text:
\begin{document}

\begin{abstract}
We study, with respect to the parameter $q\neq0$, the following  Schr\"odinger-Bopp-Podolsky system
in $\mathbb R^{3}$
\begin{equation*}
\left\{
\begin{aligned}
-&\Delta u+\omega u+q^2\phi u=|u|^{p-2}u, \\
&-\Delta \phi+a^2\Delta^2 \phi = 4\pi u^2,
\end{aligned}
\right.
\end{equation*}
where $p\in(2,3], \omega>0, a\geq0$ are fixed.
We prove, by means of the fibering approach, that the system
has no solutions {\sl at all} for large values of $q's$,
and has two {\sl radial } solutions for small $q's$. We give also
qualitative properties about  the energy level of the solutions and
 a variational characterization of these  extremals values of $q$.
Our results  recover and improve some results in \cite{GaeDa, Ruiz}.
\end{abstract}

\maketitle

\section{Introduction}

In the recent paper \cite{GaeDa} for the first time in the mathematical
literature the following system in $\mathbb R^{3}$ has been studied
\begin{equation}\label{p}
\left\{
\begin{aligned}
-&\Delta u+\omega u+q^2\phi u=|u|^{p-2}u, \\
&-\Delta \phi+a^2\Delta^2 \phi = 4\pi u^2,
\end{aligned}
\right.
\end{equation}
where $a,\omega> 0$, $q\neq0$ and $p\in (2,6)$. The system appears when one\ look for stationary
solutions $u(x)e^{i\omega t}$ of the Schr\"odinger equation coupled with the Bopp-Podolski
Lagrangian of the electromagnetic field, in the purely electrostatic situation.
Here $u$ represents the modulus of the wave function and $\phi$ the electrostatic potential.
From a physical point of view, the parameter $q$ has the meaning of the electric charge and $a$ is the parameter of
the Bopp-Podolski term.

In the cited paper, it has been shown that the problem can be addressed variationally. Indeed introducing the Hilbert space  $$\mathcal{D}:=\left\{\phi\in {D}^{1,2}(\mathbb{R}^3):\ \Delta \phi\in L^2(\mathbb{R}^3)\right\}$$
normed by $$\|\phi\|^{2}_{\mathcal D} = a^{2}\|\Delta \phi\|_{2}^{2} + \|\nabla \phi \|_{2}^{2},$$
it can be proved that
for every $u\in H^{1}(\mathbb R^{3})$ there is a unique solution $\phi_{u}\in \mathcal D$ of the second equation
in the system, that is satisfying
	\begin{equation}\label{solu}
-\Delta\phi_u+	a^2\Delta^2\phi_u=4\pi u^2.
	\end{equation}
Moreover it turns out that
$$\phi_{u} = \frac{1 - e^{-|\cdot|/a}}{|\cdot|} *u^{2}.$$
Observe from \eqref{solu} that, for every $ u\in H^{1}(\mathbb R^{3})$,
\begin{equation*}
4\pi \int \phi_u u^2=\|\phi_u\|^2_{\mathcal{D}},
\end{equation*}
which will be used throughout the paper.

By using the classical by now {\sl reduction argument }
one is led to study, equivalently,  the single equation
\begin{equation}\label{eq:equacao}
-\Delta u+\omega u+q^2\phi_{u} u=|u|^{p-2}u \quad \text{ in } \ \mathbb R^{3}
\end{equation}
containing the  nonlocal term $\phi_{u} u$.
Then whenever from now on we speak of solution of the system \eqref{p} we mean just the solution $u$
of the above equation since $\phi=\phi_{u}$ is univocally determined.
To the equation \eqref{eq:equacao} is related the energy functional
\begin{equation*}
\mathcal{J}_q(u)=\frac{1}{2}\|u\|^2+\frac{q^2}{4}\int \phi_u u^2-\frac{1}{p}\|u\|_p^p, \quad u\in H^{1}(\mathbb R^{3})
\end{equation*}
which is well defined and $C^{1}$. In this way we are simply reduced to find critical points of $\mathcal J_{q}$.

We are denoting (here and throughout the paper) by $\|u \|_{p}$ the $L^{p}-$norm and by
$$\|u \|^{2} =\|\nabla u\|^{2} + \omega\|u\|_{2}^{2}$$
the (squared) norm in $H^{1}(\mathbb R^{3})$, being $\omega$ a fixed positive constant.

\medskip

In \cite[Theorem 1.1 and Theorem 1.2]{GaeDa} it is proved that
if $p\in (3,6)$, then problem \eqref{p} admits a solution for every $q\neq0$.  On the other hand, if $p\in(2,3]$
the existence of a solution is proved just for $q\neq0$ sufficiently small.
As we can see, there is a difference in the result depending on the range where $p$ varies.
Indeed in the case of $p's$ ``small'' the value of $q$ may prevent the existence of critical points
for the functional $\mathcal J_{q}$.

\medskip

Of course, if $a=0$  system \eqref{p} reduces to the so called Schr\"odinger-Poisson system
in $\mathbb R^{3}$
\begin{equation}\label{eq:sp}
\left\{
\begin{aligned}
-&\Delta u+\omega u+q^2\phi u=|u|^{p-2}u, \\
&-\Delta \phi = 4\pi u^2
\end{aligned}
\right.
\end{equation}
or, equivalently, to
$$-\Delta u+\omega u+q^2\phi_{u}^{\textrm SP} u=|u|^{p-2}u, $$
where  now
$$\phi_{u}^{\textrm SP} = \frac{1}{|\cdot|}* u^{2}
.$$
In the mathematical literature there is a huge number of papers concerning this last problem.
However we cite here just \cite{BF} where the authors for the first time introduced
the {\sl reduction method} which allows to study a single equation
instead of a system, and \cite{Ruiz} where the author studies the problem depending on the parameter
$q^{2}$. In particular Ruiz in \cite{Ruiz}, among other results, shows that, in the case $p\in(2,3)$
 the system \eqref{eq:sp} has two  radial solutions for small $q^{2}$ and has no solutions
 at all, that is radial or not, for $q^{2}\ge 1/4$. See also \cite{RS} for similar results related to the problem in bounded domains.

\bigskip

Motivated by the cited papers \cite{GaeDa,Ruiz}, our aim here is to understand in a more satisfactory way
the existence of solutions for \eqref{p}, or \eqref{eq:equacao}, namely
the behaviour of $\mathcal J_{q}$ for what concerns its critical points in the case $p\in(2,3]$
and how they are influenced by the value of  $q$.

We prove two type of results.
The first one concerns with the smallness of $q^{2}$ as a necessary condition
in order to have a nontrivial solution of the problem (the sufficiency being proved in \cite{GaeDa}).
Indeed we show that for $q^{2}$ suitably large the problem has no solutions at all.
See Theorem \ref{th:1} below.

The second result concerns the existence of solutions for $q^{2}$ small.
However, due to the technique used (we borrow some ideas from \cite{Ruiz}),
 we are able to state such a result just in the radial
case: in this case obtaining two solutions
(in spite of the result in \cite{GaeDa} which states the existence of one solution in the nonradial case).
See Theorem \ref{th:2} below

\medskip

Before stating rigorously the results, we
observe that in the problem is appearing the positive parameter $q^{2}$.
In view of this, the results are  stated and proved
for simplicity just for $q>0$, being understood that they are valid by changing $q$ with $|q|$.

More specifically, under the assumption $p\in(2, 3]$ our main results in this work are the following
\begin{theorem}\label{th:1}
There exists $q^{*}>0$ such that, for every $q>q^{*}$ the problem admits only the trivial solution.
\end{theorem}

\begin{theorem}\label{th:2}
There exist $\varepsilon>0,q_0^{*}>0$ satisfying $q_{0}^{*}+\varepsilon<q^{*}$
(with $q^{*}$ given in Theorem \ref{th:1})
such that, for every   $q\in (0,q_0^*+\varepsilon)$ the problem
has two radial solutions.
\end{theorem}

Few comments on these results are in order.

As we already said, we are reduced to find critical points of $\mathcal J_{q}$.
We remark explicitly  that no Pohozaev identity is involved in proving the nonexistence result in Theorem \ref{th:1}:
it just follows due to the properties of the fibering map.

To prove  Theorem \ref{th:2} we will use the Mountain Pass Theorem
on the space of radial functions, that is in $H^{1}_{r}(\mathbb R^{3})$.
We take advantage of the smallness of $q$ to prove that the energy functional has
the Mountain Pass Geometry. However, in contrast to \cite{GaeDa} where the
condition of $q$ small was used in order to find a function where $\mathcal J_{q}$
is negative (and then apply the Mountain Pass Theorem in a standard way), the value $q_{0}^{*}$ we find here is a threshold: for $q<q_{0}^{*}$
there is  a function in $H^{1}_{r}(\mathbb R^{3})$ where the functional is negative, while for $q\geq q_{0}^{*}$
the functional is non-negative. Hence  the argument employed in both papers \cite{GaeDa,Ruiz} do not work
for  $q> q_0^*$, nevertheless also in this case we exhibit here a Mountain Pass structure.

As shown in \cite{GaeDa}, the solutions we find in Theorem \ref{th:2} are classical
and positive by  the Maximum Principle.

Our results holds for any fixed $a\geq0$.
We also notice that, in case $a=0$
we do not need Lemma \ref{regu} and the inequality in  Proposition \ref{ine1}
just follows (up to some positive constant)
by multiplying the second equation in \eqref{eq:sp} by $|u|$ and integrating, which is the relation used in \cite{Ruiz}.
In this sense our result recovers the one in \cite{Ruiz} and  even a better understanding of the fiber
maps is given here, since the Mountain Pass structure for the functional related to \eqref{eq:sp}
holds although the functional is  non-negative for $q\geq q_{0}^{*}$.

Actually  we deduce Theorem \ref{th:1}
and Theorem \ref{th:2} as consequences of the following result which gives
additional information on the solutions.
\begin{theorem}\label{T1}

Let $p\in (2,3], a\geq0$ be fixed. There exist positive numbers $\varepsilon, q_{0}, q_{0}^{*}$ satisfying $q_{0}^{*}+\varepsilon<q^{*}$ such that:
\begin{enumerate}
\item[1.] for each $q>q^{*}$ the functional $\mathcal J_{q}$ has no critical points in $H^{1}(\mathbb R^{3})$
other than the zero function; \medskip
\item[2.]for each $q\in (0,q_0^*+\varepsilon)$ the functional $\mathcal J_{q}$ has two nontrivial critical points $u_{q},w_{q}\in H^{1}_{r}(\mathbb R^{3})$ where $w_{q}$ is a Mountain Pass critical point with
 $$\mathcal{J}_q(w_q)>\max\{ 0, \mathcal{J}_q(u_q)\}.

 $$
More specifically, \medskip
	\begin{enumerate}[label=(\roman*),ref=\roman*]
		\item\label{i1T1} if $q\in (0,q_0^*]$, then $u_q$ is a global minimum, with
$$\mathcal{J}_q(u_q)<0\ \text{ if }\  q\in (0,q_0^*), \quad \mathcal J_{q_{0}^{*}}(u_{q_{0}}^{*})=0;$$
				\item\label{i2T1} if $q\in (q_0^*,q_0^*+\varepsilon)$, then $u_q$ is a local minimum
				with
				$$\mathcal{J}_q(u_q)>0.$$

	\end{enumerate}
\end{enumerate}
\end{theorem}

\medskip

As we can see, whenever $q=q_{0}^{*}$, then $\mathcal J_{q_{0}^{*}}$ is non-negative and we find a global
minimiser at zero energy; then an additional work is needed in order to show that this is not the zero function.
This result is new also in the case $a=0$.

Concerning the  {\sl extremal values} $q_{0}^{*}$ and $q^{*}$ we say here they have a variational
characterisation (see Section \ref{sec:prelim}).

Furthermore, although we were not able to prove it, it seems plausible that for all $q\in (0,q^*)$, the system \eqref{p} has two positive solutions satisfying the properties in Theorem \ref{T1} and $q^*$ is in fact a bifurcation parameter where the two solutions (the local minimum and the Mountain Pass type solution) collapses.

\medskip

We point out finally that similar results have been obtained in some nonlinear problems depending on a parameter
in some recent papers: see Il'yasov and Silva \cite{YaKa}, Silva and Macedo \cite{KaAb}.

\medskip

This paper is organised as follows.

In Section \ref{sec:prelim}
some preliminaries and technical results (true in the general nonradial setting) are given.
As a byproduct of these results, the proof of item 1. in Theorem \ref{T1} follows, see
Corollary \ref{nehari}.

In Section \ref{sec:PS} the important Proposition \ref{MPS} is proved. It concerns with radial
functions and is fundamental in proving our result.

Finally, the proof of Theorem \ref{T1} is completed in Section \ref{sec:final}.

\medskip

As a matter of notations, we use the generic letters $C,C',\ldots$ to denote a positive
constant, usually related to Sobolev embedding, whose value may also change
from line to line: no confusion should arise.

\section{Preliminaries and technical  results}\label{sec:prelim}

In \cite{GaeDa} some properties of the solution $\phi_{u}$ are found.
 However to deal with the case $p\in(2,3]$
we need also the next ones. Of course this applies just for $a\neq0.$

\begin{lemma}\label{regu}
 For each $u\in H^1(\mathbb{R}^3)$ we have
 \begin{enumerate}[label=(\roman*),ref=\roman*]
 \item\label{1regu} $\Delta\phi_u\in \mathcal D$, \medskip
 \item\label{2regu} $a^{2}\Delta\phi_u\le \phi_u.$
 \end{enumerate}

\end{lemma}

\begin{proof}
Let us fix $u\in H^{1}(\mathbb R^{3})$ and let $\psi_{u}: = -\Delta \phi_{u}$. Then
\begin{equation*}
-a^{2}\Delta \psi_{u} +\psi_{u}= 4\pi u^{2}
\end{equation*}
Since $u^{2}\in L^{2}(\mathbb R^{3})$, by standard results

we have $\psi_{u}\in H^{2}(\mathbb R^{3})$
and in particular,
$$\psi_{u}\in L^{6}(\mathbb R^{3}), \  \nabla \psi_{u} \in L^{2}(\mathbb R^{3}) \quad \text {and } \quad  \Delta \psi_{u}\in L^{2}(\mathbb R^{3}).$$
This gives	 $\Delta\phi_{u} \in  D^{1,2}(\mathbb R^{3})$ and $\Delta^{2}\phi_{u}\in L^{2}(\mathbb R^{3})$, namely
$\Delta\phi_{u}\in \mathcal D$ proving \eqref{1regu}.

On the other hand, if we set $v=-a^2\Delta\phi_u+\phi_u$,  then
	\begin{equation*}
	-\Delta v=4\pi u^2 \ge 0
	\end{equation*}
	and $v\in {D}^{1,2}(\mathbb{R}^3)$ and is continuous.
	Define $\Omega^-=\{x\in \mathbb{R}^3:\ v(x)<0\}$ and  suppose that $\Omega^-\neq \emptyset$. Once $v$ is continuous, the set $\Omega^-$ is  open. Let $v^-=\max\{-v,0\}$. It follows that
	\begin{equation*}
	-\int _{\Omega^-}|\nabla v|^2=\int \nabla v\nabla v^-\ge 0,
	\end{equation*}
	which is a contradiction, therefore, $\Omega^-=\emptyset$ and $a^{2}\Delta\phi_u\le {\phi_u}$ in $\mathbb R^{3}$, proving \eqref{2regu}.
\end{proof}

The next result will be useful to get a generalisation of \cite[Formula (19)]{Ruiz} to the case $a\neq0$.
 \begin{proposition}\label{ine1} There holds
	\begin{equation*}
\int |u|^3\le \frac{1}{\pi}\|\phi_u\|_{\mathcal{D}}\|\nabla u\|_2, \quad \forall\,  u\in H^1(\mathbb{R}^3).
	\end{equation*}
\end{proposition}

\begin{proof}

For $u\in H^{1}(\mathbb R^{3})$ fixed, let us consider equation \eqref{solu}.
	 Since by  Lemma \ref{regu} it holds in particular that $\nabla \Delta\phi_{u}\in L^{2}(\mathbb R^{3})$,
	 by multiplying the  equation \eqref{solu} by $|u|\in H^{1}(\mathbb R^{3}) $ and integrating we get
	\begin{eqnarray}\label{diffeq1}
	4\pi\int |u|^3&=&a^2\int \nabla (-\Delta\phi_u)\nabla |u|+\int \nabla \phi_u\nabla|u| \nonumber\\
	&\le& a^2\|\nabla(-\Delta\phi_u)\|_2\|\nabla u\|_2+\|\nabla \phi_u\|_2\|\nabla u\|_2 \nonumber\\
	&=&\left(a^2\|\nabla(-\Delta\phi_u)\|_2+\|\nabla \phi_u\|_2\right)\|\nabla u\|_2\nonumber \\
	&\le& \left(a^2\|\nabla(\Delta\phi_u)\|_2+\|\phi_u\|_{\mathcal{D}}\right)\| \nabla u\|_2.
	\end{eqnarray}
	On the other hand by multiplying  \eqref{solu} by $\Delta\phi_{u}\in \mathcal D$
        and  making use of   \eqref{2regu} of Lemma \ref{regu} we get
	\begin{eqnarray*}
	a^2\|\nabla (\Delta\phi_u)\|_2^2&=&4\pi\int \Delta\phi_u u^2-\int \nabla \phi_u \nabla(\Delta\phi_u) \nonumber\\
	&\le&  \frac{1}{a^{2}}\|\phi_u\|_{\mathcal{D}}^2+\frac{\varepsilon^2}{2} \|\nabla(\Delta\phi_u)\|_2^2+\frac{1}{2\varepsilon^2}\|\nabla \phi_u\|_2^2 \nonumber\\
	&\le&  \frac{1}{a^{2}}\|\phi_u\|_{\mathcal{D}}^2+\frac{\varepsilon^2}{2} \|\nabla(\Delta\phi_u)\|_2^2+\frac{1}{2\varepsilon^2}\| \phi_u\|_{\mathcal{D}}^2.
	\end{eqnarray*}
	By choosing  $\varepsilon=a$ above
	we conclude that, for all $u\in H^{1}(\mathbb R^{3})$,
	\begin{equation}\label{diffeq3}
	a^2\|\nabla (\Delta\phi_u)\|_2^2\le\frac{2}{a^{2}}\|\phi_u\|_{\mathcal{D}}^2+\frac{1}{a^2}\|\phi_u\|_{\mathcal{D}}^2=\frac{3}{a^2}\|\phi_u\|_{\mathcal{D}}^2.
	\end{equation}
	From \eqref{diffeq1} and \eqref{diffeq3} we conclude that
	\begin{equation*}
	\int |u|^3\le \frac{1}{\pi}\|\phi_u\|_{\mathcal{D}}\|\nabla u\|_2,
	\end{equation*}
	completing the proof.
\end{proof}

We conclude this section by showing a first simple property of the energy functional.
The next result says that the functional $\mathcal J_{q}$ has a strict local minimum in $0$,
uniformly in $q$. However to have the complete Mountain Pass structure
$q$ has to be small, as it will be shown in Corollary \ref{zeroenergylevel}.
\begin{proposition}\label{MPGAN} There exist $\rho>0$ and $M>0$ such that
$$\forall q\in \mathbb R, u \in H^{1}(\mathbb R^{3}) \text{ with } \|u\|=\rho :  \  \mathcal{J}_q(u)\ge M.$$
\end{proposition}
\begin{proof} Since
	\begin{equation*}
	\mathcal{J}_q(u)\ge \frac{1}{2}\|u\|^2-\frac{1}{p}\|u\|_p^p\ge \frac{1}{2}\|u\|^2-C \|u\|^p,
	\end{equation*}
	the conclusion easily follows.
\end{proof}

In this Section we establish some notations and study the geometry of the functional $\mathcal{J}_q$.  We observe that $\phi_{tu}=t^2\phi_u$ and therefore, if $\psi_{q,u}:[0,\infty)\to \mathbb{R}$ is defined by $\psi_{q,u}(t)=\mathcal{J}_q(tu)$, we have that
\begin{equation*}
\psi_{q,u}(t)=\frac{t^2}{2}\| u\|^2+\frac{q^2t^4}{4}\int \phi_u u^2-\frac{t^p}{p}\|u\|_p^p.
\end{equation*}
Whenever $q$ and $u$ are fixed, we will use for brevity also the notation $\psi:=\psi_{q,u}$.
A simple analysis shows that
\begin{proposition}\label{fiberingmaps} For each $q\in\mathbb{R}\setminus\{0\}$ and $u\in H^1(\mathbb{R}^3)\setminus\{0\}$, there are only three possibilities for the graph of $ \psi$:
	\begin{enumerate}[label=(\roman*),ref=\roman*]
		\item\label{i1fibering} the function $\psi$ has only two critical points when $t>0$, to wit, $0<t_q^-(u)<t_q^+(u)$. Moreover, $t_q^-(u)$ is a local maximum with $\psi''(t_q^-(u))<0$ and $t_q^+(u)$ is a local minimum with $\psi''(t_q^+(u))>0$; \medskip
		\item\label{i2fibering} the function  $\psi$ has only one critical point when $t>0$ at the value $t_q(u)$. Moreover, $\psi''(t_q(u))=0$ and  $\psi$ is increasing;  \medskip
		\item\label{i3fibering} the function  $\psi$ is increasing and has no critical points.
	\end{enumerate}
\end{proposition}
It is important to notice that \eqref{i1fibering} happens for $q$ small, and \eqref{i3fibering} for $q$ large.

Let us consider the Nehari manifold associated with the functional $\mathcal{J}_q$, that is
\begin{equation*}
\mathcal{N}_q=\{u\in H^1(\mathbb{R}^3)\setminus\{0\}:\ \psi_{q,u}'(1)=0\}.
\end{equation*}
Note that, for $u\in \mathcal N_{q}$:
$$\|u\|^{2}\leq \|u\|^{2} +q^{2}\int \phi_{u} u^{2}\leq C \|u\|^{p}$$
and then all the Nehari manifolds are bounded away from zero uniformly in $q$, in the sense that
\begin{equation}\label{eq:bddaway}
\exists\, \widetilde C>0 \ \text{ such that for all }q\in \mathbb R, \ u\in \mathcal N_{q} : \|u\|\geq \widetilde C.
\end{equation}

Moreover since
\begin{equation*}
\mathcal{N}_q=\mathcal{N}_q^+\cup \mathcal{N}_q^0\cup \mathcal{N}_q^-,
\end{equation*}
where
\begin{equation*}
\mathcal{N}_q^+=\{u\in \mathcal{N}_q:\ \psi''(1)>0\},
\end{equation*}
\begin{equation*}
\mathcal{N}_q^0=\{u\in \mathcal{N}_q:\ \psi''(1)=0\},
\end{equation*}
\begin{equation*}
\mathcal{N}_q^-=\{u\in \mathcal{N}_q:\ \psi''(1)<0\}.
\end{equation*}
as an application of the Implicit Function Theorem one has the following:
\begin{proposition}\label{minnehari} If $\mathcal{N}_q^+,\mathcal{N}_q^-\neq \emptyset$, then $\mathcal{N}_q^+,\mathcal{N}_q^-\neq \emptyset$ are $C^1$ manifolds of codimension $1$ in $H^1(\mathbb{R}^3)$. Moreover, $u\in \mathcal{N}_q^+\cup\mathcal{N}_q^-$ is a critical point for the functional $\mathcal{J}_q$ if and only if $u$ is a critical point of the constrained functional $(\mathcal{J}_q)_{|\mathcal{N}_q^+\cup\mathcal{N}_q^-}: \mathcal{N}_q^+\cup\mathcal{N}_q^-\to \mathbb{R}$.
\end{proposition}

Note that, fixed $u\in H^{1}(\mathbb R^{3})\setminus\{0\}$ we have $tu\in \mathcal{N}_q^0$ if, and only if
$\psi'_{q,tu}(1) = \psi''_{q,tu}(1)=0$
, i.e. the following system of equations is satisfied:
\begin{equation}\label{extremal}
\left\{
\begin{aligned}

&t\|u\|^2+q^2t^3\int \phi_u u^2-t^{p-1}\|u\|_p^p = 0, \\

 &\|u\|^2+3q^2t^2\int \phi_u u^2-(p-1)t^{p-2}\|u\|_p^p = 0.
\end{aligned}
\right.
\end{equation}
We can solve the system \eqref{extremal} with respect to the variables $q$ and $t$ to obtain a unique solution given by
\begin{equation*}
t(u) =\Big( \frac{2\|u\|^{2}}{(4-p) \|u\|_{p}^{p}}\Big)^{1/(p-2)}
\end{equation*}
and
\begin{equation}\label{extremalfun}
q(u)=C_p\frac{\|u\|_p^{p/(p-2)}}{\|u\|^{(4-p)/(p-2)} \|\phi_u\|_\mathcal{D}},
\qquad C_{p}=\frac{2 (p-2)^{1/2} \pi^{1/2}(4-p)^{(4-p)/2(p-2)}}{2^{1/(p-2)}}.
\end{equation}

In addition the solutions $q(u)$ and $t(u)$ are related by
\begin{equation*}
t(u)=\left(\frac{2q^2(u)}{4\pi(p-2)}\frac{\|\phi_{u}\|_{\mathcal D}^{2}}{\|u\|_p^p}\right)^{{1}/(p-4)}.
\end{equation*}
 Define the extremal value (see Il'yasov \cite{ilyasENMM})
\begin{equation*}
q^*=\sup\left\{q(u):\ u\in H^1(\mathbb{R}^3)\setminus\{0\} \right\}.
\end{equation*}

\begin{lemma}\label{extremalparamater} The function $H^1(\mathbb{R}^3)\setminus\{0\}\ni u\mapsto q(u)$ defined in \eqref{extremalfun} is $0$-homogeneous. Moreover $q^*<\infty$.
\end{lemma}

\begin{proof} That $q(u)$ is zero homogeneous is obvious. Let us prove that $q^*<\infty$. Indeed, once $p\in (2,3]$ we have from the interpolation inequality that, for all $u\in H^{1}(\mathbb R^{3})$ we have
	\begin{equation}\label{rt}
	\|u\|_p^p\le \|u\|^{6-2p}\|u\|_3^{3p-6}.
		\end{equation}
Combining the inequality \eqref{rt} with the Proposition \ref{ine1} we conclude that
\begin{equation}\label{rt1}
\|u\|_p^p\le  C\|u\|^{6-2p}\|u\|^{(3p-6)/3}\|\phi_u\|_\mathcal{D}^{(3p-6)/3}=C\|u\|^{4-p}\|\phi_u\|_\mathcal{D}^{p-2}.
\end{equation}
for some constant $C>0$. It follows from \eqref{rt1} that
\begin{equation*}
q(u)\le C\frac{\|u\|^{(4-p)/(p-2)}\|\phi_u\|_\mathcal{D}}{\|u\|^{(4-p)/(p-2)} \|\phi_u\|_\mathcal{D}}\le C
\end{equation*}
completing the proof.
\end{proof}

Another extremal value  which will be important for us is the one such that, for larger values of the parameter, the functional
is always non-negative. Let us start by   fixing $u\in H^{1}(\mathbb R^{3})\setminus\{0\}$ and
considering the system
\begin{equation*}
\left\{
\begin{aligned}
\psi_{q_0,u}(t_0)= \frac{t_0^2}{2}\|u\|^2+ q_0^2\frac{t_0^4}{4}\int \phi_u u^2-\frac{t_0^{p}}{p}\|u\|_p^p &= 0, \\
\psi'_{q_0,u}(t_0)= t_0\| u\|^2+q_0^2t_0^3\int \phi_u u^2-t_0^{p-1}\|u\|_p^p &= 0.
\end{aligned}
\right.
\end{equation*}
One can solve this system with respect to the variables $t_0$ and $q_0$ to obtain the unique solution given by
\begin{equation}\label{zeroenergy}
q_0(u)=C_{0,p}\frac{\|u\|_p^{p/(p-2)}}{\|u\|^{(4-p)/(p-2)} \|\phi_u\|_\mathcal{D}}, \qquad
C_{0,p} =\frac{2^{3/2} (p-2)^{1/2} \pi^{1/2}(4-p)^{(4-p)/2(p-2)}}{p^{1/(p-2)}}
\end{equation}
and $t_{0}(u)$ is given by
\begin{equation*}
t_0(u)=\left(\frac{pq_0^2(u)}{2(p-2)}\frac{\|\phi_u\|_{\mathcal D}^2}{\|u\|_p^p}\right)^{1/(p-4)}.
\end{equation*}
Observe that $C_{0,p}<C_p$, where $C_{p}$
is the one appearing in \eqref{extremalfun}. Then $q_{0}(u)< q(u)$. Define the extremal value as
\begin{equation*}
q_0^*=\sup\left\{q_0(u):\ u\in H^1(\mathbb{R}^3)\setminus\{0\} \right\}.
\end{equation*}

\begin{remark}\label{extremalparameter1}Once $q_0(u)$ is a multiple of $q(u)$, Lemma \ref{extremalparamater} also holds true for the function $q_0$.
\end{remark}

The solutions $q(u)$ and $q_0(u)$ given in \eqref{extremalfun}  and \eqref{zeroenergy} have the following geometrical interpretation which can be proved starting from the Proposition \ref{fiberingmaps}.
\begin{proposition}\label{fiberingvariation} For each $u\in H^1(\mathbb{R}^3)\setminus\{0\}$ there holds:
	\begin{enumerate}[label=(\roman*),ref=\roman*]
		\item\label{i1} $q(u)$ is the unique parameter $q>0$ for which the fiber map $\psi_{q,u}$ has a critical point with second derivative zero at $t(u)$. Moreover, if $0<q<q(u)$, then $\psi_{q,u}$ satisfies \eqref{i1fibering} of the Proposition \ref{fiberingmaps} while if $q>q(u)$, then $\psi_{q,u}$ satisfies \eqref{i3fibering} of the Proposition \ref{fiberingmaps}. \medskip
		\item\label{i2} $q_0(u)$ is the unique parameter $q>0$ for which the fiber map $\psi_{q,u}$ has a critical point with zero energy at $t_0(u)$. Moreover, if $0<q<q_0(u)$, then $\inf_{t>0}\psi_{q,u}(t)<0$ while if $q>q_{0}(u)$, then $\inf_{t>0}\psi_{q,u}(t)=0$.
	\end{enumerate}
\end{proposition}

Moreover the parameter $q_0^*$ has the geometrical interpretation that for each $q\in (0,q_0^*)$, there exists at least one $ u\in H^1(\mathbb{R}^3)\setminus\{0\}$ for which $\mathcal{J}_q(u)<0$, while if $q\ge q_0^*$, then $\mathcal{J}_q(u)\ge 0$ for all $ u\in H^1(\mathbb{R}^3)$. In  both works \cite{GaeDa,Ruiz}  the necessity of small values of $q$ was imposed in order to show that there exists a function where the functional is negative, in such a way that
$\mathcal{J}_q$ possesses a Mountain Pass Geometry.
Therefore, the argument employed in both papers do not work
for  $q> q_0^*$.

The above proposition has the following important consequences.
\begin{corollary}\label{nehari}

	 If $q>q^*$ the functional $\mathcal{J}_q$ has no critical points other then the zero function. Moreover if $q<q^*$, then $\mathcal{N}_q^-\neq\emptyset$ and $\mathcal{N}_q^+\neq\emptyset$.
\end{corollary}

In particular item 1. of Theorem \ref{T1} is proved.
\begin{proof}

It is sufficient to show that, for each  $u\in H^{1}(\mathbb R^{3})\setminus\{0\}$,
the function $\psi_{q,u}$ has no critical points for $q>q^{*}$.
Actually this is a consequence of the inequalities $q(u)\le q^*<q$ and
\eqref{i1} of  Proposition \ref{fiberingvariation}.

Now assume that $q<q^*$. From the definition of $q^*$, there exists $u\in H^1(\mathbb{R}^3)\setminus\{0\}$ such that $q<q(u)<q^*$. Therefore, from \eqref{i1} of Proposition \ref{fiberingvariation} we conclude that $\mathcal{N}_q^-\neq\emptyset$ and $\mathcal{N}_q^+\neq\emptyset$.
\end{proof}

\begin{corollary}\label{zeroenergylevel} For each $q\ge q_0^*$, there holds $\mathcal{J}_q(u)\ge 0$ for all $u\in H^1(\mathbb{R}^3)$. Moreover, if $q<q_0^*$, then there exists $u\in H^1(\mathbb{R}^3)$ such that $\mathcal{J}_q(u)<0$.
\end{corollary}

\begin{proof} Indeed, assume that  $q\ge q_0^*$. It follows that $q>q_0(u)$ for each  $u\in H^1(\mathbb{R}^3)\setminus\{0\}$ and from item \eqref{i2} of Proposition \ref{fiberingvariation}  there holds $\inf_{t>0}\psi_{q,u}(t)=0$. Therefore $\mathcal{J}_q(u)\ge 0$.

	Now assume that $q<q_0^*$. From the definition of $q_0^*$, there exists $w\in H^1(\mathbb{R}^3)\setminus\{0\}$ such that $q<q_0(w)<q_0^*$. Threfore, from \eqref{i2} of Proposition \ref{fiberingvariation} we conclude that $\inf_{t>0}\psi_{q,w}(t)<0$ and hence there exists $t>0$ such that if $u:= tw$, it holds $\mathcal{J}_q(u)<0$.
\end{proof}

Let us conclude this section with the following important result.

\begin{proposition}\label{N0ene} There exists a positive constant $m$ such that
$$\forall q\in \mathbb R, u\in \mathcal{N}_q^0 :  \mathcal{J}_q(u)\ge m.$$
\end{proposition}
\begin{proof} From the equations \eqref{extremal} with $t=1$ we have that
	\begin{equation*}
	\left\{
	\begin{aligned}
	\|u\|^2+q^2\int \phi_u u^2-\|u\|_p^p &= 0, \\
	\|u\|^2+3q^2\int \phi_u u^2-(p-1)\|u\|_p^p &= 0.
	\end{aligned}
	\right.
	\end{equation*}
It follows that $$q^{2}\int\phi_{u}u^{2}=\|u\|_{p}^{p} - \|u\|^{2} \quad \text{ and }\quad \|u\|_{p}^{p} = \frac{2}{4-p}\|u\|^{2}$$
so that $\mathcal{J}_q(u)=\frac{p-2}{4p}\|u\|^2$ for each $u\in \mathcal{N}_q^0$.
From \eqref{eq:bddaway}  the proof is completed.
\end{proof}

It is worth to point out that all that we have done in this section does not use the radial setting,
and clearly these results also holds in $H^{1}_{r}(\mathbb R^{3})$.

\section{Global Minima and (PS) Sequences for $\mathcal{J}_q$}\label{sec:PS}
In this section we prove the following result. Here and in the next section is fundamental
to work with radial functions.
\begin{proposition}\label{MPS} There holds:
	\begin{enumerate}[label=(\roman*),ref=\roman*]
		\item\label{iMPS} for each $q\in (0,q_0^*)$ we have that $-\infty<\inf_{u\in H^1_{r}(\mathbb{R}^3)}\mathcal{J}_q(u)<0$;
		\item\label{iiMPS} for each $q> 0$, if $\{u_n\}\subset H_r^1(\mathbb{R}^3)$ is a sequence such that $\mathcal{J}'_q(u_n)\to 0$ as $n\to \infty$, then $\{u_n\}$ is convergent, up to subsequences.
	\end{enumerate}
\end{proposition}

\begin{proof}

Let us show \eqref{iMPS}.
 Indeed, from the Corollary \ref{zeroenergylevel} we know that $\inf_{u\in H^1_r(\mathbb{R}^3)}\mathcal{J}_q(u)<0$. We claim that $-\infty<\inf_{u\in H^1_{r}(\mathbb{R}^3)}\mathcal{J}_q(u)$. In fact, given $\varepsilon>0$ such that $D:=\frac{q^2}{16\pi}-\varepsilon^4>0$ , by  Proposition \ref{ine1}, we have that for each  $u\in H^1_r(\mathbb{R}^3)$ there holds
\begin{eqnarray}
\mathcal{J}_q(u)&=&\frac{1}{4}\|\nabla u\|_2^2+\frac{1}{4}\|\nabla u\|_2^2+\frac{1}{2}\| u\|_2^2+\frac{q^2}{4}\int \phi_u u^2-\frac{1}{p}\|u\|_p^p \nonumber\\
                &\ge& \frac{1}{4}\|\nabla u\|_2^2+D\|\phi_u\|_\mathcal{D}^2+\frac{1}{2}\| u\|_2^2+\frac{\pi\varepsilon^2}{4}\|u\|_3^3-\frac{1}{p}\|u\|_p^p\nonumber\\
                &=&\frac{1}{4}\|u\|^{2}+D\|\phi_{u}\|_{\mathcal D}^{2}+\int f(u) \label{M2}
\end{eqnarray}
where
\begin{equation*}\label{ft}
f(t)=\frac{1}{4}t^2+\frac{\pi\varepsilon^2}{4}t^3-\frac{1}{p}t^p,\quad \forall\, t>0.
\end{equation*}
A simple analysis shows that $I:=\inf_{t>0}f(t)>-\infty$ and if $f(t)<0$ for some $t>0$, then $f^{-1}((-\infty,0))=(\alpha,\beta)$, where $0<\alpha<\beta<\infty$.

If $I\ge 0$, being $D_q>0$,  from  \eqref{M2} we conclude that $-\infty<\inf_{u\in H^1_r(\mathbb{R}^3)}\mathcal{J}_q(u)$.

If $I<0$ then
\begin{equation}\label{eq:bo}
\mathcal J_{q}(u)\geq\frac{1}{4}\|u\|^{2}+D_{q}\|\phi_{u}\|_{\mathcal D}^{2}+I\,{\sl meas}(A)
\end{equation}
where $A=\{ x\in \mathbb R^{3}: u(x)\in (\alpha,\beta)\}$.
If there exists a sequence $\{u_n\}\subset H^1_r(\mathbb{R}^3)$ such that $\mathcal{J}_q(u_n)\to -\infty$ as $n\to \infty$,
then $\|u_{n}\|\to+\infty$.
Moreover by \eqref{eq:bo}

we can assume without loss of generality that
\begin{equation}\label{F1}
\frac{1}{4}\|u_n\|^2<|I| \,{\sl meas}(A_{n}),\quad \forall\, n\in \mathbb N.
\end{equation}
By the result of Strauss \cite{Strau} we know that there exist a positive constant $C$ such that
\begin{equation}\label{stra}
|u(x)|\le C|x|^{-1}\|u\|,\quad \forall\,u\in H^1_r(\mathbb{R}^3).
\end{equation}
Define $\rho_n=\sup\{|x|:\ x\in A_n\}$ and observe from the inequalities \eqref{F1} and \eqref{stra} that, for every $x\in \mathbb R^{3}$
with $|x|=\rho_{n}$ it holds,
\begin{equation*}
0< \alpha\le u_{n}(x)\le C\rho_n^{-1}\|u_n\|\le 2C\rho_n^{-1}(|I| {\sl meas}(A_n))^{1/2},
\end{equation*}
and hence, for some $C'>0$, we deduce
\begin{equation}\label{F2}
C'\rho_n\le {\sl meas}(A_n)^{1/2}.
\end{equation}
 Similar to the deduction of \eqref{F1}, we can assume without loss of generality that
\begin{equation*}
D\|\phi_{u_n}\|_\mathcal{D}^2<|I|{\sl meas}(A_n),\quad \forall\, n\in \mathbb N,
\end{equation*}
and hence, once that the function $(0,\infty)\ni t\mapsto (1-e^{-t/a})/t$ is decreasing we conclude that
\begin{eqnarray*}
|I| {\sl meas}(A_n)&>&D\|\phi_{u_n}\|_\mathcal{D}^2 \\
          &=&\int\int\frac{1-e^{-\frac{|x-y|}{a}}}{|x-y|}u^2_n(x)u^2_n(y) \\
          &\ge& \int_{A_n}\int_{A_n}\frac{1-e^{-\frac{|x-y|}{a}}}{|x-y|}u^2_n(x)u^2_n(y) \\
          &\ge&\frac{1-e^{-\frac{2\rho_n}{a}}}{2\rho_n}\alpha^4 {\sl meas}(A_n)^2,
\end{eqnarray*}
which implies that
\begin{equation}\label{F22}
\frac{|I|}{\alpha^{4}}\ge \frac{1-e^{-\frac{2\rho_n}{a}}}{2\rho_n}{\sl meas}(A_n),\quad \forall n\in \mathbb N,
\end{equation}
Observe from \eqref{F2} that $|A_n|\to \infty$ as $n\to \infty$ and from \eqref{F22} that $\rho_n\to \infty$ as $n\to \infty$. Combining \eqref{F2} with \eqref{F22} we obtain that
\begin{equation*}
C''\ge (1-e^{-\frac{2\rho_n}{a}})\rho_n,
\end{equation*}
for some $C''>0$,
which is clearly a contradiction and therefore
\eqref{iMPS} is proved.

Let us show \eqref{iiMPS}.

From the convergence $\mathcal{J}'_q(u_n)\to 0$ as $n\to \infty$, we can assume without loss of generality that
\begin{equation}\label{Q0}
\mathcal{J}'_q(u_n)[u_n]\le \|u_n\|,\ \forall n\in \mathbb N.
\end{equation}
On the other hand, from  Proposition \ref{ine1} we have that
\begin{equation*}
\|u_n\|_3^3\le \frac{1}{\pi}\left(\frac{1}{\varepsilon^2}\|\nabla u_n\|_2^2+\varepsilon^2\|\phi_{u_n}\|_\mathcal{D}^2\right),
\end{equation*}
where $\varepsilon>0$ is chosen now such that $\frac{q^2}{4\pi}-\frac{\varepsilon^4}{2}>0$.
It follows that, for all $n\in \mathbb N$:
\begin{eqnarray}\label{Q1}
\mathcal{J}_q'(u_n)[u_n]&=&\|\nabla u_n\|^2+\|u_n\|_2^2+\frac{q^2}{4\pi}\|\phi_{u_n}\|_\mathcal{D}^2-\|u_n\|_p^p \nonumber\\
                      &\ge& \frac{1}{2}\|u_n\|^2+\frac{1}{2}\|u_n\|_2^2+\frac{\pi\varepsilon^2}{2}\|u_n\|_3^3-\frac{\varepsilon^4}{2}\|\phi_{u_n}\|_\mathcal{D}^2+\frac{q^2}{4\pi}\|\phi_{u_n}\|_\mathcal{D}^2-\|u_n\|_p^p \nonumber\\
                      &=& \frac{1}{2}\|u_n\|^2+\left(\frac{q^2}{4\pi}-\frac{\varepsilon^4}{2}\right)	\|\phi_{u_n}\|_\mathcal{D}^2+\frac{1}{2}\|u_n\|_2^2+\frac{\pi\varepsilon^2}{2}\|u_n\|_3^3-\|u_n\|_p^p \nonumber\\
                      &=& \frac{1}{2}\|u_n\|^2+\left(\frac{q^2}{4\pi}-\frac{\varepsilon^4}{2}\right)	\|\phi_{u_n}\|_\mathcal{D}^2+\int g(u_n),
\end{eqnarray}
where $g(t)=t^2/2+\frac{\pi\varepsilon^2}{2}t^3-t^p$ for $t>0$. We combine \eqref{Q0} with \eqref{Q1} to conclude that
\begin{equation*}
\|u_n\|\ge  \frac{1}{2}\|u_n\|^2+\left(\frac{q^2}{4\pi}-\frac{\varepsilon^4}{2}\right)	\|\phi_{u_n}\|_\mathcal{D}^2+\int g(u_n).
\end{equation*}
We conclude as in the proof of \eqref{iMPS} that $\{u_n\}$ is bounded. Once we know that $\{u_n\}$ is bounded, standard arguments (observe that the analogous of \cite[Lemma 2.1]{Ruiz} is valid) produce a convergent subsequence.

\end{proof}

\begin{remark}\label{rem:qn}
Note that \eqref{iiMPS} in the Proposition \ref{MPS} can be extended in the following way:
if $q_{n} \to q$ and  if $\{u_n\}\subset H_r^1(\mathbb{R}^3)$ is a sequence such that $\mathcal{J}'_{q_{n}}(u_n)\to 0$ as $n\to \infty$, then $\{u_n\}$ is convergent, up to subsequences.
This follows due to the smooth dependence of $\mathcal J_{q}'$ on $q$.
\end{remark}

\section{Existence of Two radial Solutions} \label{sec:final}
In this section we prove item 2. of Theorem \ref{T1}.

\begin{proposition}\label{beforeq_0} For each $q\in (0,q_0^*)$ there exists a global minimum $u_q$ such that $\mathcal{J}_q(u_q)<0$.
\end{proposition}
\begin{proof} It follows from the Proposition \ref{MPS} and Ekeland's Variational Principle.
\end{proof}
Now we prove the existence of a local minimizer for $\mathcal{J}_q$ when $q$ is near $q_0^*$. To do so, we first prove the existence of a global minimizer for the functional $\mathcal{J}_{q_0^*}$.
\begin{corollary}\label{Mq_0}
There exists a global minimizer $u_{q_0^*}\neq 0$ of $\mathcal J_{q_{0}^{*}}$
such that
$ \mathcal{J}_{q_0^*}(u_{q_0^*})=0$.
\end{corollary}
\begin{proof} Indeed, suppose that $q_n\uparrow q_0^*$ as $n\to \infty$. From the Proposition \ref{beforeq_0}, for each $n$, there exists $u_n:= u_{q_n}$ such that $u_n$ is a global minimum for $\mathcal{J}_{q_n}$ and $\mathcal{J}_{q_n}(u_n)<0$. It follows that $\mathcal{J}'_{q_n}(u_n)=0$ for each $n$ and,
being all the Nehari manifolds bounded away from zero uniformly in $q$, by \eqref{eq:bddaway} it is
$\|u_n\|\ge \widetilde C$ for each $n$. From \eqref{iiMPS} in Proposition \ref{MPS}
(see Remark \ref{rem:qn}) we conclude that $u_n\to u\neq 0$. By $\mathcal{J}_{q_n}(u_{n})<0$ for each $n$, we conclude that $\mathcal{J}_{q_0^*}(u)\le 0$ and from  Corollary \ref{zeroenergylevel} it follows that $\mathcal{J}_{q_0^*}(u)=0$. Then it is sufficient to set $u_{q_0^*}:= u$ and  the proof is completed.
\end{proof}
\begin{remark}\label{RMK} From the definition of $q_0^*$ and the Corollary \ref{Mq_0} it follows that $q_0(u_{q_0^*})=q_0^*$. Moreover $q_0^*<q(u_{q_0^*})$.
\end{remark}
For $q>0$, define
\begin{equation*}
\widehat{\mathcal{J}}_q:=\inf \left\{\mathcal{J}_q(u):\ u\in \mathcal{N}_q^+\cup \mathcal{N}_q^0 \right\}.
\end{equation*}
Observe that
\begin{equation}\label{eq:infimo}
\widehat{\mathcal{J}}_q=\inf_{u\in H^1_r(\mathbb{R}^3)}\mathcal{J}_q(u)\quad \forall q\in (0,q_0^*]
\end{equation}
 and from  Corollary \ref{zeroenergylevel} there holds  $\widehat{\mathcal{J}}_q\ge 0$ for $q \geq q_0^*$.
\begin{proposition}\label{enerynear} Given $\delta>0$, there exists $\varepsilon>0$ such that for each $q\in (q_0^*,q_0^*+\varepsilon)$ there holds $\widehat{\mathcal{J}}_q<\delta$.
\end{proposition}
\begin{proof} Indeed, let $u_{q_0^*}\in \mathcal N_{q}^{+}$ be given as in  Corollary \ref{Mq_0}. Observe that if $q\downarrow q_0^*$, then $\mathcal{J}_q(u_{q_0^*})\to \mathcal{J}_{q_0^*}(u_{q_0^*})=0$. Moreover, once $q_0^*<q(u_{q_0^*})$, it follows that there exists $\varepsilon_1>0$ such that $q_0^*+\varepsilon_1<q(u_{q_0^*})$. From  Proposition \ref{fiberingmaps} and \eqref{i2} in Proposition \ref{fiberingvariation}, for each $q\in (q_0^*,q_0^*+\varepsilon_1)$, there exists $t_q^+(u_{q_0^*})$  such that $t_q^+(u_{q_0^*})u_{q_0^*}\in \mathcal{N}_q^+$. Note that $t_q^+(u_{q_0^*})\to 1$ as $q\downarrow q_0^*$ and therefore
	\begin{equation*}
	\mathcal{J}_q(u_{q_0^*})\le \mathcal{J}_q(t_q^+(u_{q_0^*})u_{q_0^*})\to \mathcal{J}_{q_0^*}(u_{q_0^*})=0,\ q\downarrow q_0^*.
	\end{equation*}
If $\varepsilon_2>0$ is choosen in such a way that $\mathcal{J}_q(t_q^+(u_{q_0^*})u_{q_0^*})<\delta$ for each $q\in (q_0^*,q_0^*+\varepsilon_2)$, then we set $\varepsilon=\min\{\varepsilon_1,\varepsilon_2\}$ and the proof is completed.
\end{proof}
Let us recall that by
Proposition \ref{MPGAN}, there exist positive constants $\rho,M$ such that $\mathcal{J}_q(u)\ge M$ for each $\|u\|=\rho$. We can assume without loss of generality that $\rho<\widetilde C$, where $\widetilde C$ is
such that
$$\text{ for all }q\in \mathbb R, \ u\in \mathcal N_{q} : \|u\|\geq \widetilde C,  $$
(see \eqref{eq:bddaway}).

We choose $\delta>0$  in the Proposition \ref{enerynear} in such a way that
\begin{equation}\label{eq:delta}
\delta<\min\{M,m\},
\end{equation}
where $m$ is the
positive constant such that, by  Proposition \ref{N0ene},
 $$\forall q\in \mathbb R, u\in \mathcal{N}_q^0 :  \mathcal{J}_q(u)\ge m.$$
Let $\varepsilon>0$ be as in Proposition \ref{enerynear} in  correspondence of the above fixed
$\delta>0$.

\begin{proposition}\label{EKE1} There holds
	\begin{equation*}
	\inf\left\{\mathcal{J}_q(u):\ \|u\|\ge \rho\right\}=\widehat{\mathcal{J}}_q,\quad \forall\, q\in (q_0^*,q_0^*+\varepsilon).
	\end{equation*}
\end{proposition}
\begin{proof} First observe from the inequality $\rho<\widetilde C$ that $\inf\{\mathcal{J}_q(u):\|u\|\ge \rho\}\le \widehat{\mathcal{J}}_q$. We claim that the equality holds. Indeed, by one hand, if the fiber map $\psi_{q,u}$ satisfies \eqref{i2fibering} or \eqref{i3fibering} of the Proposition \ref{fiberingmaps}, then $\inf_{t>\rho} \psi_{q,u}(t)=M$. On the other hand, if the fiber map $\psi_{q,u}$ satisfies \eqref{i1fibering} of the Proposition \ref{fiberingmaps}, then $\inf_{t>\rho} \psi_{q,u}(t)\ge\widehat{\mathcal{J}}_q $. Once
	$M>\delta >\widehat{\mathcal{J}}_q$ the proof is completed.
\end{proof}
\begin{corollary}\label{LocM}
For each $q\in (q_0^*,q_0^*+\varepsilon)$ there holds: there exists $u_q\in \mathcal{N}_q^+$ such that $\mathcal{J}_q(u_q)=\widehat{\mathcal{J}}_q$. In particular $\mathcal J_{q}(u_{q})>0$ and  $\|u_q\|\ge \widetilde C>\rho$.
\end{corollary}
\begin{proof}

	Fix $q\in (q_0^*,q_0^*+\varepsilon)$ and let $\{u_n\}\subset \mathcal N_{q}^{+}\cup \mathcal N_{q}^{0}$ be a minimising sequence for $\widehat{\mathcal J}_{q}<\delta$ by Proposition \ref{enerynear}.  Since $m>\delta$  and, by Proposition \ref{N0ene}, $\mathcal J_{q} (u)\geq m$ on $\mathcal N_{q}^{0}$, we can assume that $\{u_{n}\}\subset \mathcal N_{q}^{+}$ and hence, by the Ekeland's Variational Principle,
also that $\mathcal{J}'_q(u_n)\to 0$.

	  We conclude from the Proposition \ref{MPS}  that $u_n\to u$ in $H_r^1(\mathbb{R}^3)$ with $\|u\|\ge \widetilde C> \rho$.
	Setting $u_q:= u$ clearly we obtain that  $u_q\in \mathcal{N}_q^+$ and $\mathcal{J}_q(u_q)=\widehat{\mathcal{J}}_q$. Due to the definition of $q_0^*$ and the fact that $q>q_0^*$, we conclude that $\mathcal J_{q}(u_{q})>0$.
\end{proof}
We observe two properties of  the function $(0,q_0^*+\varepsilon)\ni q\mapsto\widehat{\mathcal{J}}_q$.
\begin{lemma}\label{contidecre} The function $(0,q_0^*+\varepsilon)\ni q\mapsto\widehat{\mathcal{J}}_q$ is increasing and continuous.
\end{lemma}
\begin{proof}
Indeed, suppose that $q<q'$. From  Proposition \ref{beforeq_0}, Corollary \ref{Mq_0} and the Corollary \ref{LocM}
 there exists $u_{q'}$ such that $\widehat{\mathcal{J}}_{q'}=\mathcal{J}_{q'}(u_{q'})$.

 If $q'\in (q_0^*,q_0^*+\varepsilon)$, from the Corollary \ref{LocM} it is $\|u_{q'}\|\ge \widetilde C>\rho$ and hence from the Proposition \ref{EKE1} we obtain
	\begin{equation*}
	\widehat{\mathcal{J}}_{q}\le \mathcal{J}_{q}(u_{q'})<\mathcal{J}_{q'}(u_{q'})=\widehat{\mathcal{J}}_{q'}.
	\end{equation*}
	If $q'\in (0,q_{0}^{*}]$ the lemma follows by \eqref{eq:infimo}.

\medskip

Now we prove that $(0,q_0^*+\varepsilon)\ni q\mapsto\widehat{\mathcal{J}}_q$ is continuous. In fact, suppose that $q_n\uparrow q\in (0,q_0^*+\varepsilon)$. From  Proposition \ref{beforeq_0}, Corollary \ref{Mq_0} and  Corollary \ref{LocM}, for each $n$, there exists $u_n:= u_{q_n}$ such that 	$\widehat{\mathcal{J}}_{q_n}= \mathcal{J}_{q_n}(u_n)$. Similar to the proof of  Corollary \ref{Mq_0} we may assume that $u_n\to u\neq 0$.

As before, if $q\in (0,q_{0}^{*}]$ the lemma holds due to   \eqref{eq:infimo}.

If $q\in (q_0^*,q_0^*+\varepsilon)$ observe from  Corollary \ref{LocM} that $\|u\|>\rho$. We claim that $\widehat{\mathcal{J}}_{q_n}\to \widehat{\mathcal{J}}_q$ as $n\to \infty$. Indeed, once $(0,q_0^*+\varepsilon)\ni q\mapsto\widehat{\mathcal{J}}_q$ is increasing, we can assume that $\widehat{\mathcal{J}}_{q_n}< \widehat{\mathcal{J}}_q$ for each $n$ and $\widehat{\mathcal{J}}_{q_n}\to \mathcal{J}_q(u) \le\widehat{\mathcal{J}}_q$ as $n\to \infty$, which implies that $\mathcal{J}_q(u) =\widehat{\mathcal{J}}_q$.

Now suppose that $q_n\downarrow q\in (0,q_0^*+\varepsilon)$. Once $(0,q_0^*+\varepsilon)\ni q\mapsto\widehat{\mathcal{J}}_q$ is increasing, we can assume that $\widehat{\mathcal{J}}_{q}<\widehat{\mathcal{J}}_{q_{n}}$ and $\widehat{\mathcal{J}}_q\leq \lim_{n\to \infty}\widehat{\mathcal{J}}_{q_n}$. Choose $u_q$ such that $\widehat{\mathcal{J}}_q=\mathcal{J}_q(u_q)$ ($\|u\|>\rho$ in case $q\in (q_0^*,q_0^*+\varepsilon)$) and observe that $\widehat{\mathcal{J}}_q\le \lim_{n\to \infty}\widehat{\mathcal{J}}_{q_n}\le\lim_{n\to \infty}\mathcal{J}_{q_n}(u_q)=\widehat{\mathcal{J}}_q$.
\end{proof}

Now we turn our attention to the second solution.

Let $q\in (0,q_{0}^{*})$.
 As a consequence of the Corollary \ref{zeroenergylevel} we have that $$\Gamma_{q}=\{\gamma\in C([0,1],H_r^1(\mathbb{R}^3)):\ \gamma(0)=0,\ \mathcal{J}_{q}(\gamma(1))<0 \}$$ is non-empty. Define the Mountain Pass level
\begin{equation*}
c_{q}:= \inf_{\gamma\in \Gamma_{q}}\max_{t\in [0,1]}\mathcal{J}_{q}(\gamma(t))> 0.
\end{equation*}

By Proposition \ref{MPGAN} and Proposition \ref{MPS} we deduce the following
\begin{proposition}\label{MPG} For each $q\in (0,q_0^*)$ there exists $w_q\in H^1_r(\mathbb{R}^3)\setminus\{0\}$ such that $\mathcal{J}_q(w_q)=c_q$ and $\mathcal{J}'_q(w_q)=0$. In particular $\mathcal J_{q}(w_{q})>\mathcal J_{q}(u_{q})\in(-\infty,0).$
\end{proposition}

Let now $q\in (q_0^*,q_0^*+\varepsilon)$,
where $\varepsilon>0$ is the one fixed in correspondence of $\delta$ in \eqref{eq:delta}.

Let  $u_q\in \mathcal{N}_q^+$ (by Proposition \ref{LocM}) such that $\mathcal{J}_q(u_q)=\widehat{\mathcal{J}}_q$. Define
\begin{equation*}
d_q=\inf_{\gamma\in \Gamma_q}\max_{t\in [0,1]}\mathcal{J}(\gamma(t)),
\end{equation*}
	where $\Gamma_{q}=\{\gamma\in C([0,1],H_r^1(\mathbb{R}^3)):\ \gamma(0)=0,\ \gamma(1)=u_q \}$.
\begin{proposition}\label{MPCP} For each $q\in (q_0^*,q_0^*+\varepsilon)$ there exists $w_q\in H^1_r(\mathbb{R}^3)\setminus\{0\}$ such that $\mathcal{J}_q(w_q)=d_q$ and $\mathcal{J}'_q(w_q)=0$. In particular $\mathcal J_{q}(w_{q})>\mathcal J_{q}(u_{q})$.
\end{proposition}
\begin{proof} Indeed, we combine  Proposition \ref{MPGAN} with the inequality $M>\delta\ge \widehat{\mathcal{J}}_q =\mathcal{J}_q(u_q)$ to obtain a Mountain Pass Geometry for the functional $\mathcal{J}_q$. The proof follows from \eqref{iiMPS} in Proposition \ref{MPS}.
\end{proof}

\medskip

Now we conclude the proof of item 2. of Theorem  \ref{T1}.

Let $\varepsilon$ be given as in the Proposition \ref{EKE1}. The existence of  the minimum
$u_q$ follows from Proposition \ref{beforeq_0}, Corollary \ref{Mq_0}, Corollary \ref{LocM}.
The existence of a Mountain Pass critical point $w_{q}$ satisfying $\mathcal J_{q}(w_{q})>\max\{ 0, \mathcal J_{q}(u_{q})\}$ follows by Proposition \ref{MPG} and Proposition \ref{MPCP}.
That $u_{q}$ and $w_{q}$ are actually critical points of $\mathcal J_{q}$ follows by Proposition \ref{minnehari}.


TITLE: Is it possible that two measurements will determine if the labels are right?
QUESTION [3 upvotes]: Suppose we have $6$ identically looking cubes that weigh  $1,\dots,6$ grams, respectively. Someone has labeled them 1g, 2g, 3g, 4g, 5g,6g however he might have labeled them incorrectly.
Suppose you have a scale and you can't feel the weight by your hand, is it possible to use the scale twice to tell if the labels are correct?
Edit
The scale can only tell you if two side are equal, or that which side is heavier. The answer by @Evan means that we are allow to measure the weight of the set of cubes. Sorry for the confusion I've made.

REPLY [3 votes]: Compare (6) and (1,2,3). If scales are not equal we know some labels are wrong. If they are equal, we know 6 is 6, set {1,2,3} really contains cubes with weights {1,2,3} (may be wrong labelled), set {4,5} really contains cubes weighted 4 and 5.
Compare (6, 1) and (5, 3). This time we expect right side to be heavier. !!!

Left side weights at least 7 (because we know real 6 in on the left). And the right side can weight 8 only if it has heaviest from sets {1,2,3} and {4, 5}, that is 5 and 3.


text:
\begin{document}
\maketitle

\begin{abstract}
For a packet erasure broadcast channel with three receivers, we propose a new coding algorithm that makes use of feedback to dynamically adapt the code. Our algorithm is throughput optimal, and we conjecture that it also achieves an asymptotically optimal average decoding delay at the receivers. We consider heavy traffic asymptotics, where the load factor $\rho$ approaches 1 from below with either the arrival rate ($\lambda$) or the channel parameter ($\mu$) being fixed at a number less than 1. We verify through simulations that our algorithm achieves an asymptotically optimal decoding delay of $O\left(\frac1{1-\rho}\right)$.
\end{abstract}

\section{Introduction}
Reliable communication over packet erasure channels is a well studied problem. Several solutions have been proposed, each with its own requirements, merits and issues. In this work, we consider a three-receiver packet erasure broadcast channel with feedback and address questions of throughput and decoding delay at the receivers.

To communicate over a packet erasure broadcast channel, one can use the random linear network coding solution of \cite{desmondallerton}, where the sender transmits a random linear combination of all packets that have arrived so far. Digital fountain codes (\cite{ltcodes,raptor}) form another approach to this problem. These solutions use coding to ensure that with high probability, the transmitted packet will have what we call the \emph{\bf innovation guarantee property}, \ie, it will be \emph{innovative}\footnote{An innovative packet is a coded packet whose coefficient vector is outside the span of previously received packets' coefficient vectors.} to every receiver that receives it, except if the receiver already knows as much as the sender. Thus, every successful reception brings a unit of new information. Such schemes achieve 100\% throughput.

However, both fountain codes and random linear network coding perform block-based encoding. In general, the receiver may not be able to extract the original packets from the coded packets till the entire block has been received. This leads to a decoding delay, which is a problem for real-time packet streaming applications such as video. Ideally we want a code that would allow playback even before the full block is received. In other words, we are interested in minimizing the average per-packet delay. Related questions have been studied by \cite{eminthesis}, \cite{sujay}, \cite{rtoblivious} and \cite{keller}.

With full feedback, the optimal scheme over a point-to-point packet erasure channel is Automatic Repeat reQuest (ARQ) -- the sender simply retransmits a packet upon erasure. This scheme also has the advantage of being a sliding window approach as opposed to a block-based approach. Although it achieves 100\% throughput and optimal packet delay, ARQ does not extend to broadcast-mode links. On the other hand, network coding readily extends to broadcast-mode links.

Reference \cite{ARQISIT} proposes a scheme that uses feedback to acknowledge degrees of freedom instead of original packets, thus combining the benefits of network coding and ARQ. This new framework allows the sender to dynamically adapt its code to incorporate receivers' states of knowledge. This fact was used to design a queue management algorithm called the drop-when-seen algorithm that minimizes the sender's queue size, along with a coding module that provides 100\% throughput. Another related reference is \cite{tran}, where the authors combine an acknowledgment scheme with network coding. Here, the main focus is to maximize the throughput. In contrast to these works, our current work focuses on achieiving the best possible decoding delay for all receivers, while maintaining optimal throughput.

By decoding delay of a receiver, we mean the time elapsed between the arrival of a packet into the sender's queue and its getting decoded by the receiver under consideration, averaged over the packets in the long run in a packet streaming scenario. This is different from but related to the notion of delay used in \cite{keller}.

For the special case of a packet erasure broadcast channel with only two receivers, reference \cite{durvy} proposes a feedback-based coding algorithm that not only achieves 100\% throughput, but also guarantees that every successful innovative reception will cause the receiver to decode a new packet. We call this property \emph{instantaneous decodability}. Instantaneous decodability and 100\% throughput are both desirable goals. However, this approach does not extend to the case of more than two receivers. With prior knowledge of the erasure pattern, \cite{keller} gives an offline algorithm that achieves optimal delay and throughput for the case of three receivers. However, in the online case, even with only three receivers, \cite{durvy} shows through an example that it is not possible to simultaneously guarantee instantaneous decodability as well as throughput optimality.

In the light of this example, our current work aims for a relaxed version of instantaneous decodability while still retaining the requirement of optimal throughput. Our relaxation of the condition is as follows. Let $\lambda$ and $\mu$ be the arrival rate and the channel quality parameter respectively. Let $\rho\triangleq \lambda/\mu$ be the load factor. We consider asymptotics when the load factor on the system tends to 1 (\ie, 100\%), while keeping either $\lambda$ or $\mu$ fixed at a number less than 1. The optimal throughput requirement means that the queue of undelivered packets is stable for all values of $\rho$ less than 1. Our new requirement on decoding delay is that the growth of the average decoding delay as $\rho\rightarrow 1$ should be the same as for the single receiver case. The expected per-packet delay of a receiver in a system with more than one receiver is clearly lower bounded by the corresponding quantity for a single-receiver system. Thus, instead of optimal decoding delay, we aim to guarantee asymptotically optimal decoding delay. The motivation is that in most practical systems, delay becomes a critical issue only when the system starts approaching its full capacity. When the load on the system is well within its capacity, the delay is usually small and hence not an issue. For the case of two receivers, it can be shown that this relaxed requirement is satisfied by the scheme in \cite{durvy} due to the instantaneous decodability property, \ie, the scheme achieves the asymptotically optimal average decoding delay per packet for the two-receiver case.

In our current work, we provide a new coding module for the case of three receivers that achieves optimal throughput. We conjecture that at the same time it also achieves an asymptotically optimal decoding delay in the following sense. With a single receiver, the optimal scheme is ARQ with no coding and we show that this achieves an expected per-packet delay at the sender of $\Theta\left(\frac{1}{1-\rho}\right)$. For the three-receiver system, we conjecture that our scheme also achieves a delay of $O\left(\frac{1}{1-\rho}\right)$, and thus meets the lower bound in an asymptotic sense. We have verified this behavior through simulations for values of $\rho$ that are very close to 1. Our scheme thus achieves feedback-based control of the decoding delay, along the lines suggested in \cite{desmondfeedback}. We believe that our approach can be extended to an arbitrary number of receivers as well.

\section{Preliminaries}

\subsection{The setup}
The setup is the same as in \cite{ARQISIT}. Time is slotted. Packets arrive into the sender's queue according to a Bernoulli process of rate $\lambda$. The sender wants to broadcast this stream to three receivers over a packet erasure broadcast channel. The sender has one queue with no preset size constraints. The channel accepts one packet per slot. Each receiver either receives this packet with no errors (with probability $\mu$) or an erasure occurs (with probability $(1-\mu)$). Erasures occur independently across receivers and across slots and can be detected by receivers. There is perfect feedback in each slot. Figure \ref{earlyarrival} shows the timing of events in a slot. In particular, we assume that the sender finds out whether the receivers received the previous slot's transmission before selecting the current slot's transmission.

\begin{figure}
	\centering
		\includegraphics[width=0.45\textwidth]{EarlyArrival.eps}
		\caption{Relative timing of arrival, service and departure points within a slot}\label{earlyarrival}
\end{figure}

\subsection{The lower bound}\label{lowerbound}
The expected per-packet delay for the single receiver case is clearly a lower bound for the corresponding quantity at one of the receivers in a three-receiver system. We will compute this lower bound in this section. Figure \ref{markovchain} shows the Markov chain for the queue size. If $\rho:=\frac\lambda\mu<1$, then the chain is positive recurrent and its steady state distribution can be found (\cite{hunterbook}). Based on this, the steady state expected queue size can be computed to be $\frac{\rho(1-\mu)}{(1-\rho)}=\Theta\left(\frac1{1-\rho}\right)$. Now, if $\rho<1$, then the system is stable and Little's law can be applied to show that the expected per-packet delay in the single receiver system is also $\Theta\left(\frac1{1-\rho}\right)$.

\begin{figure}
	\centering
		\includegraphics[width=0.45\textwidth]{MarkovChain.eps}
		\caption{Markov chain for the sender's queue size -- single receiver case. Here $\bar{\lambda}:=(1-\lambda)$ and $\bar{\mu}:=(1-\mu)$. }\label{markovchain}
\end{figure}

\subsection{Representing knowledge}
We treat packets as vectors over some finite field. Throughout this paper, we consider a single source that generates a stream of packets. The $k^{th}$ packet that the source generates is said to have an \emph{index} $k$ and is denoted as $\mathbf{p_k}$. We assume that the sender uses only linear codes, \ie, the transmission is some linear combination of packets. The linear combination is uniquely specified using the vector of coefficients used to form it.
With linear codes, the state of knowledge of a node after receiving some set of linear combinations has a vector space structure. This is because the node can compute any linear combination whose coefficient vector is within the linear span of the coefficient vectors of previously received linear combinations. This leads to the following definition of \emph{knowledge space} which we restate from \cite{ARQISIT}.

\begin{definition}[Knowledge of a node]
	The \emph{knowledge of a node} is the set of all linear combinations of original packets that it can compute, based on the information it has received so far. The coefficient vectors of these linear combinations form a vector space called the \emph{knowledge space} of the node. The dimension of this vector space is called the \emph{rank} of the node.
\end{definition}

We next restate the definition of a node ``seeing'' a message packet from \cite{ARQISIT}. A node is said to have \emph{seen} a message packet $\mathbf{p}$ if it has received enough information to compute a linear combination of the form $(\mathbf{p} + \mathbf{q})$, where $\mathbf{q}$ is itself a linear combination involving only packets with an index greater than that of $\mathbf{p}$. (Decoding implies seeing, as we can pick $\mathbf{q}=\mathbf{0}$.) The number of packets seen by a node is precisely the rank of the node (see \cite{ARQISIT} for more details).

We introduce a new notion of packets that a node has ``heard of''. A node is said to have heard of a packet if it knows some linear combination involving that packet.

\section{The new coding module}
Our coding module works in the Galois field of size 3. At the beginning of every slot, the module has to decide what linear combination to transmit. Since there is full feedback, the module is fully aware of the current state of knowledge of each of the three receivers. Thus, it can compute the rank of each receiver. We denote the highest rank among the three receivers as $m$. Our coding module maintains an invariant that the transmission will never involve a packet whose index is greater than $(m+1)$. We denote the receiver(s) whose rank is $m$ as the leader(s). We consider three cases:
\subsection{All three receivers are leaders}
In this case, all three receivers have a rank of $m$ which means each has seen $m$ packets. If $\mathbf{p_{m+1}}$ has not arrived yet, the module does nothing. Otherwise, since all transmissions have involved only the set of packets up to $\mathbf{p_{m+1}}$, there is exactly one unseen packet for each receiver within this set. This could be a different packet for each of the three receivers. The coding module selects a linear combination that if received successfully by any receiver, will reveal to that receiver its unseen packet, thereby guaranteeing innovation. This is done by simply forming a linear combination involving only the unseen packets of the three receivers. It can be verified that with a field of size 3, it is always possible to choose coefficients such that innovation is guaranteed for all three receivers.

\subsection{There are two leaders}
It can be shown by induction that at all times, at least one leader would have decoded all packets from 1 to $m$. Now, when there are two leaders, if exactly one leader has decoded all packets 1 to $m$, then the coding module performs the operations of case 3, treating this leader as the unique leader. If both leaders have decoded packets 1 to $m$, then module does the following.

If $\mathbf{p_{m+1}}$ has not arrived yet, the module transmits the oldest undecoded packet of the non-leader (if there are packets that the non-leader has heard of but not yet decoded, then they are preferred). Suppose $\mathbf{p_{m+1}}$ has arrived. Now, if it has already been decoded by the non-leader, then the module sends the sum of $\mathbf{p_{m+1}}$ and the oldest undecoded packet of the non-leader (again, if there are packets that the non-leader has heard of but not yet decoded, then they are preferred). Otherwise, the module sends $\mathbf{p_{m+1}}$ by itself.

\subsection{Unique leader}
In this case, the module computes the following sets for the two non-leaders:

$H_1$:= Set of packets heard of by first non-leader

$H_2$:= Set of packets heard of by second non-leader

$D_1$:= Set of packets decoded by first non-leader

$D_2$:= Set of packets decoded by second non-leader

Note that $D_1\subseteq H_1$ and $D_2\subseteq H_2$. We also define a universe set $U$ consisting of packets $\mathbf{p_1}$ to $\mathbf{p_m}$, and also $\mathbf{p_{m+1}}$ if it has arrived. In this setting, the following sets partition the universe (refer to Figure \ref{venn}):
\begin{itemize}
\item $S_1=D_1\cap D_2$
\item $S_2=D_1\cap(H_2\backslash D_2)$
\item $S_3=D_2\cap(H_1\backslash D_1)$
\item $S_4=(H_1\backslash D_1)\cap (H_2\backslash D_2)$
\item $S_5=D_1\backslash H_2$
\item $S_6=D_2\backslash H_1$
\item $S_7=(H_1\backslash D_1)\backslash H_2$
\item $S_8=(H_2\backslash D_2)\backslash H_1$
\item $S_9=U\backslash(H_1\cup H_2)$
\end{itemize}

\begin{figure}
	\centering
		\includegraphics[width=0.45\textwidth]{venndiag.eps}
		\caption{Sets used by the coding module}\label{venn}
\end{figure}

The coding module picks a linear combination depending on which of these sets $\mathbf{p_{m+1}}$ falls in, as follows:

{\it Case 1 -- $\mathbf{p_{m+1}}$ has not arrived:}
Check if $S_4$ is non-empty. If it is, then send the oldest packet in $S_4$. Otherwise, check if both $S_2$ and $S_3$ are non-empty. If they are, pick the oldest packet from each, and send their sum. If not, try the following pairs of sets: $S_3$ and $S_5$, else $S_2$ and $S_6$, else $S_5$ and $S_6$. If none of these pairs of sets work, then send the oldest packet in $S_7$ if it is non-empty. If not, try $S_8$, $S_9$, $S_2$, $S_3$, $S_5$ and $S_6$ in that order. If all of these are empty, then send nothing.

{\it Case 2 -- $\mathbf{p_{m+1}}\in S_1$:} This is identical to case 1, except that $\mathbf{p_{m+1}}$ must also be added to the linear combination that case 1 suggests.

{\it Case 3 -- $\mathbf{p_{m+1}}\in S_2$:} Send $\mathbf{p_{m+1}}$ added to another packet. The other packet is chosen to be the oldest packet in the first non-empty set in the following sets, tested in this particular order: $S_3$, $S_4$, $S_6$, $S_8$, $S_7$ and then $S_9$.

{\it Case 4 -- $\mathbf{p_{m+1}}\in S_3$:} This is similar to the $S_2$ case (using symmetry) -- test $S_2$, $S_4$, $S_5$, $S_7$, $S_8$ and then $S_9$.

{\it Case 5 -- $\mathbf{p_{m+1}}\in S_4$:} Send $\mathbf{p_{m+1}}$ as it is.

{\it Case 6 -- $\mathbf{p_{m+1}}\in S_5$:} Send $\mathbf{p_{m+1}}$ added to another packet. The other packet is chosen to be the oldest packet in the first non-empty set in the following sets, tested in the following order: $S_3$, $S_6$, $S_4$, $S_8$, $S_7$ and then $S_9$.

{\it Case 7 -- $\mathbf{p_{m+1}}\in S_6$:} This is similar to the $S_5$ case (using symmetry) -- test $S_2$, $S_5$, $S_4$, $S_7$, $S_8$ and then $S_9$.

{\it Case 8 -- $\mathbf{p_{m+1}}\in S_7$:} Send $\mathbf{p_{m+1}}$ as it is.

{\it Case 9 -- $\mathbf{p_{m+1}}\in S_8$:} Send $\mathbf{p_{m+1}}$ as it is.

{\it Case 10 -- $\mathbf{p_{m+1}}\in S_9$:} Send $\mathbf{p_{m+1}}$ as it is.

In all these cases, the coefficient for the chosen packets must be selected to be either 1 or 2, in such a way that the resulting linear combination is innovative to any receiver that receives it, except if the receiver already knows all that the sender knows. It can be shown that such a choice is always possible with a field of size 3.

\begin{remark}
\emph{We conjecture based on the simulations that the algorithm maintains the following invariant -- at most one of $H_1\backslash D_1$ and $H_2\backslash D_2$ is non-empty at any given time. If proved to be true, this observation can be used to simplify the algorithm.}
\end{remark}

\section{The intuition behind the algorithm}
The main idea behind the algorithm is to first of all guarantee innovation. It can be shown that the linear combination computed by this coding module is indeed innovative to any receiver that receives it. In addition to this requirement however, the module also tries to cause each receiver that has a successful reception to decode as many packets as possible.

An interesting property of this algorithm is that the transmitted linear combination always has at most two undecoded packets involved in it from any receiver's point of view. In other words, every transmission is essentially either an uncoded packet or the sum of two packets. This property leads to a nice structure in the knowledge space of the receivers, using which, we present a strategy to control the extent to which packets get mixed with each other, thereby controlling the decoding delay.

In order to explain this structure, we define the following relation. The ground set $G$ of the relation contains all packets that have arrived at the sender so far, along with a fictitious all-zero packet that is known to all receivers even before transmission begins. The relation is defined with respect to a specific receiver. Two packets $\mathbf{p_x}\in G$ and $\mathbf{p_y}\in G$ are defined to be related to each other if the receiver knows at least one of $\mathbf{p_x}+\mathbf{p_y}$ and $\mathbf{p_x}+2\mathbf{p_y}$.

Now, a packet added with two times the same packet gives $\mathbf{0}$ which is trivially known to the receiver. Hence, the relation is reflexive. It is also symmetric since addition is a commutative operation. Now, for any $\mathbf{p_x}, \mathbf{p_y}, \mathbf{p_z}$ in $G$, if a receiver knows $\mathbf{p_x}+ \alpha\mathbf{p_y}$ and $\mathbf{p_y}+ \beta\mathbf{p_z}$, then it can compute either  $\mathbf{p_x}+\mathbf{p_z}$ or $\mathbf{p_x}+2\mathbf{p_z}$ by canceling out the $\mathbf{p_y}$, for $\alpha=1$ or 2 and $\beta=1$ or 2. Therefore the relation is also transitive and is thus an equivalence relation. It defines a partition on the ground set, namely the equivalence classes, which provide a structured way to represent the knowledge of the node. It can be seen that the class containing the all-zero packet is precisely the set of decoded packets ($D_1$ or $D_2$). Packets that have not been involved in any of the successfully received linear combinations so far will form singleton equivalence classes. These correspond to the packets that the receiver has not heard of $(U\backslash H_1$ or $U\backslash H_2$).

We say a class is nontrivial if it is neither a singleton class nor the class of decoded packets. Thus, nontrivial classes contain the packets that have been heard of but not decoded. Revealing any packet in a class will reveal the entire class to the node. The number of nontrivial classes is thus the number of packets that the node needs to know in order to decode all packets it has heard of. This number is thus a measure of how far away a node is from decoding all packets it has heard of. We call this number the \emph{deficit of the node}.

For instance, revealing a packet from $H_1\backslash D_1$ will allow the entire class containing that packet to be decoded by receiver 1. The algorithm ensures that a packet from $H_1\backslash D_1$ or $H_2\backslash D_2$ is revealed whenever possible, as opposed to a packet that the receiver has not heard of. This ensures that the deficit is reduced whenever possible. As a result, the deficit drops to zero frequently, thereby causing the node to decode packets.

\section{Performance of the algorithm}
\subsection{Throughput optimality}
The algorithm has been designed in such a way that innovation is guaranteed to all the receivers whenever possible. Packet $\mathbf{p_{m+1}}$ is always included in the linear combination if it has arrived, in order to guarantee innovation to the leader. If both the other receivers have also not decoded it, then sending $\mathbf{p_{m+1}}$ by itself satisfies the innovation guarantee. This happens in cases 5, 8, 9 and 10.

If however, some receiver has already decoded it as in the other cases, then another packet is included in the linear combination that the receiver has not yet decoded, thereby ensuring innovation. While choosing such a packet, preference is given to packets that the receiver has heard of, as revealing such a packet will cause several packets to be decoded at once.

If $\mathbf{p_{m+1}}$ has not yet arrived, then the leader is already satisfied. For the other two receivers, the transmission is selected in such a way that it simultaneously reveals an undecoded packet to both of them whenever possible.

We can show that in all these cases, over a field of size 3, the coefficients can also be chosen carefully to guarantee innovation for all those who receive the linear combination successfully. This discussion is summarized in the following theorem.
\begin{theorem}\label{innovation_for_modified}
\it The coding module satisfies the innovation guarantee property.
\end{theorem}
This means that the algorithm achieves optimal throughput, \ie, for all $\rho<1$, the decoding delay and the queue at the sender will remain stable.

\subsection{Decoding delay}
\begin{figure}
  \begin{center}
	\includegraphics[width=0.5\textwidth]{normal.eps}
  \end{center}
  \caption{Decoding delay as load approaches capacity}
  \label{normal}
\end{figure}

We now study the delay experienced by an arbitrary arrival before it gets decoded by one of the receivers, say receiver 1. We consider a system where $\mu$ is fixed at 0.5. The value of $\rho$ is varied as follows: 0.95, 0.97, 0.98, 0.99 and 0.995. We plot the expected decoding delay per packet averaged across the three receivers, as a function of $\left(\frac{1}{1-\rho}\right)$ in Figure \ref{normal}. We also plot the log of the same quantities in Figure \ref{loglog}. The value of the delay is averaged over $500000$ time slots for the first three points and $10^6$ time slots for the last two points.

Figure \ref{normal} shows that the growth of the expected decoding delay is linear in $\left(\frac{1}{1-\rho}\right)$ as $\rho$ approaches 1. Figure \ref{loglog} confirms this behavior -- we can see that the slope on the plot of the logarithm of these quantities is indeed close to 1. This observation leads to the following conjecture:

\begin{figure}[t!]
  \begin{center}
	\includegraphics[width=0.5\textwidth]{loglog.eps}
  \end{center}
  \caption{Decoding delay -- log-log plot}
  \label{loglog}
\end{figure}

\begin{conjecture}\label{conj1}
For the newly proposed coding module, the expected decoding delay per packet from one particular receiver's point of view grows as $O\left(\frac{1}{1-\rho}\right)$, which is asymptotically optimal.
\end{conjecture}

\subsection{Queue size}
The queue update rule is as follows -- the sender drops a packet if all the receivers have decoded it. This means that by Little's law, the expected queue size will be proportional to the time a packet spends in the system before it is decoded. Thus, if the expected decoding delay is indeed $O\left(\frac1{1-\rho}\right)$ as conjectured above, then the drop-when-decoded queue update rule will ensure that the expected queue size at the sender is $O\left(\frac1{1-\rho}\right)$, which is asymptotically optimal.

\section{Conclusions}\label{conc}
For a three receiver packet erasure broadcast channel with feedback, we have proposed a new coding scheme that makes use of the feedback to dynamically adapt the code. As argued earlier, $\Theta\left(\frac1{1-\rho}\right)$ is an asymptotic lower bound on the decoding delay. We have observed through simulations that this lower bound seems to be achieved by our scheme, which would imply the asymptotic optimality of our coding module in terms of decoding delay. We conjecture that this is indeed true. All these delay benefits are obtained without compromising on throughput. If the conjecture is true, then the expected queue size of undecoded packets is also $O\left(\frac1{1-\rho}\right)$, which is asymptotically optimal. Thus, our scheme also simplifies the queue management at the sender. In the future, we wish to extend this approach to an arbitrary number of receivers. Also, we wish to make the algorithm robust to delays and erasures in the feedback.
\bibliographystyle{IEEEtran}
\bibliography{References}
\end{document}

config: arxiv