\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 16, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/16\hfil Multiple solutions] {Multiple solutions for a q-Laplacian equation on an annulus} \author[S. Tai, J. Wang \hfil EJDE-2012/16\hfilneg] {Shijian Tai, Jiangtao Wang} % in alphabetical order \address{Shijian Tai \newline Shenzhen Experimental Education Group, Shenzhen, 5182028, China} \email{363846618@qq.com} \address{Jiangtao Wang \newline School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, 430073, China} \email{wjtao1983@yahoo.com.cn} \thanks{Submitted November 7, 2011. Published January 24, 2012.} \subjclass[2000]{35J40} \keywords{Ground state; minimizer; nonradial function; q-Laplacian; \hfill\break\indent Rayleigh quotient} \begin{abstract} In this article, we study the q-Laplacian equation $$ -\Delta_{q}u=\big||x|-2\big|^{a}u^{p-1},\quad 1<|x|<3 , $$ where $\Delta_{q}u=\operatorname{div}(|\nabla u|^{q-2} \nabla u)$ and $q>1$. We prove that the problem has two solutions when $a$ is large, and has two additional solutions when $p$ is close to the critical Sobolev exponent $q^{*}=\frac{Nq}{N-q}$. A symmetry-breaking phenomenon appears which shows that the least-energy solution cannot be radial function. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} This article concerns the q-Laplacian equation \begin{equation}\label{1.1} \begin{gathered} -\Delta_{q}u=\Phi_{a}u^{p-1} \quad \text{in } \Omega,\\ u>0 \quad \text{in } \Omega,\\ u=0 \quad \text{on } \partial \Omega, \end{gathered} \end{equation} where $\Delta_{q}u=div(|\nabla u|^{q-2}\nabla u)$, $\Omega=\{x\in \mathbb{R}^N |1<|x|<3\}$ is an annulus in $\mathbb{R}^{N}$, $N\geq 3$, $a>0$, $p>q>1$ and $\Phi_{a}$ is the radial function $$ \Phi_{a}(x)=\big||x|-2\big|^{a}. $$ Equation \eqref{1.1} is an extension of the problem \begin{equation}\label{1} \begin{gathered} -\Delta_{q}u=|x|^{a}u^{p-1} \quad \text{in } |x|<1,\\ u=0 \quad \text{on } |x|=1. \end{gathered} \end{equation} Equation \eqref{1} can be seen as a natural extension to the annular domain $\Omega$ of the celebrated H\'{e}non equation with Dirichlet boundary conditions \begin{equation}\label{1.2} \begin{gathered} -\Delta u=|x|^{a}u^{p-1} \quad \text{in } |x|<1,\\ u=0 \quad \text{on } |x|=1. \end{gathered} \end{equation} This equation was proposed by H\'{e}non in \cite{h} when he studied rotating stellar structures. For $a>0$, $2
0$.
Ni \cite{n} proved that the infimum
\begin{equation}\label{1.4}
\inf_{0\neq u\in H^{1}_{0,\rm rad}(B)}\frac{\int_{B}|\nabla
u|^{2}dx}{\big(\int_{B}|x|^{a}|u|^{p}dx\big)^{2/p}}
\end{equation}
is attained for any $2 < p < 2^{*}+2a/(N-2)$ by a function in
$H^{1}_{0,\rm rad}(B)$, the space of radial $H^{1}_0(B)$ functions.
Therefore, radial solutions of \eqref{1.2} also exist for
supercritical exponents $p$. Indeed, $H^{1}_{0,\rm rad}(B)$ shows a
power-like decay away from the origin (as a result of the Strauss
Lemma, see \cite{am,s2}) that combines with the weight $|x|^{a}$ and
provides the compactness of the embedding $H^{1}_{0,\rm rad}(B)\subset
L^{p}(B)$ for any $2 < p < 2^{*}+2a/(N-2)$.
When $a>0$, Smets, Su and Willem obtained some symmetry-breaking
results for \eqref{1.2} in \cite{ssw}. They proved that minimizers
of \eqref{1.3} could not be radial, at least for $a$ sufficiently
large. Consequently, \eqref{1.2} had at least two solutions when $a$
was large (see also \cite{sw}).
Serra \cite{s1} proved that \eqref{1.2} had at least one
nonradial solution when $p=2^{*}$, and in \cite{bs} Badiale and
Serra obtained the existence of more than one solutions to
\eqref{1.2} also for some supercritical values of $p$. These
solutions are nonradial and they are obtained by minimization under
suitable symmetry constraints.
Cao and Peng \cite{cp} proved that, for $p$
sufficiently close to $2^{*}$, the ground-state solutions of
\eqref{1.2} possessed a unique maximum point whose distance from
$\partial B$ tended to zero as $p\to 2^{*}$. And they also
proved the same results in the case of q-laplacian of the H\'{e}non
equation (see \cite{cpy}).
This result was improved in \cite{p}, where multi-bump
solutions for the H\'{e}non equation with almost critical Sobolev
exponent $p$ were found, by applying a finite-dimensional reduction.
These solutions are not radial, though they are invariant under the
action of suitable subgroups of $O(N)$, and they concentrate at
boundary points of the unit ball $B$ in $\mathbb{R}^{N}$ as $p\to
2^{*}$. However, the role of $a$ is a static one (for more results
for $p\approx 2^{*}$, see also \cite{ps}).
When the weight disappeared; i.e. $a=0$, Brezis and Nirenberg
proved in \cite{bn} that the ground state solution of $-\Delta
u=u^{p}$ in $H^{1}_0(\Omega)$ was not a radial function. Actually,
they proved that both a radial and a nonradial (positive)
solution arise as $p\approx 2^{*}$.
When the weight in \eqref{1.1} disappeares; i.e. $a=0$, Li
and Zhou \cite{lz} proved that existence of multiple solutions to
the p-Laplacian type elliptic problem
\begin{equation}\label{1.5}
\begin{gathered}
-\Delta_{p} u(x)=f(x,u) \quad \text{in } \Omega,\\
u=0 \quad \text{on } \partial\Omega,
\end{gathered}
\end{equation}
where $\Omega$ was a bounded domain in $\mathbb{R}^{N}$ ($ N > 1 $) with
smooth boundary $\partial\Omega$, and $f(x,u)$ went asymptotically
in $u$ to $|u|^{p-2}u$ at infinity.
When $q=2$ in \eqref{1.1}, Calanchi, Secchi and
Terraneo \cite{cst} obtained multiple solutions for a H\'{e}non-like
equation on $1<|x|<3$. For more results about asymptotic estimates
for solutions of the H\'{e}non equation with $a$ large, one
can see \cite{bw1,bw2}.
This paper is mainly motivated by \cite{cst}. We want to extend the
results in \cite{cst} to a general $q$-Laplacian problem. We
consider the critical points of
\begin{equation}\label{1.6}
R_{a,p}=\frac{\int_{\Omega}|\nabla
u|^qdx}{\big(\int_{\Omega}\Phi_{a}|
u|^{p}dx\big)^{q/p}},\quad
u\in W^{1,q}_0(\Omega)\backslash\{0\},
\end{equation}
which is the Rayleigh quotient associated with \eqref{1.1}.
Our main results are as follows.
\begin{theorem}\label{thm1}
Assume that $p\in(q,q^{*})$. For $a $ large enough, any ground
state $u_{a ,p}$ is a nonradial function.
\end{theorem}
\begin{theorem}\label{thm2}
Assume that $a >0$. For $p$ close to $q^{*}$ the quotient
$R_{a ,p}$ has at least two nonradial local minima.
\end{theorem}
\begin{theorem}\label{thm3}
There exist $\overline{a }>0$ and $q<\overline{p} 0,\; y\in \mathbb{R}^{N}.
$$
We set
$$
U^{i}_{\varepsilon}(x)=\varepsilon^{-\frac{N-q}{q}}U
\Big(\frac{x-x_{i}}{{\varepsilon^{(q-1)/q}}}\Big)
=\frac{1}{(\varepsilon+|x-x_{i}|^{\frac{q}{q-1}})^{\frac{N-q}{q}}}
$$
and denote by $\psi_{i}(i=0,1)$ two cut-off functions such that
$0\leq\psi_{i}\leq 1$,
$|\nabla\psi_{i} |\leq C|\ln\varepsilon|$ for
some constant $C>0$, and
$$
\psi_{i}=\begin{cases}
1 & \text{if } |x-x_{i}|< \frac{1}{2|\ln\varepsilon|},\\
0 & \text{if } |x-x_{i}|\geq \frac{1}{|\ln\varepsilon|}.
\end{cases}
$$
The following lemma shows that the truncated functions
\begin{equation}\label{2.10}
u^{i}_{\varepsilon}=\psi_{i}(x)U^{i}_{\varepsilon}(x), \quad i=0,1,
\end{equation}
are almost minimizers for $S_{0,q^{*}}$. Since it is an easy
modification of the arguments of \cite{cp}, we omit the proof of
this fact.
\begin{lemma}\label{lem2.5}
If $a > 0$, then
\begin{equation}\label{2.11}
\lim_{p\to
q^{*}}R_{a,p}(u^{i}_{\varepsilon})=S_{o,q^{*}}+K(\varepsilon),
\end{equation}
with $\lim_{\varepsilon\to 0}K(\varepsilon)=0$.
\end{lemma}
As a direct consequence of Lemma \ref{lem2.5}, we obtain the following
result.
\begin{corollary}\label{lem2.6}
$S_{0,q^{*}}=S_{\alpha,q^{*}}$.
\end{corollary}
\begin{proof}
On one hand, $S_{0,q^{*}}\leq S_{a,q^{*}}$ since $\Phi_{a}(|x|)\leq 1$.
On the other hand by Lemma \ref{lem2.5}, we have
$$
R_{a,q^{*}}(u^{i}_{\varepsilon})
=\lim_{p\to q^{*}}R_{a,p}(u^{i}_{\varepsilon})
=S_{0,q^{*}}+K(\varepsilon),
$$
which implies that $S_{0,q^{*}}+K(\varepsilon)\geq S_{a,q^{*}}$ for
every $\varepsilon>0$. Letting $\varepsilon\to 0$, we infer
$S_{0,q^{*}}\geq S_{a,q^{*}}$. Therefore $S_{0,q^{*}}= S_{a,q^{*}}$.
\end{proof}
Now we are ready to prove Theorem \ref{thm2}.
\begin{proof}[Proof of Theorem \ref{thm2}]
Let $u_{a,p}$ be a ground state solution. Let us suppose that it
concentrates on the outer boundary. The infimun of $R_{a,p}$ on
$\bar{\Lambda}$ is attained. However it cannot be attained on the
boundary $\partial \Lambda = \Sigma$. In fact, from Lemma
\ref{lem2.4}, we obtain
$$
\inf_{\Sigma}R_{a,p}(u)> S_{0,q^{*}}+\delta, \quad\text{as }
p\to q^{*}
$$
and
$$
\inf_{\Lambda}R_{a,p}(u)\leq R_{a,p}(u^{1}_{\varepsilon})\to
S_{0,q^{*}}+K_1(\varepsilon), \quad \text{as } p\to q^{*},
$$
since $u^{1}_{\varepsilon} \in \Lambda$ for $\varepsilon$
sufficiently small. Then the infimum is attained in an interior
point of $\Lambda$ and is therefore a critical point of $R_{a,p}$.
\end{proof}
\subsection{Proof of Theorem \ref{thm3}}
Now we prove the existence of a third nonradial solution, in the
previous section we proved the existence of two solutions of
\eqref{1.1} which were local minima of Rayleigh quotient for $p$ near
$q^{*}$. We would expect another critical point of $R_{a,p}$ located
in some sense between these minimum points.
For $\varepsilon $ sufficiently small let
$u^{i}_{\varepsilon}=\psi_{i}(x)U^{i}_{\varepsilon}(x),
i\in\{0,1\}$, be defined as in \eqref{2.10}. We will verify that
$R_{a,p}$ has the mountain-pass geometry.
Let us introduce the mountain-pass level
\begin{equation}\label{c}
c=c(a,p)=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}R_{a,p}(\gamma(t)),
\end{equation}
where
$$
\Gamma=\big\{\gamma\in
C([0,1],W^{1,q}_0(\Omega)):
\gamma(0) =u^{0}_{\varepsilon},\gamma(1)=u^{1}_{\varepsilon}\big\}
$$
is the set of continuous paths joining $u^{0}_{\varepsilon}$ with
$u^{1}_{\varepsilon}$. We claim that c is a critical value for
$R_{a,p}$.
We start to prove that $c$ is larger, uniformly with respect to
$\varepsilon$, than the values of the functional $R_{a,p}$ at the
points $u^{0}_{\varepsilon}$ and $u^{1}_{\varepsilon}$.
\begin{lemma}\label{lem2.8}
Set $M_{\varepsilon}=\max\{R_{a,p}(u^{0}_{\varepsilon}),
R_{a,p}(u^{1}_{\varepsilon})\}$.
There exists $\sigma > 0$ such that $c \geq M_{\varepsilon}+ \sigma$
uniformly with respect to $\varepsilon$.
\end{lemma}
\begin{proof}
We prove that there exists $\sigma$ such that for all $\gamma \in
\Gamma$,
$$
\max R_{a,p}(\gamma(t))\geq M_{\varepsilon}+ \sigma.
$$
A simple continuity argument shows that for every
$\gamma \in \Gamma$ there exists $t_{\gamma}$ such that
$\gamma(t_{\gamma}) \in \Sigma$, where
$$
\Sigma=\big\{u\in W^{1,p}_0(\Omega)\backslash\{0\}:
\int_{\Omega^{-}}|\nabla u|^qdx=\int_{\Omega^{+}}|\nabla
u|^qdx\big\}.
$$
In fact the map
$$
t \in [0,1] \mapsto\int_{\Omega^{+}}|\nabla
\gamma(t)|^qdx-\int_{\Omega^{-}}|\nabla \gamma(t)|^qdx
$$
is continuous and it takes a negative value at $t=0$ and a positive
value at $t=1$. It follows from Lemma \ref{lem2.4} that for $p$ near
$q^{*}$ there exists $\delta > 0$ such that
$$
\max_{t\in [0,1]}R_{a,p}(\gamma(t))\geq R_{a,p}(\gamma(t_{\gamma}))
\geq \inf_{u\in\Sigma}R_{a,p}(u) \geq S_{0,q^{*}}+\delta.
$$
On the other hand, for $\varepsilon$ small enough, we have
$$
M_{\varepsilon} < S_{0,q^{*}}+\frac{\delta}{2}.
$$
This completes the proof.
\end{proof}
By the previous estimates, we can show that c is a critical level
for $R_{a,p}$. As a result a further nonradial solution to
\eqref{1.1}
arises.
\begin{proof}[Proof of Theorem \ref{thm3}]
By the previous results, we can apply a deformation argument (see
\cite{am,s3}) to prove that $c$ is a critical level and it is
achieved ( since the PS condition is satisfied ) by a function
$\upsilon$. By the asymptotic estimate \eqref{2.4} for the radial
level $S_{a,p}^{\rm rad}$, we know that there exists a constant $C$
independent of $p$ such that
$$
S_{a,p}^{\rm rad} \geq Ca^{q-1+\frac{q}{p}}.
$$
Particularly, we obtain $ S_{a,p}^{\rm rad} \to +\infty as a
\to +\infty. $ Therefore we can choose $a_0$ such that
$$
S_{a,p}^{\rm rad}\geq 3S_{0,q^{*}} \quad\forall a\geq a_0.
$$
Define $\zeta\in\Gamma$ by
$\zeta(t)=tu^{1}_{\varepsilon}+(1-t)u^{0}_{\varepsilon}$ for all $t
\in [0,1]$, and let $\tau \in [0,1]$ be such that
$$
R_{a,p}(\zeta(\tau))=\max_{t\in[0,1]}R_{a,p}(\zeta(t)).
$$
Noting that $u^{0}_{\varepsilon}$ and $u^{1}_{\varepsilon}$ have
disjoint support one has, for $\varepsilon$ small enough, we have
\begin{align*}
R_{a,p}(\upsilon)
&=c\leq R_{a,p}(\zeta(\tau))\\
&= \frac{\int_{\Omega}|\nabla(\tau u^{1}_{\varepsilon}
+(1-\tau)u^{0}_{\varepsilon})|^qdx}
{\big(\int_{\Omega}\Phi_{a}|\tau u^{1}_{\varepsilon}
+(1-\tau)u^{0}_{\varepsilon}|^{p}dx\big)^{q/p}}\\
&= \frac{\int_{\Omega}\tau^q|\nabla u^{1}_{\varepsilon}|^qdx+
\int_{\Omega}(1-\tau)^q |\nabla u^{0}_{\varepsilon}|^qdx}
{\big(\tau^{p} \int_{\Omega}\Phi_{a}|u^{1}_{\varepsilon}|^{p}dx
+(1-\tau)^{p} \int_{\Omega}\Phi_{a}|u^{0}_{\varepsilon}|^{p}dx
\big)^{q/p}}\\
&\leq \frac{\tau^q\int_{\Omega}|\nabla u^{1}_{\varepsilon}|^qdx}
{\big(\tau^{p}\int_{\Omega}\Phi_{a}|u^{1}_{\varepsilon}|^{p}dx\big)^{q/p}}+
\frac{(1-\tau)^q\int_{\Omega} |\nabla u^{0}_{\varepsilon}|^qdx}
{\big((1-\tau)^{p} \int_{\Omega}\Phi_{a}|u^{0}_{\varepsilon}|^{p} dx\big)^{q/p}}\\
&= R_{a,p}(u^{1}_{\varepsilon})+R_{a,p}(u^{0}_{\varepsilon})\\
&\leq 2M_{\varepsilon}< 3S_{0,q^{*}}\\
&\leq S^{\rm rad}_{a,p}.
\end{align*}
\end{proof}
\section{Behavior of the ground-state solution for $a$ large}
In this section, we mainly analyze a ground state solution as
$a\to +\infty$. Even in this case this solution tends to
``concentrate" at the boundary $\partial\Omega$.
However, this concentration is much weaker than concentration
as $p\to q^{*}$.
We use the notation
$C(\rho_1,\rho_2)=\{x\in \mathbb{R}^{N}| \rho_1 <|x| < \rho_2 \}$.
Let $\delta$ be sufficiently small (say $\delta< \frac{1}{2}$ )
and $\varphi$ be a smooth cut-off function such
that $0 \leq \varphi \leq 1$ with
\begin{equation}\label{3}
\varphi(x)=\begin{cases}
1 , & x\in C(1,1+\delta)\cup C(3-\delta,3),\\
0 , & x\in C(2-\delta,2+\delta).
\end{cases}
\end{equation}
From now on, since $p\in(q,q^{*})$ is fixed we denote a ground state
solution of problem \eqref{1.1} $u_{a,p}$ with $u_{a}$.
\begin{lemma}\label{lem3.1}
Let $u_{a}$ be such that $R_{a,p}(u_{a})= S_{a,p}$. If $\varphi$ is
defined in \eqref{3}, then
\begin{equation}\label{3.1}
R_{a,p}(\varphi u_{a})= S_{a,p}+o(S_{a,p}) as a \to
+\infty.
\end{equation}
\end{lemma}
\begin{proof}
By the homogeneity of $R_{a,p}$, we may assume $\int_{\Omega}|\nabla
u_{a}|^qdx=1$. We will prove it by two steps.
Step 1. We claim that
\begin{equation}\label{3.2}
\int_{\Omega}\Phi_{a}(\varphi
u_{a})^{p}dx=\int_{\Omega}\Phi_{a}u_{a}^{p}dx+o\Big(\int_{\Omega}\Phi_{a}u_{a}^{p}dx\Big).
\end{equation}
Actually, if we assume
$$
\limsup_{\alpha\to
\infty}\frac{\int_{\Omega}\Phi_{a}u_{a}^{p}(1-\varphi^{p})dx}
{\int_{\Omega}\Phi_{a}u_{a}^{p}dx}=b>0,
$$
which implies that, up to some subsequence,
$$
\frac{\int_{\Omega}\Phi_{a}u_{a}^{p}(1-\varphi^{p})dx}
{\int_{\Omega}\Phi_{a}u_{a}^{p}dx}>\frac{b}{2}>0.
$$
Since $1-\varphi^{p}\equiv0$ on $C(1,1+\delta)\cup C(3-\delta,3)$,
we have
\begin{align*}
\int_{\Omega}\Phi_{a}u_{a}^{p}(1-\varphi^{p})dx
&= \int_{C(1+\delta,3-\delta)}\Phi_{a}u_{a}^{p}(1-\varphi^{p})dx\\
&\leq (1-\delta)^{a}\int_{\Omega}u_{a}^{p}(1-\varphi^{p})dx\\
&\leq (1-\delta)^{a}\int_{\Omega}u_{a}^{p}dx.
\end{align*}
Hence,
$$
\int_{\Omega}u_{a}^{p}dx\geq(1-\delta)^{-a}
\int_{\Omega}\Phi_{a}u_{a}^{p}(1-\varphi^{p})dx.
$$
Thus,
\begin{equation}\label{4}
\frac{\int_{\Omega}u_{a}^{p}dx}{\int_{\Omega}\Phi_{a}u_{a}^{p}dx}
\geq(1-\delta)^{-a}\frac{\int_{\Omega}\Phi_{a}u_{a}^{p}(1-\varphi^{p})dx}{\int_{\Omega}\Phi_{a}u_{a}^{p}dx}
\geq(1-\delta)^{-a}\frac{b}{2}.
\end{equation}
Since $S_{a,p}^{p/q}=(\int_{\Omega}\Phi_{a}u_{a}^{p}dx)^{-1}$,
\eqref{4} can be written as
$$
S_{a,p}^{p/q} \geq
\frac{b}{2}\frac{(1-\delta)^{-a}}{\int_{\Omega}u_{a}^{p}dx}
\geq \frac{b}{2} (1-\delta)^{-a}S_{0,p}^{p/q},
$$
where
$$
S_{0,p}=\inf_{u\neq0}\frac{\int_{\Omega}|\nabla
u|^qdx}{\big(\int_{\Omega}u^{p}dx\big)^{q/p}}.
$$
On the other hand, from \eqref{2.2}, it follows that
$$
S_{a,p}^{p/q}\leq Ca^{p-\frac{pN}{q}+N},
$$
which gives a contradiction for $a$ large. Hence \eqref{3.2} is
true.
Step 2. Now we prove that
\begin{equation}\label{3.3}
\int_{\Omega}|\nabla(\varphi u_{a})|^qdx=\int_{\Omega}|\nabla
u_{a}|^qdx+o(1)=1+o(1).
\end{equation}
It is easy to prove that $u_{a}$ satisfies the problem
\begin{equation}\label{3.4}
\begin{gathered}
-\Delta_{q}u_{a}=S_{a,p}^{p/q}\Phi_{a}u_{a}^{p-1} \quad \text{in }
\Omega,\\
u_{a}>0 \quad \text{in } \Omega,\\
u_{a}=0 \quad \text{on } \partial\Omega.
\end{gathered}
\end{equation}
Since $\int_{\Omega}|\nabla u_{a}|^qdx=1$, up to subsequences,
as $a\to \infty$, we have that:
$$
\text{$u_{a}\to u$ weakly in $W_0^{1,q}(\Omega)$, and
strongly in $L^{s}(\Omega)$ \a. e. in $\Omega$}.
$$
Now we prove that $u=0$. In fact, multiplying problem \eqref{3.4}
by a smooth function $\phi$ with $\operatorname{supp}\phi\Subset \Omega$ and
integrate, we obtain
\begin{equation}\label{3.5}
\int_{\Omega}|\nabla u_{a}|^{q-2}\nabla u_{a}\nabla \phi
dx=\int_{\Omega}S_{a,p}^{p/q}\Phi_{a}u_{a}^{p-1}\phi dx \to
0, \quad \text{as } a\to +\infty,
\end{equation}
since, by \eqref{2.2}, $S_{a,p}^{p/q}\Phi_{a} \to 0$
uniformly on $\operatorname{supp}\phi$ and $u_{a}$ is uniformly
bounded in $L^{s}$ for $qq$. As $a \to \infty$, there exist two constants
$C_1$, $C_2$ depending on $p$ such that
\begin{equation}\label{2.4}
0
0$. We get
\begin{equation}\label{3.9}
\begin{split}
&R_{a,p}(\varphi u_{a})\\
&= \frac{\int_{\Omega}|\nabla
u_{a,1}|^qdx+\int_{\Omega}|\nabla u_{a,2}|^qdx}
{\big(\lambda_{a}\int_{\Omega}\Phi_{a}u_{a,2}^{p}dx
+\int_{\Omega}\Phi_{a}u_{a,2}^{p}dx\big)^{q/p}}\\
&= \frac{\int_{\Omega}|\nabla
u_{a,1}|^qdx}{(\lambda_{a}+1)^{q/p}
\big(\int_{\Omega}\Phi_{a}u_{a,2}^{p}dx\big)^{q/p}}
+\frac{\int_{\Omega}|\nabla
u_{a,2}|^qdx}{(\lambda_{a}+1)^{q/p}
\big(\int_{\Omega}\Phi_{a}u_{a,2}^{p}dx\big)^{q/p}}\\
&= \frac{\lambda_{a}^{q/p}\int_{\Omega}|\nabla
u_{a,1}|^qdx}{(\lambda_{a}+1)^{q/p}
\big(\int_{\Omega}\Phi_{a}u_{a,1}^{p}dx\big)^{q/p}}
+\frac{\int_{\Omega}|\nabla
u_{a,2}|^qdx}{(\lambda_{a}+1)^{q/p}
\big(\int_{\Omega}\Phi_{a}u_{a,2}^{p}dx\big)^{q/p}}.
\end{split}
\end{equation}
By the definition of $S_{a,p}$ each quotient $R_{a,p}(u_{a,1})$ and
$R_{a,p}(u_{a,2})$ in the last term is greater than or equal to
$S_{a,p}$. Therefore by Lemma \ref{lem3.1} and equation \eqref{3.8}
we obtain
\begin{equation}\label{3.10}
S_{a,p}+o(S_{a,p})\geq
\frac{1+\lambda_{a}^{q/p}}{(\lambda_{a}+1)^{q/p}}S_{a,p}.
\end{equation}
We notice that the function $g(x)=\frac{1+x^{q/p}}{(x+1)^{q/p}}$ is
strictly greater than $1$ for every $x>0$, $g(0)=1$ and
$g(x)\to 1$ as $x\to +\infty$. Further it is
increasing in $[0,1]$ and decreasing in $ [ 1, +\infty)$ and
$\max_{x>0}g(x)=g(1)=2^{1-q/p}$. Let $L\in \Lambda$ and $\{a_{n}\}$
a sequence such that $\lambda_{a_{n}}\to L$ as $n\to
+\infty$. Taking to the limit in \eqref{3.10}, we obtain that
$1\geq\frac{1+L^{q/p}}{(L+1)^{q/p}}$ and so either $L=+\infty$ or
$L=0$.
\end{proof}
\begin{corollary}\label{cor3.3}
With the notation of Proposition \ref{prop3.2}, for any sequence
$\{a_{n}\}$ such that $\lambda_{a_{n}}\to 0$ one has
\begin{equation}\label{3.11}
\lim_{n\to +\infty}\frac{\int_{\Omega}|\nabla u_{a_{n},1}
|^qdx}{\int_{\Omega}|\nabla u_{a_{n},2} |^qdx} = 0 .
\end{equation}
\end{corollary}
\begin{proof}
Denote
$$
\frac{\int_{\Omega}|\nabla u_{a,1}
|^qdx}{\int_{\Omega}|\nabla u_{a,2} |^qdx}=\kappa_{a}
$$
and suppose that
$\limsup_{n\to \infty}\kappa_{a_{n}}>0$.
Passing to subsequences, $\kappa_{a_{n}}>\kappa>0$ for some
$\kappa$. Therefore we have
\begin{align*}
S_{a_{n},p}+o(S_{a_{n},p})
&= \frac{\int_{\Omega}|\nabla u_{a_{n},1}|^qdx+\int_{\Omega}|\nabla u_{a_{n},2}|^qdx}
{\big(\int_{\Omega}\Phi_{a_{n}}u_{a_{n},1}^{p}dx
+\int_{\Omega}\Phi_{a_{n}}u_{a_{n},2}^{p}dx\big) ^{q/p}}\\
&= \frac{(1+\kappa_{a_{n}})\int_{\Omega}|\nabla u_{a_{n},2}|^qdx}
{(\int_{\Omega}\Phi_{a_{n}}u_{a_{n},2}^{p}dx)^{q/p}
(1+\lambda_{a_{n}})^{q/p}}\\
&\geq R_{a_{n},p}(u_{a_{n},2})\frac{1+\kappa}{1+o(1)}\\
&\geq (1+\kappa)S_{a_{n},p}+o(S_{a_{n},p}),
\end{align*}
which is a contradiction. Therefore,
$$
\kappa_{a_{n}}= \frac{\int_{\Omega}|\nabla u_{a_{n},1}
|^qdx}{\int_{\Omega}|\nabla u_{a_{n},2} |^qdx}\to 0.
$$
\end{proof}
The following result is an immediate consequence of the previous
results.
\begin{proposition}\label{prop3.4}
For any $a_{n}$ such that $\lambda_{a_{n}}\to 0$,
\begin{equation}\label{3.12}
\lim_{n\to +\infty}\int_{\Omega}|\nabla u_{a_{n},1}|^q dx=0 .
\end{equation}
\end{proposition}
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\end{document}