\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 146, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2006/146\hfil On a Brezis-Nirenberg Type Problem]
{On a Brezis-Nirenberg type problem}
\author[Florin Catrina\hfil EJDE-2006/146\hfilneg]
{Florin Catrina}
\address{Department of Mathematics and Computer Science\\
St. John's University\\
Queens, New York 11439, USA}
\email{catrinaf@stjohns.edu}
\thanks{Submitted February 17, 2006. Published November 26, 2006.}
\subjclass[2000]{35J25, 35J70, 35J60}
\keywords{Positive solution; ground state; symmetry breaking}
\begin{abstract}
In this note we discuss the existence and symmetry breaking of
least energy solutions for certain weighted elliptic equations
in the unit ball in $\mathbb{R}^N$, with zero Dirichlet boundary conditions.
We prove a multiplicity result, which answers one of the questions
we left open in \cite{cala} regarding a Brezis-Nirenberg type problem.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{question}[theorem]{Question}
\section{Introduction}\label{S:intr}
Over the last four decades, a large amount of work has been done
on existence and qualitative properties of solutions for
semi-linear elliptic problems. A significant proportion of these
studies deal with positive solutions for problems that lack
compactness. The loss of compactness may be due to the existence
of limiting problems which are invariant under translations or (in
the case of critical nonlinearities) under dilations. The presence
of zeros or singularities in the coefficients, in many instances
plays a role in the form of the limiting problem. As a typical
sample of existence results on the whole of $\mathbb{R}^N$ one may
consult \cite{bieg,terr,smet}, and the references therein.
On bounded domains in $\mathbb{R}^N$ for $N\geq 3$, it was shown
by Pohozaev as early as 1965 (see \cite{poho}) that nonlinear
eigenvalue problems of the form
\begin{gather*}
-\Delta u = u^{p-1} \quad\mbox{in } \Omega, \\
u = 0 \quad \mbox{on } \partial \Omega
\end{gather*}
have no positive solution if $\Omega$ is
star-shaped and $p \geq 2^*$ with $2^*=\frac{2N}{N-2}$ the
critical Sobolev exponent.
An intriguing phenomenon was observed by Brezis and Nirenberg in \cite{brni}
relative to the problem
\begin{equation}\label{bn}
\begin{gathered}
-\Delta u = u^{2^*-1}+ \lambda u, \\
\quad u > 0 \quad \mbox{in } B,\\
u = 0 \quad \mbox{on } \partial B.
\end{gathered}
\end{equation}
where $B $ is the unit ball in $\mathbb{R}^N$ with $N \geq 3$.
The authors have proved the following theorem.
\begin{theorem} \label{bnt}
Let $\lambda_1= \lambda_1(N)$ be the first eigenvalue of $-\Delta$
with zero Dirichlet boundary condition on $B$.
Then problem (\ref{bn}) has solution if and only if
\begin{itemize}
\item[(a)] $N\geq 4$ and $\lambda \in (0, \lambda_1)$; or
\item[(b)] $N=3$ and $\lambda \in (\lambda_1/4, \lambda_1)$.
\end{itemize}
\end{theorem}
This started a flurry of work on problems in which the same phenomenon
was observed. That is, for some dimensions
$N$ the branch of solutions which bifurcates from the trivial solution
exists for all $\lambda$ between $\lambda_1$ and zero, while for
other ``critical'' dimensions this branch is bounded
away from $\lambda= 0$ (see
\cite{brdt,brpe,chge,clfm,egne1,egne2,fequ, ghyu,jaco,pucs},
and the references therein).
The present work is motivated by one of the questions we raised in \cite{cala}.
Let $B$ be the unit ball in $\mathbb{R}^N$.
Consider the problem
\begin{equation}\label{pr}
\begin{gathered}
-\mathop{\rm div}(|x|^{-2a} \nabla u) = |x|^{-bp}u^{p-1}+
\lambda |x|^{-2(a+1)+c}u, \\
u > 0 \quad \mbox{in } B, \quad u \in \mathcal{D}_{a}(B).
\end{gathered}
\end{equation}
Throughout we shall consider
\begin{equation}\label{con}
\begin{gathered}
2\leq N, \quad a < \frac{N-2}{2}, \quad a < b< a+1,\\
p = \frac{2N}{N-2(a+1-b)}, \quad 0 \frac{N-1}{p-2}$.
Then for any
\[
0 < \lambda <\Big(\frac{c}{2} Z\Big(\frac{2}{c}\sqrt{\gamma^2
- \frac{N-1}{p-2}}\Big)\Big)^2
\]
the least energy solution for the problem \eqref{pr} exists and it
is nonradial.
\end{theorem}
As the existence of radial solutions has been discussed in \cite{cala}
(see also Theorem~\ref{nicl} in the concluding remarks),
we obtain the following corollary.
\begin{corollary} \label{coro1}
In the range of parameters stated in Theorem~\ref{main},
the problem \eqref{pr} has a (higher energy) radial and a
(ground state) nonradial solution.
\end{corollary}
In the case $N \geq 3$, $a = b= 0$ and $c=2$, one has
$p = \frac{2N}{N-2}=2^*$, the critical exponent for the Sobolev
imbedding $\mathcal{D}_{0}(\mathbb{R}^N) \subset L^p(\mathbb{R}^N)$, and the problem under
study \eqref{pr} becomes exactly (\ref{bn}).
Due to a well known result of Gidas, Ni and Nirenberg (see \cite{ginn})
every solution of (\ref{bn}) is radial. Therefore in this case,
the study of the ODE obtained by symmetry reduction answers our questions
completely in form of Theorem~\ref{bnt}. Chou and Geng in \cite{chge},
extend the moving plane method of \cite{ginn}, and they obtain that
for $3 \leq N$, $0 \leq a < \frac{N-2}{2}$, every solution of \eqref{pr}
is radial. Hence, again the ODE results settle the questions.
We remark that in the range $a<0$, the method of moving planes breaks down.
It is in this range where Theorem~\ref{main} applies.
Indeed, since $2^*> p > 2$ one has from $\gamma^2 > \frac{N-1}{p-2}$ that
$\gamma^2 > \frac{N-1}{2^*-2}$. This leads to
$\sqrt{N-2}(\sqrt{N-2}-\sqrt{N-1})>2a$ which forces $a$ to be negative.
The fact that the problem may admit both radial and non-radial
solutions for certain values of the parameters is somewhat
expected in view of the symmetry breaking phenomenon of the least
energy solutions observed in \cite{cawa1}. Combined with the
existence result for radial solutions in Theorem~\ref{nicl}, the
symmetry breaking of the ground state for problem \eqref{pr}
guarantees the existence of at least two solutions. Hence our
theorem in the present article should be viewed both, as a
symmetry breaking result and a multiplicity theorem.
The main novelty in Theorem~\ref{main} is to give an explicit range
of parameters in which the ground state (least energy solution) is nonradial.
The existence of the ground state is based on Lemma 1 in \cite{brez}.
In order to apply this Lemma we need a precise estimate on the decay at
infinity of solutions of a related problem in $\mathbb{R}^N$
which we establish in Lemma~\ref{est}. For this we employ Harnack
inequality as it is presented in the
Section~{11} in the Lecture Notes \cite{lipe}. For the symmetry breaking part,
we employ the technique of \cite{smws} which originated in the work \cite{kawo}.
The paper is organized as follows.
In Section~\ref{S:prel} we introduce an equivalent problem, which we
find more convenient to work with, and we gather some preliminary results.
Section~\ref{S:exist} contains the proof of Theorem~\ref{main}.
The paper ends with a short section of concluding remarks.
\section{Preliminaries} \label{S:prel}
For $u \in \mathcal{D}_{a}(B)\setminus \{ 0\}$, consider the energy
\[
E(u)= E_{a,b,c,\lambda}(u)
= \frac{\displaystyle \int_{B} |x|^{-2a}|\nabla u|^2 -\lambda |x|^{-2(a+1)+c}u^2
\ dx}{ \displaystyle \Big(\int_{B} |x|^{-bp}|u|^{p} \ dx \Big)^{2/p}},
\]
and denote
\[
J=J(a,b,c,\lambda) = \inf_{u\in \mathcal{D}_{a}(B)\setminus \{ 0\}} E(u).
\]
Note that solutions of \eqref{pr} are critical points of $E$. Conversely,
if the infimum $J$ of $E$ is positive and it is achieved by
some function $u$ in $\mathcal{D}_{a}(B)\setminus \{ 0\}$, then after an
eventual multiplication by a constant, $u$ is solution of \eqref{pr}.
As in \cite{cawa1}, consider the cylinder
$\mathcal{C}= \mathbb{R}\times \mathbb{S}^{N-1} \subset \mathbb{R}^{N+1}$.
Define the conformal diffeomorphism
\[
\vartheta : \mathbb{R}^N \setminus\{0\} \to \mathcal{C}\quad \mbox{given by }
\vartheta (x)= \Big(-\ln |x|, \frac{x}{|x|}\Big)= (t, \theta)
\]
Note that $\vartheta$ takes $B\setminus \{0\}$ diffeomorphically into the
half cylinder
$\Omega = (0,\infty)\times \mathbb{S}^{N-1}$.
The transformation
\[
\Upsilon_a : H_a(\Omega) \to \mathcal{D}_{a}(B) \quad \Upsilon_a v = u \quad
\mbox{with } u(x)=|x|^{-\gamma}v( \vartheta(x))
\]
is a Hilbert space isomorphism.
Here $H_a(\Omega)$ is obtained by completion of smooth functions with
compact support in $\Omega$ under the norm
\[
\|v\|^2 = \int_{\Omega} |\nabla v|^2 +\gamma^2v^2 \,d\mu
\]
We remark that as a function space, $H_a(\Omega)$ is independent of $a$.
We keep the index $a$ however, to indicate which inner product is used.
If $u$ is solution of \eqref{pr} then $v= \Upsilon_a^{-1}u$ satisfies
the equation
\begin{equation}\label{cyl}
\begin{gathered}
-\Delta v + \gamma^2v = v^{p-1}+ \lambda e^{-ct}v, \\
v > 0 \quad \mbox{in } \Omega, \quad v \in H_a(\Omega)
\end{gathered}
\end{equation}
One can check that for any $v \in H_a(\Omega)$ and
$\Upsilon(v) = u \in \mathcal{D}_a(B)$, we have
\[
E(u)= F(v):=\frac{\displaystyle \int_{\Omega} |\nabla v|^2
+(\gamma^2-\lambda e^{-ct})v^2 \,d\mu}{\displaystyle
\Big( \int_{\Omega} |v|^{p} \,d\mu \Big)^{2/p}};
\]
therefore
\[
J = \inf_{u\in \mathcal{D}_a(B)\setminus \{ 0\}} E(u)
= \inf_{v\in H_a(\Omega)\setminus \{ 0\}} F(v).
\]
It is known (see \cite{cawa1}) that on the whole cylinder $\mathcal{C}$, the
functional
\[
G(u)=\frac{ \int_{\mathcal{C}} |\nabla u|^2 +\gamma^2 u^2 \,d\mu}
{\big( \int_{\mathcal{C}}|u|^{p} \,d\mu \big)^{2/p}}
\]
has a positive infimum $S$ achieved by a positive function
$U\in H^1(\mathcal{C})$ which satisfies
\begin{equation}
\label{wcy}
-\Delta U + \gamma^2U = U^{p-1},
\end{equation}
and so,
\[
S= S(N,a,b)= \inf_{u\in H^1(\mathcal{C})}G(u)=G(U)
= \Big(\int_{\mathcal{C}} U^p \ d\mu \Big)^{(p-2)/p}
\]
In \cite{cawa1} we also discussed the symmetry of $U$.
While for some values of the parameters, $S$ is achieved by the radial solutions
\[
U(t)=\Big(\frac{\gamma^2p}{2}\Big)^\frac{1}{p-2}
\Big(\cosh\big(\frac{p-2}{2}\gamma t\big) \Big)^{-\frac{2}{p-2}},
\]
for other values we proved that the ground state is not radial
anymore (see \cite{fesc} for an improvement of the range obtained
in \cite{cawa1}). We have however that eventually after a
translation we can assume that $U(t,\theta)=U(-t, \theta)$ for any
$t \in \mathbb{R}$ and $\theta \in \mathbb{S}^{N-1}$.
We shall prove Theorem~\ref{main} by showing the corresponding result
for the equivalent problem \eqref{cyl}. That is, we prove the following
result.
\begin{theorem} \label{macy}
Assume $0 < b-a < 1$, $0 < c < 2\gamma$, and $\gamma^2 > \frac{N-1}{p-2}$.
Then for any
\[0 < \lambda <\Big(\frac{c}{2} Z\Big(\frac{2}{c}\sqrt{\gamma^2
- \frac{N-1}{p-2}}\Big)\Big)^2
\]
problem \eqref{cyl} has both, a radial and a nonradial solution.
\end{theorem}
\section{Existence and Multiplicity} \label{S:exist}
We employ Lemma~1 in \cite{brez} to show that $J = \inf F$ is achieved.
Then we prove the symmetry breaking part, from which Theorem~\ref{macy} follows.
Lemma~1 in \cite{brez} translates to our situation step by step to give
the following lemma.
\begin{lemma} \label{br}
If $J~~0$ such that
$\frac{1}{C}e^{-\gamma |t|} \leq U(t,\theta)
\leq C e^{-\gamma |t|}$ for all $t\in \mathbb{R}$ and
$\theta \in \mathbb{S}^{N-1}$.
\end{lemma}
\begin{proof}
Let $U$ be a positive solution of \eqref{wcy}, which without loss of
generality we can assume even in $t$,
i.e. $U(-t, \theta)= U(t, \theta)$. For any $t\in (-\infty, \infty)$, let
\[
f(t) = \int_{\mathbb{S}^{N-1}}U(t,\theta)\ d\theta, \quad\mbox{and}\quad
h(t) = \int_{\mathbb{S}^{N-1}} U^{p-1}(t,\theta)\ d\theta.
\]
Making the necessary modifications to Moser's proof of Harnack
inequality \cite{mose}, one can prove that there is a positive
constant $C_0$ (depending on $U$) such that
\begin{equation} \label{stest}
\frac{1}{C_0} < \frac{U(t,\theta)}{f(t)} 0$ such that $\frac{1}{C_1}e^{-\gamma t} \leq f(t)$
for all $t \geq 0$.
We now prove that
\begin{equation}
\label{in0}
C_2=\int_{-\infty}^\infty e^{2\gamma r}\int_r^\infty e^{-\gamma s}h(s)
\,ds \,dr < \infty;
\end{equation}
therefore
\[
e^{\gamma t} f(t) \leq C_2, \quad \mbox{hence} \quad
f(t) \leq C_2e^{-\gamma t}.
\]
Multiply \eqref{wcy} by $U^{p-2}(t,\theta)$ to get
\[
- \frac{\Delta U^{p-1}}{p-1} +(p-2) U^{p-3}|\nabla U|^2
+ \gamma^2 U^{p-1} = U^{2p-3}.
\]
Hence
\begin{equation}
\label{ine}
-\Delta U^{p-1} + (p-1)(\gamma^2 - U^{p-2})U^{p-1} \leq 0.
\end{equation}
From (\ref{stest}) we have that for any $\varepsilon > 0$, there exists
$t_0$ sufficiently large, such that if
$t\geq t_0$ then $U^{p-2}(t,\theta) < \varepsilon$.
Let
\[
0 < \varepsilon < \gamma^2 \frac{p-2}{p-1} \quad
\mbox{and so} \quad \gamma < \alpha = \sqrt{(p-1)(\gamma^2 - \varepsilon)}.
\]
Integrate on ${\mathbb{S}^{N-1}}$ in (\ref{ine}) to obtain, for $t \geq t_0$,
$ -h_{tt}(t)+\alpha^2 h(t) < 0$;
i.e.,
\[
-\frac{d}{dt}\big(e^{-2\alpha t} \frac{d}{dt}(e^{\alpha t} h(t))\big) <0,
\]
hence
$e^{-2\alpha t} \frac{d}{dt}(e^{\alpha t} h(t))$ is increasing for $t \geq t_0$.
If $\frac{d}{dt}(e^{\alpha t} h(t)) = A > 0$ for some
$t = t_1 \geq t_0$, we obtain
\[
\frac{d}{dt}(e^{\alpha t} h(t)) > A e^{2\alpha t}, \quad \mbox{i.e.} \quad
h(t) > \frac{A}{2\alpha} e^{\alpha t} + B e^{-\alpha t},
\]
for all $t> t_1$. But this contradicts the fact that $h$ is bounded.
Therefore, $\frac{d}{dt}(e^{\alpha t} h(t)) \leq 0$, and
so $e^{\alpha t} h(t)$ is non-increasing.
Letting $A = e^{\alpha t_0} h(t_0)$, we obtain $h(t) \leq A e^{-\alpha t}$
for all $t\geq t_0$.
Since $\gamma < \alpha$ we have that
\[
\int_{-\infty}^\infty e^{\gamma s}h(s) \,ds = B < \infty, \quad
\mbox{i.e.} \quad
\int_{-\infty}^\infty e^{-\gamma s}h(s)\int_{-\infty}^s e^{2\gamma r}
\,dr \,ds = \frac{B}{2\gamma}=C_2 < \infty.
\]
Reversing the order of integration, we obtain (\ref{in0}).
The lemma now follows from (\ref{stest}) by taking $C = \max\{C_0C_1, C_0C_2\}$.
\end{proof}
\begin{lemma} \label{est}
For any $0 < b-a< 1$, $0 0$, we have $J< S$.
\end{lemma}
\begin{proof}
We construct test functions $v$ such that $J \leq F(v) < S$.
Let $\varphi \in C_0^\infty (\Omega)$ be a cut-off function, with $0\leq \varphi (t,\theta) \leq 1$,
$\varphi (t,\theta)\equiv 1 $ for $t \geq 1$, and
$|\nabla \varphi (t,\theta)| \leq 2$ for all $(t,\theta) \in \Omega$. For $\tau > 0$, let
$U_\tau(t,\theta) = U(t-\tau,\theta)$ and take $v_\tau = \varphi U_\tau$.
Then
\begin{align*}
F(v_\tau)
& = \frac{\displaystyle
\int_\Omega U_\tau^2 |\nabla \varphi|^2 + \nabla U_\tau \cdot \nabla (U_\tau\varphi^2)
+ \gamma^2 U_\tau^2\varphi^2 -\lambda e^{-ct} U_\tau^2 \varphi^2}
{\displaystyle \Big(\int_\Omega U_\tau^p\varphi^p\Big)^{2/p}}
\\
& = \frac{\displaystyle \int_\Omega U_\tau^2( |\nabla \varphi|^2 -\lambda e^{-ct}
\varphi^2)+ U_\tau\varphi^2(-\Delta U_\tau
+ \gamma^2 U_\tau)}
{\displaystyle \Big( \int_\Omega U_\tau^p\varphi^p\Big)^{2/p}}
\\
& = \frac{\displaystyle \int_\Omega U_\tau^2
( |\nabla \varphi|^2 -\lambda e^{-ct} \varphi^2)+ U_\tau^p\varphi^2
}{\displaystyle\Big( \int_\Omega U_\tau^p\varphi^p\Big)^{2/p}}\,.
\end{align*}
We use the following estimates
\begin{align*}
& \int_\Omega U_\tau^2( |\nabla \varphi|^2 -\lambda e^{-ct} \varphi^2)
+ U_\tau^p\varphi^2 \\
& \leq \|U_\tau \|_p^p - \int_{\mathcal{C} \setminus \Omega} U_\tau^p
+ 4 \int_{[0,1]\times{\mathbb{S}^{N-1}} } U_\tau^2 -
\int_{[1,\infty) \times{\mathbb{S}^{N-1}}} \lambda e^{-ct} U_\tau^2 \\
& \leq \|U_\tau \|_p^p - C_1 \exp(-p\gamma\tau) +C_2 \exp(-2\gamma\tau)
- C_3 \exp(-c\tau)
\end{align*}
and
\[
\int_\Omega U_\tau^p\varphi^p \geq \|U_\tau \|_p^p
- \int_{(-\infty,1]\times {\mathbb{S}^{N-1}}} U_\tau^p \geq
\|U_\tau \|_p^p - C_4 \exp(-p\gamma \tau ),
\]
with $C_1$, $C_2$, $C_3$, and $C_4$ positive constants independent of $\tau$.
Therefore, if $\tau$ is sufficiently large, and since $c < 2\gamma$ we have
\[
J \leq F(v_\tau) < \frac{\|U_\tau \|_p^p}{(\|U_\tau \|_p^p)^{2/p}}= S\,.
\]
\end{proof}
Under conditions (\ref{con}) and $c< 2\gamma$, from Lemma~\ref{br} and
Lemma~\ref{est} we have that for $\lambda > 0$, $J$ is achieved.
If we also assume
$\lambda < \lambda_1(N,a,c)= (\frac{c}{2}Z(\frac{2\gamma}{c}))^2$, then
$J$ is positive and after an eventual multiplication by a constant, the function that achieves $J$ solves \eqref{cyl}.
The following lemma will prove that for certain values of $\gamma$
and $\lambda$, the least energy solutions obtained above are nonradial.
The argument is similar to that of \cite{smws}, or Theorem~3, part c)
in \cite{kawo}.
\begin{lemma} \label{nrd}
If the least energy solution is $V= V(t)$ (independent of $\theta$), then
\[
0 \geq \int_0^\infty V_t^2 + \left(-\frac{ N-1}{p-2}+\gamma^2
- \lambda e^{-ct}\right) V^2
\]
\end{lemma}
\begin{proof}
For some $h\in H_a(\Omega)$, let $f(s) = F(V+sh)$. Since $V$ is a local
minimum of $F$, it follows that $f''(0) \geq 0$.
This implies
\begin{equation} \label{sder}
\begin{aligned}
& 2\Big(\int_\Omega |\nabla h|^2 + (\gamma^2 - \lambda e^{-ct})h^2\Big)
\Big( \int_\Omega V^p\Big)^{2/p} \\
& \geq 2\Big(\int_\Omega |\nabla V|^2 + (\gamma^2 - \lambda e^{-ct})V^2\Big) \\
& \Big[ (p-1)\Big(\int_\Omega V^p\Big)^{\frac{2}{p}-1}
\Big(\int_\Omega V^{p-2}h^2\Big)
- (p-2) \Big(\int_\Omega V^p\Big)^{\frac{2}{p}-2}
\Big(\int_\Omega V^{p-1}h\Big)^2\Big]
\end{aligned}
\end{equation}
By taking $h(t,\theta)= V(t)Y(\theta)$, with $Y(\theta)$ a first harmonic,
from
\[
-\Delta_{\mathbb{S}^{N-1}} Y = (N-1) Y, \quad \mbox{and} \quad
\int_{\mathbb{S}^{N-1}} Y(\theta) \ d\theta= 0
\]
we get that (\ref{sder}) simplifies to
\[
\int_0^\infty V_t^2 + (N-1+\gamma^2 - \lambda e^{-ct})V^2 \geq (p-1)
\int_0^\infty V_t^2 + (\gamma^2 - \lambda e^{-ct})V^2
\]
from which the conclusion of the lemma follows.
\end{proof}
For $\gamma^2 > \frac{N-1}{p-2}$ and
$0 < \lambda < \tilde{\lambda_1}(N,a,b,c) = (\frac{c}{2}Z(\frac{2}{c}
\sqrt{\gamma^2 - \frac{N-1}{p-2}}))^2$ we have that
\[
\int_0^\infty w_t^2+ (-\frac{ N-1}{p-2}+\gamma^2 - \lambda e^{-ct}) w^2 >0,
\]
for any radial $w = w(t) \in H_a(\Omega)$. Therefore in this range,
and assuming $0 2\gamma$ and $\lambda \in (\mu_1, \lambda_1)$.
\end{itemize}
\end{theorem}
Note that when looking only for radial solutions of \eqref{pr},
one can assume a slightly larger range for $b$, and the variational
setting works the same way (see \cite{cala}) for
\[
-\gamma = a - \frac{N-2}{2} < b < a+1.
\]
In closing we make several remarks. We believe that the existence
of nonradial solutions is due to symmetry breaking, where the
least energy solutions bifurcate from the radial solutions of
\eqref{cyl} as $b-a$ is fixed and $\gamma$ increases.
It would be interesting to find the exact values
$\lambda = \lambda(N,a,b,c)$ where bifurcation occurs.
It is not difficult to see that in this range of the
parameters one can construct multi-bump solutions (similar to \cite{cawa2}),
and hence to conclude that the number of essentially distinct
solutions of \eqref{cyl} tends to infinity as $\gamma$ increases to infinity.
For questions of a similar nature on a related problem in $\mathbb{S}^3$
(but qualitatively different solutions) see \cite{brpe}.
It remains an interesting problem to investigate what happens in the
``critical'' case $b=a$ (therefore $p=2^*$) for $N \geq 3$,
and the situation when $c \geq 2\gamma$. We emphasize the fact that the
nonexistence results for $c \geq 2\gamma$ stated in
Theorem~\ref{nicl} refer only to radial solutions of \eqref{pr}.
One would like to either have an existence theorem, or to be able
to extend the nonexistence results to nonradial
functions.
\subsection*{Acknowledgment}
The author would like to thank the referee for useful
comments and suggestions.
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\end{document}
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