_\tau\Big).$$
Hence we get $$\tilde{\eta}_{E_\alpha}(N^5_\epsilon(d),g_N)=\eta_{E_\alpha}(N^5_\epsilon(d),g_N)-1\cdot \eta(N^5_\epsilon(d),g_N)=-\eta(\Sigma^5_\epsilon(d),g_\Sigma)_\tau$$
Now since the metrics on $\Sigma^5_\epsilon(d)$ and $W^6_\epsilon(d)$ both are $\tau$-invariant and of $scal>0$, applying Theorem \ref{thm:vanish}, we see that the equivariant index and kernel vanish. Using the fixed point formula from above we get $\eta(\Sigma^5_\epsilon(d))_\tau=2\sum_{i=1}^d a_{\{p_i\}}$, where the $p_i$ are the isolated fixed points of the action of $\tau$ on $W^6_\epsilon(d)$. Finally, using Proposition \ref{prop:localcontr} to compute $a_{\{p_i\}}=2^{-n}$ with $n=3$, we get the desired result.

\end{proof}

\end{proposition}

\section{Moduli space of Riemannian metrics}

For a general treatment on moduli spaces, see \cite{TW15}.

Let $M$ be a closed smooth manifold and $\mathcal{R}(M)$ the set of Riemannian metrics on $M$, equipped with the $C^\infty$-topology of uniform convergence of all the derivatives. By taking pull-backs of metrics, we get an action of the group of diffeomorphisms Diff($M$) on $\mathcal{R}(M)$. The quotient space $\mathcal{M}(M):=\mathcal{R}(M)/\text{Diff}(M)$ is called the \textit{moduli space} of $M$. If we restrict to metrics of nonnegative sectional curvature or positive Ricci curvature, we obtain $\mathcal{R}_{\text{sec}\geq 0}(M)$ and $\mathcal{R}_{\text{Ric}> 0}(M)$ respectively. Restricting on isometry classes of metrics of nonnegative sectional curvature or positive Ricci curvature, we obtain corresponding moduli spaces $\mathcal{M}_{\text{sec}\geq 0}(M)$ and $\mathcal{M}_{\text{Ric}> 0}(M)$.

For technical reasons, we will actually need to work in the quotient space $\mathcal{R}_{sec\geq 0}(M)/\mathcal{D}$, where $\mathcal{D}\subset \text{Diff}(M)$ is the subgroup of diffeomorphisms which preserve the $Spin^c$-structure on $M$.

First we need to determine the reduced eta-invariant of $(N^5_0(d),g^d)$, where $g^d$ is the modified Grove-Ziller metric from \textsection \ref{sec:metrics}. Since $H^2(N^5_0(d);\Z)\cong \Z_2$, there exists a unique flat non-trivial complex line bundle over $N^5_0(d)$ which we will denote by $\alpha$. Fix $0<\epsilon_0<1$.

\begin{lemma}\label{lemma:etainvcomp}
Let $(N^5_0(d),g^d)$ be equipped with the preferred $Spin^c$-structure and a flat connection on the associated principal $U(1)$-bundle. Then $$\tilde{\eta}_\alpha(N^5_0(d),g^d)=-\frac{d}{4}.$$

\begin{proof}
Let $$\tilde{Z}(d)=\bigcup_{0\leq \epsilon \leq \epsilon_0}N^5_\epsilon(d)$$ be equipped with the extended metric $h$ as in \textsection \ref{sec:metrics}. Let $g^d_t$ and $h_t$ denote the Cheeger deformations of $g^d$ and $h$ respectively and $t_0$ the parameter from Lemma \ref{lemma:metrics}. We construct a path in $\mathcal{R}_{scal>0}(N^5_{\epsilon_0}(d))$, connecting $(N^5_0(d),g^d)$ to $(N^5_{\epsilon_0}(d),h_{t_0}|_{N^5_{\epsilon_0}(d)})$, for which we can actually compute the reduced eta-invariant as in Proposition \ref{prop:etainveps}. Let $\phi_\epsilon:N^5_{\epsilon_0}(d)\rightarrow N^5_{\epsilon}(d)$ be a smooth family of diffeomorphisms for $0<\epsilon<\epsilon_0$.

Using Lemma \ref{lemma:metrics}, we can now define two paths of metrics of positive scalar curvature on $N^5_{\epsilon_0}(d)$ in the following way.

\begin{align*}
\tilde{\gamma}_1: & \text{ }[0,t_0]\rightarrow \mathcal{R}_{scal>0}(N^5_{\epsilon_0}(d)) \\ & \quad \text{ } t \mapsto \phi^*_0(g^d_t)
\end{align*}
and
\begin{align*}
\tilde{\gamma}_2: & \text{ } [0,\epsilon_0]\rightarrow \mathcal{R}_{scal>0}(N^5_{\epsilon_0}(d)) \\ & \quad \text{ } \epsilon \mapsto \phi^*_\epsilon(h_{t_0}|_{N^5_{\epsilon}(d)})
\end{align*}

\noindent Since $\tilde{\gamma}_1(t_0)=\phi^*_0(g^d_{t_0})=\phi^*_0(h_{t_0}|_{N^5_{0}(d)}))=\tilde{\gamma}_2(0)$, we can concatenate these two paths to get a path $\tilde{\gamma}$ in $\mathcal{R}_{scal>0}(N^5_{\epsilon_0}(d))$ with endpoints $\phi^*_0(g^d)$ and $\tilde{h}_{t_0}|_{N^5_{\epsilon_0}(d)}$. By Lemma \ref{lemma:metrics} and the discussion above it, $h_{t_0}|_{N^5_{\epsilon_0}(d)}$ lifts to a metric on $\Sigma^5_{\epsilon_0}(d)$ which extends to a metric on $W^6_{\epsilon_0}$ satisfying the conditions of Proposition \ref{prop:etainveps}. By Propostion \ref{prop:etainvsame} and using the fact that the reduced eta-invariant is preserved under pull-backs by diffeomorphisms, we finally get:

$$\tilde{\eta}_\alpha(N^5_0(d),g^d)=\tilde{\eta}_\alpha(N^5_{\epsilon_0}(d),\phi^*_0(g^d))=\tilde{\eta}_\alpha(N^5_{\epsilon_0}(d),h_{t_0}|_{N^5_{\epsilon_0}(d)})=-\frac{d}{4}.$$
\end{proof}

\end{lemma}

We can now prove our main theorem.

Fix $d\in\{0,2,4,6,8\}$, $d_0\neq d_1\in \N$ such that $d_i\equiv \pm d \text{ mod } 16$. Therefore, by Theorem \ref{thm:classification5mfds} and Lemma \ref{lemma:diffquo}, there exist diffeomorphisms $\psi_i:N^5_0(d)\rightarrow N^5_0(d_i)$ for $i=0,1$. Denote by $h_i=\psi^*_i(g^{d_i})$ the pull-back metric for $i=0,1$. Let $\mathcal{D}\subset \text{Diff}(N^5_0(d))$ denote the subgroup of diffeomorphisms preserving the $Spin^c$-structure on $N^5_0(d)$.

Since $\mathcal{D}$ has finite index in Diff$(N^5_0(d))$, if $\mathcal{R}_{sec\geq 0}(N^5_0(d))/\mathcal{D}$ has infinitely many path components, it will follow that $\mathcal{M}_{\text{sec}\geq 0}(N^5_0(d))$ has infinitely many path components as well.

Assume now that there is a path $\gamma:[0,1]\rightarrow \mathcal{R}_{sec\geq 0}(N^5_0(d))/\mathcal{D}$ connecting $[h_0]$ and $[h_1]$.  As a consequence of Ebin's slice theorem, this path can be lifted to a path $\tilde{\gamma}$ in $\mathcal{R}_{sec\geq 0}(N^5_0(d))$ such that $\tilde{\gamma}(0)=h_0$ and $\tilde{\gamma}(0)=\phi^*(h_1)$ for some $\phi\in\mathcal{D}$ (see \cite{E70} and \cite[Proposition 4.6]{CK19}). By B\"{o}hm and Wilking, the path $\tilde{\gamma}$ evolves instantly to a path in $\mathcal{R}_{Ric> 0}(N^5_0(d))$ under the Ricci flow (see \cite[Theorem A]{BW07}). If we concatenate this resulting path with the orbits of the endpoints of $\tilde{\gamma}$, we obtain a path $\gamma'$ in $\mathcal{R}_{scal> 0}(N^5_0(d))$ with endpoints $\gamma'(0)=h_0$ and $\gamma'(1)=\phi^*(h_1)$ as well.

Using Lemma \ref{lemma:etainvcomp}, we can now compute the reduced eta-invariants of the endpoints of $\gamma'$. We immediately see that $\tilde{\eta}_\alpha(N^5_0(d_0),h_0)=-\frac{d_0}{4}\neq-\frac{d_1}{4}=\tilde{\eta}_\alpha(N^5_0(d_1),\phi^*(h_1))$. Hence $h_0$ and $\phi^*(h_1)$ lie in different path components in $\mathcal{R}_{scal>0}(N^5_{0}(d))$, which contradicts the construction we have obtained from our initial assumption.

By Theorem \ref{thm:classification5mfds}, Lemma \ref{lemma:diffquo} and the above remark on the link between path components of $\mathcal{R}_{sec\geq 0}(N^5_0(d))/\mathcal{D}$ and $\mathcal{M}_{\text{sec}\geq 0}(N^5_0(d))$, it follows that the moduli space $\mathcal{M}_{\text{sec}\geq 0}(N^5_0(d))$ has infinitely many path components. If we focus on the evolved path under the Ricci flow, we see that the same argument applies to $\mathcal{R}_{Ric > 0}(N^5_0(d))/\mathcal{D}$ and hence $\mathcal{M}_{Ric > 0}(N^5_0(d))$ has infinitely many path components as well. \proofend

config: arxiv


TITLE: Easy question about subdifferential of a functional on $L^2(0,T;L^2)$
QUESTION [1 upvotes]: Define $J:L^1(0,T;L^1) \to \mathbb{R}$ by
$$J(v) = \int_0^T \int_\Omega \Psi(v)$$
where $\Psi(v) = \int_0^v \beta(s)\;ds$ where $\beta$ is a nice function that passes through the origin.
We have that $\partial J(u) = \{v \in L^2(0,T;L^2) \mid v = \beta(u) \text{ a.e}\}.$ Why does this mean that

for all $w \in L^2(0,T;L^2)$
  $$\int_0^T \int_{\Omega}(\beta(u)u + \Psi(w)) \geq \int_0^T\int_\Omega(\Psi(u) + \beta(u)w)?$$

I thought this may be due to definition of the subdifferential but it doesn't appear so..

REPLY [1 votes]: I think that as a part of being "nice", $\beta$   should be increasing. This makes $\Psi$ convex.
Let $v=\beta(u)$ and consider the  functional
$$E(w) =  \int_{0}^T \int_\Omega (v w - \Psi(w)) = \left(\int_{0}^T \int_\Omega v w \right)- J(w)$$
Its derivative is zero when  $w=u$, because then it is equal to
$$ h\mapsto \int_{0}^T \int_\Omega (v h - \beta(u) h) = 0$$
Therefore, $u$ is a point of minimum of $E$. This means
$$\int_{0}^T \int_\Omega (v u - \Psi(u))  \le \int_{0}^T \int_\Omega (v w - \Psi(w)) $$
which is the desired inequality.


TITLE: How many ways can you give out 4 oranges, 2 bananas and 2 lemons
QUESTION [0 upvotes]: How many ways can you give out 4 oranges, 2 bananas and 2 lemons between
a) 8 persons so that every person get exactly 1 fruit
Hmm I wonder if I can use "Stars and bars" for this problem? :)

REPLY [0 votes]: Choose which four people receive an orange and which two of the four remaining people receive a lemon.  The other two people must receive a banana. Hence, there are
$$\binom{8}{4}\binom{4}{2}\binom{2}{2}$$
ways to distribute the fruits so that each of the eight people receives exactly one fruit.
To compare this answer with that provided by Maria Mazur, observe that
$$\binom{8}{4}\binom{4}{2}\binom{2}{2} = \binom{8}{4}\binom{4}{2} = \frac{8!}{4!4!} \cdot \frac{4!}{2!2!} = \frac{8!}{4!2!2!}$$
The numbers in the denominator represent the number of ways we could permute the four oranges among the people who receive oranges, the number of ways we could permute the lemons among the people who receive lemons, and the number of ways we could permute the bananas among the people who receive bananas without changing which person receives which type of fruit.


TITLE: Fourier Series piecewise
QUESTION [0 upvotes]: Can someone please explain why the integral gose from -1 to 1 to (0 to 1)
how does this happen?

REPLY [1 votes]: It is a general strategy to break integrals of piecewise functions at the boundaries of the pieces.  This means that we have simple recipes for the functions in each resulting integral.  In your case,
\begin{align*}
\int_{-1}^{1} \; &\begin{cases} 0, & -1 \leq x < 0 \\ x^2, & 0 \leq x < 1 \end{cases} \, \mathrm{d}x  \\
    &= \int_{-1}^{0} \; \begin{cases} 0, & -1 \leq x < 0 \\ x^2, & 0 \leq x < 1 \end{cases} \, \mathrm{d}x + \int_{0}^{1} \; \begin{cases} 0, & -1 \leq x < 0 \\ x^2, & 0 \leq x < 1 \end{cases} \, \mathrm{d}x  \\
    &= \int_{-1}^{0} \; 0 \, \mathrm{d}x + \int_{0}^{1} \; x^2 \, \mathrm{d}x  \\
    &= 0 + \int_{0}^{1} \; x^2 \, \mathrm{d}x  \\
\end{align*}
Notice that once we split the interval of integration at the boundaries of the pieces of the definition of $f$, we only keep the one recipe in that interval.  Then the subsequent integrals are much easier to work with.