\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 229, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/229\hfil Sign-changing solutions]
{Sign-changing solutions for elliptic equations with
fast increasing weight and concave-convex nonlinearities}

\author[X. Qian, J. Chen \hfil EJDE-2017/229\hfilneg]
{Xiaotao Qian, Jianqing Chen}

\address{Xiaotao Qian (corresponding author) \newline
College of Mathematics and Computer Science \& FJKLMAA,
Fujian Normal University,
Qishan Campus, Fuzhou 350108, China. \newline
Jinshan College,
Fujian Agriculture and Forestry University,
Fuzhou 350002, China}
\email{qianxiaotao1984@163.com}

\address{Jianqing Chen \newline
College of Mathematics and Computer Science \& FJKLMAA,
Fujian Normal University,
Qishan Campus, Fuzhou 350108, China}
\email{jqchen@fjnu.edu.cn}

\dedicatory{Communicated by Paul H. Rabinowitz}

\thanks{Submitted July 13, 2017. Published September 22, 2017.}
\subjclass[2010]{35J20, 35J60}
\keywords{Sign-changing solutions; variational method; critical problem}

\begin{abstract}
 In this article, we study the problem
 \[
 -\operatorname{div}(K(x)\nabla u)=a(x)K(x)|u|^{q-2}u+b(x)K(x)|u|^{2^{\ast}-2}u,
 \quad x\in \mathbb{R}^N,
 \]
 where $2^{\ast}=2N/(N-2)$, $N\geq3$, $1<q<2$, $K(x)=\exp({|x|^{\alpha}/4})$
 with $\alpha\geq2$. Under some assumptions on the potentials $a(x)$ and $b(x)$,
 we obtain a pair of sign-changing solutions of the problem via variational
 methods and certain estimates.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks


\section{Introduction}

In this article, we consider the existence of sign-changing
solutions for the problem
\begin{equation} \label{e1.1}
-\operatorname{div}(K(x)\nabla u)=a(x)K(x)|u|^{q-2}u+b(x)K(x)|u|^{2^{\ast}-2}u,
\quad x\in \mathbb{R}^N,
\end{equation}
where $2^{\ast}=2N/(N-2)$, $N\geq3$, $1<q<2$, $K(x)=\exp({|x|^{\alpha}/4})$
 with $\alpha\geq2$.

 Our motivations of studying the equation \eqref{e1.1} relies on the fact that,
for $\alpha=q=2$, $a(x)= (N-2)/(N+2)$ and $b(x)\equiv 1$, equation \eqref{e1.1}
occurs when one tries to find self-similar solutions of the form
\[
w(t,x)=t^{{2-N}\over{N+2}}u(xt^{-1/2})
\]
for the evolution equation
\[
w_t-\Delta w=|w|^{4/(N-2)}w \quad\text{on } (0,\infty)\times \mathbb{R}^N.
\]
See \cite {es,ha} for a detailed description.

Equation \eqref{e1.1} with $q=2$, $a(x)\equiv \lambda$ and $b(x)\equiv 1$,
has been studied in \cite{he,na1,na2,qi}. We also refer to the paper
of Catrina et al.\ \cite{ca} where the authors considered the case
$q=2$, $a(x)=\lambda|x|^{\alpha-2}$ and $b(x)\equiv 1$, and showed that
the value of $\alpha$ affects the critical dimension of the problem.
Later on, Furtado et al.\ \cite{fu1} studied the equation
\begin{equation} \label{e1.2}
 -\operatorname{div}\big(K(x)\nabla u\big)=\lambda K(x)|x|^{\beta}|u|^{q-2}u
+K(x)|u|^{2^{\ast}-2}u,\quad x\in \mathbb{R}^N,
\end{equation}
where $\beta=(\alpha-2){(2^{\ast}-q)\over(2^{\ast}-2)}$. In that paper,
by using Mountain Pass Theorem, the authors obtained a positive solution
if $2<q<2^{\ast}$. Furthermore, they applied Linking Theorem to show that
problem \eqref{e1.2} when $q=2$ has a nontrivial solution for any
$\lambda\ge \lambda_1$, where $\lambda_1$ is the first positive eigenvalue of the
linear problem
\[
 -\operatorname{div}\big(K(x)\nabla u\big)=\lambda K(x)|x|^{\alpha -2}u,\quad
x\in \mathbb{R}^N.
\]
With the help of the result of \cite{ca}, namely there is no positive
solution of \eqref{e1.2} for $q=2$ and $\lambda\ge \lambda_1$, then they can conclude
that this nontrivial solution indeed is a sign-changing solution.
Recently, Furtado et al.\ \cite{fu2} obtained two nonnegative nontrivial
solutions for \eqref{e1.1} when the potential $a(x)$ has small norm in a
suitable weighted Lebesgue space.

On the other hand, for similar problems in bounded domain, Ambrosetti et al.\
 \cite{am} studied the semilinear problem
\[
-\Delta u=\lambda u^{q-1}+u^{p-1} \quad \text{in } \Omega, \; u\in H_0^1(\Omega)
\]
where $\Omega\subset\mathbb{R}^N$ is bounded, $N\geq3$, $\lambda >0$,
$1<q<2<p\leq 2^{\ast}$. They proved the existence of at least two positive
solutions if $\lambda \in (0,\lambda_0)$ for some positive $\lambda_0$.
We also refer the interested readers to \cite{al,ch2,dg,wu} where equations
with concave and convex nonlinearity on bounded domains were considered.

Motivated by the works we described above, in present paper, we try to seek
more solutions of \eqref{e1.1}. Special concern is the existence of
sign-changing solutions of \eqref{e1.1}. This kind of problem is
variational in nature. Indeed, let us denote by $H$ the Hilbert space obtained
as the closure of $C_c^\infty(\mathbb{R}^N)$ with respect to the norm
\[
\|u\|=\Big(\int_{\mathbb{R}^N} K(x)|\nabla u|^2dx\Big)^{1/2}.
\]
We also define the weighted Lebesgue spaces
\[
L_K^s(\mathbb{R}^N)=\big\{u\text{ measurable in } \mathbb{R}^N :
\|u\|_s^s=\int_{\mathbb{R}^N}K(x)|u|^s dx<\infty \big\}.
\]
It is proven in \cite{fu1} that the embedding $H \hookrightarrow L_K^r(\mathbb{R}^N)$
 is continuous for $2\leq r\leq 2^{\ast}$, and compact for $2\leq r< 2^{\ast}$.
For any $r>1$, we denote by $r'$ its conjugated exponent, that is, the unique
$r'>1$ so that $1/r+1/r'=1$. Throughout this paper, we always use the following
assumptions:

\begin{itemize}
 \item[(A1)] $a(x)>0$ and $a(x)\in L_K^{{\sigma}_q}(\mathbb{R}^N)
\cap C(\mathbb{R}^N)$ for some $(2/q) \leq \sigma_q' < ({2^{\ast}}/q)$;

 \item[(A2)] $b(x)>0$ and $b(x) \in L^\infty(\mathbb{R}^N)$;

 \item[(A3)] the set $\Omega_b^+:=\{x \in \mathbb{R}^N : b(x)>0\}$
has an interior point;

 \item[(A4)] there are $x_0 \in \mathbb{R}^N$ and $\delta>0$ such that
$B_\delta(x_0) \subset \Omega_b^+$ and
\[
|b(x)|_\infty - b(x)\leq M|x-x_0|^{\gamma} ,
\]
for a.e. $x\in B_\delta(x_0)$, with $M>0$ and $\gamma > N$.
\end{itemize}

On $H$, we define the functional
\[
I(u)=\frac12 \int_{\mathbb{R}^N}K(x)|\nabla u|^2dx
- {1\over q}\int_{\mathbb{R}^N}K(x)a(x)|u|^qdx
- \frac{1}{2^*} \int_{\mathbb{R}^N}K(x)b(x)|u|^{2^{\ast}}dx.
\]
By (A1), (A2) and the above embedding, we conclude that $I$ is well
defined and $I \in C^1(H,\mathbb{R})$. Now, it is well known that there exists
a one to one correspondence between the critical points and the weak solutions
 of \eqref{e1.1}. Here, we say $u \in H$ is a weak solution of \eqref{e1.1},
 if for any $\phi \in H$, there holds
\[
\int_{\mathbb{R}^N}K(x)\big[\nabla u \nabla {\phi} -a(x)|u|^{q-2}u \phi
- b(x)|u|^{2^{\ast}-2}u \phi \big]dx=0.
\]
Our main result is stated below.

\begin{theorem}\label{thm11}
Assume that {(A1)--(A4)}. If $N\geq 7$ and $(3N-2)/(2N-4)<q<2$, then
 there exists $M_2>0$, such that \eqref{e1.1} has at least two nonnegative
 nontrivial solutions and a pair of sign-changing solutions in $H$ for
$\|a(x)\|_{\sigma _q}<M_2$ and $\alpha >(N-2)/2$.
\end{theorem}

Since Furtado et al.\ \cite{fu2} showed that \eqref{e1.1} has at least two
nonnegative nontrivial solutions in $H$ for $\|a\|_{\sigma_q}<M_1$ with some
$M_1>0$, we will focus our attentions to find out sign-changing solutions
of \eqref{e1.1}. To this end, there are some difficulties. Firstly,
since the embedding $H \hookrightarrow L_K^{2^{\ast}}(\mathbb{R}^N)$
is not compact, the functional $I$ satisfies $(PS)$ condition only locally.
We prove that the energy level belongs to the range where $(PS)$ condition
hold by choosing a suitable test function as in \cite{ca,fu2}.
Secondly, as pointed in \cite{ch}, the Mountain Pass Theorem which was used
in \cite{fu1,fu2} is usually unable to prove the existence of sign-changing
solutions. Moreover, the Linking Theorem used in \cite{fu1} can not be
applied here becasue to $1<q<2$. Instead of the two above Theorems,
 we shall employ the separation argument for Nehari-type set of the problem,
 which has been used in \cite{ch,ta1,ta2}.
Thirdly, the potentials $a(x)$ and $b(x)$ bring much difficulty to the
 above separation argument. To overcome this difficulty, inspired by \cite{su},
we impose conditions (A1) and (A2) on the potentials $a(x)$ and $b(x)$ respectively,
 which are stronger than the corresponding ones in \cite{fu2}.

This article is organized as follows. In the next section, we give some
 notation and preliminaries. Then we prove Theorem \ref{thm11}.

\section{Preliminaries}

Throughout this paper, $E^{-1}$ denotes the dual space of a Banach space $E$.
We denote by $|\cdot |_t$, the norm of the standard Sobolev space
$L^t(\mathbb{R}^N)$. $\mathcal{D}^{1,2}(\mathbb{R}^N)$ is the closure of
$C_0^\infty(\mathbb{R}^N)$ under the norm of $\int_{\mathbb{R}^N}|\nabla \cdot|^2dx$.
$B_r(x)$ is a ball centered at $x$ with radius $r$. $\rightarrow$ denotes strong
convergence. $\rightharpoonup$ denotes weak convergence. $d,d_i$ will denote
various positive constants whose exact values are not important.
Finally, we write $\int u$, $\|a\|_{\sigma _q}$ and $|b|_\infty$ instead of
$\int_{\mathbb{R}^N}u(x)dx$, $\|a(x)\|_{\sigma _q}$ and $|b(x)|_\infty$,
 respectively.

For each $r\in[2,2^{\ast}]$, the existence of the embedding
$H \hookrightarrow L_K^r(\mathbb{R}^N)$ enables us to define
\begin{equation} \label{e2.1}
S_r = \inf\big\{\int K(x)|\nabla u|^2 : u \in H, \int K(x)|u|^r = 1\big\}.
\end{equation}
In particular, when $r={2^{\ast}}$, we only write $S:=S_{2^{\ast}}$.
It is worth pointing out that this constant is equal to the best constant
 of the embedding
 $\mathcal{D}^{1,2}(\mathbb{R}^N) \hookrightarrow L^{2^{\ast}}(\mathbb{R}^N)$,
see \cite{ca}.

By the condition (A4), we can choose $\eta > 0$ small enough such that
$B_{2\eta}(x_0) \subset B_{\delta}(x_0)$ with $x_0 \in \text{int}(\Omega_b^+)$
and $\delta > 0$. Define a cutoff function $\psi (x)$ satisfying
$\psi (x) \equiv 1$ in $B_{\eta}(x_0)$, $\psi (x) \equiv 0$ outside
$B_{2\eta}(x_0)$ and $0 \leq \psi \leq 1$.
Inspired by \cite{ca,fu2}, we consider the function
\[
u_\varepsilon(x)= K(x)^{-1/2}\psi (x)
\Big( {{1}\over {\varepsilon + |x-x_0|^2}} \Big)^{(N-2)/2},
\]
and set
\[
U_\varepsilon(x)= K(x)^{-1/2}\Big( {{1}\over {\varepsilon + |x-x_0|^2}}\Big)^{(N-2)/2},
\quad
v_\varepsilon(x)= {{u_\varepsilon(x)}\over{\|u_\varepsilon\|_{2^{\ast}}}}.
\]
Without loss of generality, we assume that $x_0 = 0$ from now on.
To prove Theorem \ref{thm11}, we first give the next three Lemmas which
will be useful later.

\begin{lemma}\label{lem21}
For $\varepsilon > 0$ small,
\begin{gather}
\int u_{\varepsilon}^{\mu}=O(1) \quad \text{if } 0<\mu< \frac{N}{N-2}, \label{e2.2}\\
\int u_{\varepsilon}^{\mu}=O\big(\varepsilon^{\frac{N}{2}-\frac{N-2}{2}\mu}
|\ln\varepsilon|\big)\quad \text{if } \mu=\frac{N}{N-2}, \label{e2.3}\\
\int u_{\varepsilon}^{\mu}=O\big( \varepsilon^{\frac{N}{2}-\frac{N-2}{2}\mu} \big)
\quad \text{if }\frac{N}{N-2}< \mu <2^{\ast}. \label{e2.4}
\end{gather}
\end{lemma}

\begin{proof}
Note that
\begin{align*}
\int u_{\varepsilon}^{\mu}
&\leq d\int_{B_{2\eta(0)}}{{dx}\over{(\varepsilon+|x|^2)}^{{(N-2)\mu/2}}}\\
&\leq d_1\int_0^{{2\eta/{\sqrt{\varepsilon}}}}{{{\varepsilon}^{N/2}\rho^{N-1}d
\rho}\over {\varepsilon}^{{(N-2)\mu}/2}(1+|\rho|^2)^{{(N-2)\mu}/2}}\,.
\end{align*}
Since $N-1-(N-2)\mu > -1$, when $0< \mu <{N/(N-2)}$, we have
\[
\int u_{\varepsilon}^{\mu} \leq d_2\int_{B_{2\eta(0)}}{{1}\over{|x|}^{{(N-2)\mu}}}
= O(1).
\]
Thus, \eqref{e2.2} holds. The proofs of \eqref{e2.3} and \eqref{e2.4}
 are similar.
\end{proof}

\begin{lemma}\label{le22}
For $\varepsilon > 0$ small, we have
\begin{gather}
\int v_{\varepsilon}^{\mu}
=\frac{\int u_{\varepsilon}^{\mu}}{\|u_{\varepsilon}\|_{2^{\ast}}^{\mu}} \nonumber\\
= O\big(\varepsilon^{(N-2)\mu/4}\big) \quad\text{if } 0< \mu <\frac{N}{N-2},
\label{e2.5}\\
= O\big(\varepsilon^{\frac{N}{2}-{{N-2}\over4}\mu}|\ln\varepsilon|\big)
 \quad \text{if } \mu=\frac{N}{N-2}, \label{e2.6}\\
= O\big( \varepsilon^{\frac{N}{2}-{{N-2}\over4}\mu} \big)
 \quad \text{if } \frac{N}{N-2}< \mu <2^{\ast} \label{e2.7}.
\end{gather}
\end{lemma}

\begin{proof}
According to \cite{ca},
\[
\|u_{\varepsilon}\|_{2^{\ast}}^{2^{\ast}}
= \int K(x)|u_{\varepsilon}|^{2^{\ast}} = {\varepsilon}^{{-N}/2}A_0+O(1),
\quad \text{if } N>2,
\]
with
\[
A_0 = \int {{1}\over{(1+|x|^2)^N}}, \quad \text{if } N>2,
\]
from which it follows that
\begin{align*}
\|u_{\varepsilon}\|_{2^{\ast}}^{\mu}
&= \big({\varepsilon}^{{-N}/2}A_0+O(1)\big)^{\mu/2^{\ast}}\\
&=d{\varepsilon}^{{-(N-2)\mu}/4}
 +O\big(\varepsilon ^{-\frac{N}{2}({{\mu}\over {2^{\ast}}}-1)} \big).
\end{align*}
This and \eqref{e2.2} imply that for $0<\mu<N/(N-2)$ and $\varepsilon$ small enough
\begin{align*}
\int v_{\varepsilon}^{\mu}
&={{{\int u_{\varepsilon}^{\mu}}\over{\|u_{\varepsilon}\|_{2^{\ast}}^{\mu}}}}\\
&={{{ {O(1)}}}\over{{d{{\varepsilon}^{{-(N-2)\mu}/4}}
+{O\big(\varepsilon ^{{-N\over2}({{\mu}\over {2^{\ast}}}-1)} \big)}}}}
=O\big({\varepsilon}^{{(N-2)\mu}/4}\big).
\end{align*}
Thus, \eqref{e2.5} follows. Similar arguments arrive at \eqref{e2.6}
 and \eqref{e2.7}.
\end{proof}

\begin{lemma}\label{lem23}
Let $w_1$ be a nonnegative nontrivial solution of \eqref{e1.1}.
For $1<q<2$ and $\varepsilon > 0$ small, then we have
\begin{gather}
 \int K(x)a(x)v_{\varepsilon}^{q}\ge d\varepsilon^{\frac{N}{2}-{{N-2}\over4}q}
+O\big(\varepsilon^{{(N-2)q}\over4}\big)
\quad \text{if }  {{{N}\over{N-2}}<q<2}, \label{e2.8} \\
 \int K(x)a(x)|w_1|v_{\varepsilon}^{q-1}
=O\big(\varepsilon^{{(N-2)(q-1)}\over4}\big), \label{e2.89}\\
\int K(x)a(x)|w_1|^{q-1}v_{\varepsilon}
=O\big(\varepsilon^{{N-2}\over4}\big), \label{e2.10}\\
\int K(x)a(x)|w_1|^{q-2}w_1v_{\varepsilon}
=O\big(\varepsilon^{{N-2}\over4}\big), \label{e2.11}\\
 \int K(x)b(x)|w_1|^{2^{\ast}-1}v_{\varepsilon}
=O\big(\varepsilon^{{N-2}\over4}\big), \label{e2.12}\\
\int K(x)b(x)|w_1|^{2^{\ast}-2}w_1v_{\varepsilon}
=O\big(\varepsilon^{{N-2}\over4}\big), \label{e2.13}\\
 \int K(x)b(x)|w_1|v_{\varepsilon}^{2^{\ast}-1}
 =O\big(\varepsilon^{{N-2}\over4}\big). \label{e2.14}
\end{gather}
\end{lemma}

\begin{proof}
We only prove part$(i)$. The rest parts of the Lemma can be proved by
a similar argument. Using  (A1), one has
\begin{align*}
&\int K(x)a(x)|u_{\varepsilon}|^q \\
&=\int_{B_{2\eta}(0)}
\frac{K(x)a(x)K(x)^{-q/2}\psi^q(x)}{(\varepsilon+|x|^2)^{q(N-2)/2}} \\
&\ge d_1\int_{B_{2\eta}(0)}{{\psi^q(x)}\over{(\varepsilon+|x|^2)^{q(N-2)/2}}} \\
&=d_1\Big(\int_{B_{2\eta}(0)}{{1}\over{(\varepsilon+|x|^2)^{q(N-2)/2}}}
+\int_{B_{2\eta}(0)}{{\psi^q(x)-1}\over{(\varepsilon+|x|^2)^{q(N-2)/2}}}\Big)\\
&=d_1\Big({\varepsilon^{{{N}\over{2}}-{{(N-2)q}\over{2}}}}
 \int_{B_{2\eta/{\sqrt{\varepsilon}}}(0)}{{1}\over{(1+|x|^2)^{q(N-2)/2}}}
+\int_{B_{2\eta}(0)}{{\psi^q(x)-1}\over{(\varepsilon+|x|^2)^{q(N-2)/2}}}\Big)\\
&=d_2{\varepsilon^{{{N}\over{2}}-{{(N-2)q}\over{2}}}}+O(1)
\end{align*}
whenever $q>N/(N-2)$. Therefore,
\begin{align*}
\int K(x)a(x)|v_{\varepsilon}|^q
&={{{\int K(x)a(x)|u_{\varepsilon}|^q}\over{\|u_{\varepsilon}\|_{2^{\ast}}^{q}}}}\\
&\ge{{{d_2\varepsilon^{{{N}\over{2}}-{{(N-2)q}\over{2}}}}+O(1)}
\over{{d_3{\varepsilon}^{{-(N-2)q}/4}}
 +{O\big(\varepsilon ^{{-N\over2}({{q}\over {2^{\ast}}}-1)} \big)}}}\\
&=d\varepsilon^{\frac{N}{2}-{{N-2}\over4}q}
 +O\big(\varepsilon^{{(N-2)q}\over4}\big).
\end{align*}
Hence, we obtain \eqref{e2.8} holds.
\end{proof}

\section{Existence of sign-changing solutions}

Following Tarantello \cite{ta2} and Chen \cite{ch}, we first decompose
the Nehari-type set of the considered problem, then consider minimization
problems of $I$ on its proper subset. Set
\[
\Lambda=\{u \in H: \langle I'(u),u \rangle = 0\}.
\]
Consider the following three subsets of $ \Lambda$:
\begin{gather*}
 \Lambda_0=\{u\in\Lambda:(2-q)\|u\|^2-(2^{\ast}-q)\int K(x)b(x)|u|^{2^{\ast}}=0\}, \\
\Lambda^+=\{u\in\Lambda:(2-q)\|u\|^2-(2^{\ast}-q)\int K(x)b(x)|u|^{2^{\ast}}>0\}, \\
\Lambda^-=\{u\in\Lambda:(2-q)\|u\|^2-(2^{\ast}-q)\int K(x)b(x)|u|^{2^{\ast}}<0\}.
\end{gather*}
Furthermore, if we denote
\[
\overline{M}=\Big({{2-q}\over{2^{\ast}-q}}\Big)^{{2-q}\over{2^{\ast}-2}}
\Big({{2^{\ast}-2}\over{2^{\ast}-q}}\Big)
S^{\frac{N}{2}-{N\over4}q}S_{q\sigma_q'}^{q/2}|b|_{\infty}^{{q-2}\over{2^{\ast}-2}},
\]
we indeed get that for $\|a\|_{\sigma_q}<\overline{M}$ the following
minimization problems:
\[
c_0=\inf_{u \in {\Lambda^+}} I(u)\quad \text{and} \quad
c_1=\inf_{u\in\Lambda^-}I(u)
\]
attain their infimum at $u_0$ and $u_1$, respectively. Additionally, $u_0$
and $u_1$ are nonnegative nontrivial solutions of \eqref{e1.1}.
Next, we start establishing the existence of sign-changing solutions of \eqref{e1.1}.

\subsection{Some lemmas}

For every $u\in H$ and $u\neq0$, we set
\[
t_{\rm max}=\Big[{{(2-q)\|u\|^2}\over{(2^{\ast}-q)\int K(x)b(x)|u|^{2^{\ast}}}}
\Big]^{1\over{2^{\ast}-2}}.
\]
Then we have the following result.

\begin{lemma}\label{lem31}
 Let $\|a\|_{\sigma_q}<\overline{M}$. For every $u\in H$ and $u\neq0$, we have
\begin{itemize}
\item[(i)]   there exists a unique $t^+=t^+(u)>t_{\rm max}>0$ such that
 $t^+u\in \Lambda^-$  and $I(t^+u)=\max_{t\geq t_{\rm max}}I(tu)$.

\item[(ii)] there exists a unique $0<t^-=t^-(u)<t_{\rm max}$ such that
$t^-u\in \Lambda^+$ and $I(t^-u)=\min_{0\le t\le t^+} I(tu)$.
\end{itemize}
\end{lemma}

\begin{proof}
From direct computations, we have
\[
{{\partial I}\over{\partial t}}(tu)
=t^{q-1}\Big(t^{2-q}\|u\|^2-t^{2^{\ast}-q}\int K(x)b(x)|u|^{2^{\ast}}-
\int K(x)a(x)|u|^q\Big).
\]
Let
\[
\varphi(t)=t^{2-q}\|u\|^2-t^{2^{\ast}-q}\int K(x)b(x)|u|^{2^{\ast}}
-\int K(x)a(x)|u|^q.
\]
 By (A1), (A2) and easy calculations show that
$\lim_{t\to 0^+}\varphi(t)=-\int K(x)a(x)|u|^q<0$ and
$\lim_{t\to +\infty}\varphi(t)=-\infty$. In addition, $\varphi(t)$
is concave and attains its maximum at the point $t_{\rm max}$.
Also
\begin{align*}
\varphi(t_{\rm max})
&=\Big({{2-q}\over{2^{\ast}-q}}\Big)^{{2-q}\over{2^{\ast}-2}}
\Big({{2^{\ast}-2}\over{2^{\ast}-q}}\Big)
\Big[{{\|u\|^{2(2^{\ast}-q)}}\over{({\int K(x)b(x)|u|^{2^{\ast}}})^{(2-q)}}}
 \Big]^{{N-2}\over{4}} \\
&\quad - \int K(x)a(x)|u|^q.
\end{align*}
From (A1), (A2) and \eqref{e2.1}, it is easily verified that
\begin{align*}
\varphi(t_{\rm max})
&\geq {\Big({{2-q}\over{2^{\ast}-q}}\Big)^{{2-q}\over{2^{\ast}-2}}
\Big({{2^{\ast}-2}\over{2^{\ast}-q}}\Big)
S^{N(2-q)/4}\|u\|^q|b|_{\infty}^{{q-2}\over{2^{\ast}-2}}}-\int K(x)a(x)|u|^q\\
&\ge \Big[{\Big({{2-q}\over{2^{\ast}-q}}\Big)^{{2-q}\over{2^{\ast}-2}}
\Big({{2^{\ast}-2}\over{2^{\ast}-q}}\Big)
S^{N(2-q)/4}|b|_{\infty}^{{q-2}\over{2^{\ast}-2}}}
-\|a\|_{\sigma_q}S_{q\sigma_q'}^{-q/2}\Big]\|u\|^q.
\end{align*}
Thus, for $\|a\|_{\sigma_q}<\overline{M}$, we have $\varphi(t_{\rm max})>0$.
It then follows that $\varphi(t)$ has exactly two points $0<t^-<t_{\rm max}<t^+$
such that
\[
\varphi(t^+)=0=\varphi(t^-) \quad \text{and} \quad \varphi'(t^+)<0<\varphi'(t^-).
\]
Equivalently, we obtain $t^+u\in \Lambda^-$ and $t^-u\in \Lambda^+$.
 Also $I(t^+u)\geq I(tu)$,
for any $t\geq t_{max}$ and $I(t^-u)\leq I(tu)$, for any $t\in [0,t^+]$.
\end{proof}


\begin{lemma}\label{lem32}
Let $\|a\|_{\sigma_q}<\overline{M}$, then $\Lambda_0=\{0\}$.
\end{lemma}

\begin{proof}
Suppose to the contrary, there exists $w\in \Lambda_0$, $w\neq0$ such that
$(2-q)\|w\|^2-(2^{\ast}-q)\int K(x)b(x)|w|^{2^{\ast}}=0$.
Combining this with \eqref{e2.1}, we can obtain that
 $\|w\|\geq \big({{2-q}\over{2^{\ast}-q}}\big)^{(N-2)/4}|b|_{\infty}^{(2-N)/4}S^{N/4} $.
 On the other hand, we infer from $w\in \Lambda$ that
\begin{align*}
0&=\|w\|^2-\int K(x)a(x)|w|^q-\int K(x)b(x)|w|^{2^{\ast}}\\
 &\geq {\Big({{2^{\ast}-2}\over{2^{\ast}-q}}\Big)}\|w\|^2
 -\|a\|_{\sigma_q}S_{q\sigma_q'}^{-q/2}\|w\|^q\\
 &\geq \|w\|^q{\Big[{{2^{\ast}-2}\over{2^{\ast}-q}}
\Big({{2-q}\over{2^{\ast}-q}}\Big)^{(N-2)(2-q)/4}|b|_{\infty}^{(q-2)/(2^{\ast}-2)}
 S^{N(2-q)/4}-\|a\|_{\sigma_q}S_{q\sigma_q'}^{-q/2}\Big]}>0,
\end{align*}
which is a contradiction. This completes the proof.
\end{proof}

\begin{lemma}\label{lem33}
Let $\|a\|_{\sigma_q}<\overline{M}$. Given $u\in \Lambda^-$, there are
$\rho_u>0$ and a differential function $g_{\rho_u}:B_{\rho_u}(0)\to\mathbb{R}^+$
 defined for $w\in H$, $w\in B_{\rho_u}(0)$ such that
\begin{itemize}
\item[(i)] $ g_{\rho_u}(0)=1,\quad g_{\rho_u}(w)(u+w)\in\Lambda^-$,

\item[(ii)]
\begin{align*}
&\langle g_{\rho_u}'(0),\phi \rangle\\
&=\Big(-2\int K(x)\nabla u\nabla \phi
 +2^{\ast}\int K(x)b(x)|u|^{2^{\ast}-2}u\phi\\
&\quad +q\int K(x)a(x)|u|^{q-2}u\phi\Big)
\big/\Big((2-q)\|u\|^2-(2^{\ast}-q)
 \int K(x)b(x)|u|^{2^{\ast}}\Big).
\end{align*}
\end{itemize}
\end{lemma}

\begin{proof}
Define $F:\mathbb{R}\times H \to \mathbb{R}$ by:
\[
F(t,w)=t^{2-q}\|u+w\|^2-t^{2^{\ast}-q}\int K(x)b(x)|u+w|^{2^{\ast}}\\
 -\int K(x)a(x)|u+w|^q.
\]
In view of $u\in \Lambda^-\subset \Lambda$, we obtain $F(1,0)=0$ and
\[
F_t(1,0)=(2-q)\|u\|^2-(2^{\ast}-q)\int K(x)b(x)|u|^{2^{\ast}}<0.
\]
Using Implicit function Theorem for $F$ at the point $(1,0)$, we know
that there is $\bar\varepsilon>0$ so that for
$w\in H$, $\|w\|<\bar\varepsilon$, the equation $F(t,w)=0$ has a unique
solution $t=g_{\rho_u}(w)>0$ with $g_{\rho_u}(0)=1$.
Since $F(g_{\rho_u}(w),w)=0$ for $w\in H$, $\|w\|<\bar\varepsilon$, we have
\begin{align*}
&g_{\rho_u}^{2-q}(w)\|u+w\|^2-g_{\rho_u}^{2^{\ast}-q}(w)\int{K(x)b(x)
 |u+w|^{2^{\ast}}}
-\int{K(x)a(x)|u+w|^{q}}\\
&=\Big(\|g_{\rho_u}(w)(u+w)\|^2-\int{K(x)b(x)|g_{\rho_u}(w)(u+w)|^{2^{\ast}}}\\
&\quad  -\int{K(x)a(x)|g_{\rho_u}(w)(u+w)|^{q}}\Big)
\big/\Big(g_{\rho_u}^{q}(w)\Big)
=0,
\end{align*}
namely,
$g_{\rho_u}(w)(u+w)\in \Lambda$ for all $w\in H$ with $\|w\|<\bar\varepsilon$.
Since $F_t(1,0)<0$ and
\begin{align*}
&F_t(g_{\rho_u}(w),w)\\
&=(2-q)g_{\rho_u}^{1-q}(w)\|u+w\|^2
-(2^{\ast}-q) g_{\rho_u}^{2^{\ast}-q-1}(w)\int {K(x)b(x)|u+w|^{2^{\ast}}}\\
&={{(2-q)\|g_{\rho_u}(w)(u+w)\|^2
-(2^{\ast}-q)\int {K(x)b(x)|g_{\rho_u}(w)(u+w)|^{2^{\ast}}}}
\over{g_{\rho_u}^{1+q}(w)}},
\end{align*}
we can choose $\varepsilon>0$ small enough $(\varepsilon<\bar\varepsilon)$
such that for $w\in H$ and $\|w\|<\varepsilon$,
$$
(2-q)\|g_{\rho_u}(w)(u+w)\|^2
-(2^{\ast}-q)\int {K(x)b(x)|g_{\rho_u}(w)(u+w)|^{2^{\ast}}}<0,
$$
which means that
$$
g_{\rho_u}(w)(u+w)\in \Lambda^-,\quad \text{for all }
w\in H,\ \|w\|<\varepsilon.
$$
Moreover, for any $\phi \in H, \ r>0$, we have
\begin{align*}
&F(1,0+r\phi)-F(1,0)\\
&=\int K(x)|\nabla (u+r\phi)|^2-\int K(x)b(x)|u+r\phi|^{2^{\ast}}
 -\int K(x)a(x)|u+r\phi|^q \\
&\quad -\int K(x)|\nabla u|^2+\int K(x)b(x)|u|^{2^{\ast}}+\int K(x)a(x)|u|^q\\
&=\int K(x)(2r\nabla u\nabla\phi+r^2|\nabla\phi|^2)
-\int {K(x)b(x)\Big(|u+r\phi|^{2^{\ast}}-|u|^{2^{\ast}}\Big)}\\
&\quad -\int {K(x)a(x)\Big(|u+r\phi|^{q}-|u|^{q}\Big)}
\end{align*}
and so
\begin{align*}
\langle F_w,\phi\rangle |_{t=1, w=0} 
&=\lim_{r\to 0} {{F(1,0+r\phi)-F(1,0)}\over{r}}\\
&=2\int K(x)\nabla u\nabla\phi -2^{\ast}\int {K(x)b(x)|u|^{2^{\ast}-2}u\phi} \\
&\quad -q\int {K(x)a(x)|u|^{q-2}u\phi}.
\end{align*}
Therefore,
\begin{align*}
&\langle g_{\rho_u}'(0),\phi \rangle\\
&=  -{{\langle F_w,\phi\rangle}\over{F_t}}\big|_{t=1 ,w=0}\\
&={{{{-2\int K(x)\nabla u\nabla \phi}+2^{\ast}\int K(x)b(x)|u|^{2^{\ast}-2}u\phi
 +q\int K(x)a(x)|u|^{q-2}u\phi}\over{(2-q)\|u\|^2-(2^{\ast}-q)
 \int K(x)b(x)|u|^{2^{\ast}}}}}.
\end{align*}
This completes the proof.
\end{proof}

\subsection{Existence results}

We are now in a position to prove Theorem \ref{thm11}. To this end, we need
 to make comparisons among some  minimization problems. Set
\begin{gather*}
\Lambda_1^-=\{u=u^+-u^-\in\Lambda: u^+\in\Lambda^-\}, \\
\Lambda_2^-=\{u=u^+-u^-\in\Lambda: -u^-\in\Lambda^-\},
\end{gather*}
where $u^{+}=\max \{u,0\}$ and $u^{-}=u^{+}-u$. Define
\begin{gather*} \label{e3.2}
\beta_1=\inf_{u\in \Lambda_1^-}I(u), \\
\label{e3.3}
\beta_2=\inf_{u\in \Lambda_2^-}I(u).
\end{gather*}

\begin{lemma}\label{lem34}
Let $\|a\|_{\sigma_q}<\overline{M}$, then $\Lambda_1^-$ and $\Lambda_2^-$ are closed.
\end{lemma}

\begin{proof}
Let $\{u_n\}$ be a sequence in $\Lambda_1^-$ with $u_n\to u_0$.
It then follows from $ \{u_n\}\subset\Lambda_1^-\subset\Lambda $ that
\begin{align*}
\|u_0\|^2
&= \lim_{n\to\infty}\|u_n\|^2\\
&= \lim_{n\to\infty}\Big[\int{K(x)a(x)|u_n|^{q}}
 + \int {K(x)b(x)|u_n|^{2^{\ast}}}\Big]\\
&= \int{K(x)a(x)|u_0|^{q}}+ \int {K(x)b(x)|u_0|^{2^{\ast}}}
\end{align*}
and
\begin{align*}
&(2-q)\|u_0^{+}\|^2-(2^{\ast}-q)\int {K(x)b(x)|u_0^{+}|^{2^{\ast}}}\\
&=\lim_{n\to\infty}\Big[(2-q)\|u_n^{+}\|^2-(2^{\ast}-q)
\int {K(x)b(x)|u_n^{+}|^{2^{\ast}}}\Big]
\le 0,
\end{align*}
namely,
$u_0\in \Lambda$ and $u_0^{+}\in\Lambda^-\cup\Lambda_0$.

Since there exists a positive $d_1$ such that $\|u^{+}\|\ge d_1>0$ for all
$u\in\Lambda_1^-$, we know $u_0^+\ne0$. Combining this with Lemma \ref{lem32},
for $\|a\|_{\sigma_q}<\overline{M}$, we have
$u_0^{+}\notin \Lambda_0$. In turn, $u_0^{+}\in\Lambda^-$ and hence,
$u_0\in \Lambda_1^-$. Thus, $\Lambda_1^-$ is closed for
 $\|a\|_{\sigma_q}<\overline{M}$. The same argument can prove that
$\Lambda_2^-$ is closed. The proof of Lemma \ref{lem34} is complete.
\end{proof}

\begin{lemma}\label{lem35}
(i) If $\beta_1<c_1$, then the minimization problem \eqref{e3.2}
 achieves its infimum at a point which defines
a sign-changing critical point of $I$.

(ii) If $\beta_2<c_1$, then the same conclusion follows for the minimization
problem \eqref{e3.3}.
\end{lemma}

\begin{proof}
We only prove (i). Part (ii) of the lemma can be proved by a similar argument.
By Lemma \ref{lem34}, we can use Ekeland variational principle to
construct a minimizing sequence $\{u_n\}\subset\Lambda_1^-$
with the following properties:
\begin{itemize}
\item[(1)] $I(u_n)\to\beta_1$,
\item[(2)] $I(z)\geq I(u_n)-{1\over n}\|u_n-z\|$ for all $z\in \Lambda_1^-$.
\end{itemize}
Firstly, we claim that $\|u_n^-\|\geq d >0$. Indeed, if to the contrary,
there is a subsequence (still denoted by $\{u_n^-\}$) such that
$\|u_n^-\| \to 0$, then
\[
\beta_1+o(1)=I(u_n)=I(u_n^+)+I(-u_n^-)\geq c_1+o(1),
\]
which is a contradiction with assumption $\beta_1<c_1$.
Secondly, we claim $I'(u_n)\to 0$ in $H^{-1}$. Indeed, set
 $0 <\rho <\rho_n \equiv \rho_{u_n}$, $g_n^{\pm}\equiv g_{u_n}^{\pm}$,
 where $\rho_{u_n}$ and $g_{u_n}^{\pm}$ are given by Lemma \ref{lem33}
 so that for $v_{\rho}=\rho v$ with $\|v\|=1$, there holds
\[
z_{\rho}=g_n^+(v_{\rho})(u_n-v_{\rho})^+ - g_n^-(v_{\rho})(u_n-v_{\rho})^-
\in \Lambda_1^-.
\]
Consequently,
\begin{equation} \label{e3.4}
\begin{aligned}
{{1\over n}}\|z_{\rho}-u_n\|
&\geq \langle I'(u_n),u_n-z_{\rho}\rangle+o(1)\|z_{\rho}-u_n\|  \\
&=\langle I'(u_n),u_n-v_{\rho}-z_{\rho}\rangle+\rho\langle I'(u_n),v\rangle+o(1)\|z_{\rho}-u_n\| \\
&=(1-g_n^+(v_{\rho}))\langle I'(u_n),(u_n-v_{\rho})^+\rangle+\rho\langle I'(u_n),v\rangle \\
&\quad -(1-g_n^-(v_{\rho}))\langle I'(u_n),(u_n-v_{\rho})^-\rangle+o(1)\|z_{\rho}-u_n\|.
\end{aligned}
\end{equation}
It is trivial to show $\{u_n^{+}\}$ is bounded, and so we may assume that
$u_n^{+} \rightharpoonup w_0^+$ in $H$ for some $w_0^+\in H$.
Since $\{u_n\}\subset \Lambda_1^-$, one has
\[
(2^{\ast}-2)\|u_n^{+}\|^2-(2^{\ast}-q)\int {K(x)a(x)|u_n^{+}|^{q}}>0.
\]
This together with $\lim_{n\to\infty}\int K(x)a(x)|u_n^+|^q
= \int K(x)a(x)|w_0^+|^q$ (see \cite{fu2}) imply
\[
(2^{\ast}-2)\liminf_{n\to \infty}\|u_n^{+}\|^2-(2^{\ast}-q)
\int {K(x)a(x)|w_0^{+}|^{q}}\ge0.
\]
At this point, we show that for $\|a\|_{\sigma_q}<\overline{M}$,
\begin{equation} \label{e3.51}
(2^{\ast}-2)\liminf_{n\to \infty}\|u_n^{+}\|^2-(2^{\ast}-q)
\int {K(x)a(x)|w_0^{+}|^{q}}>0.
\end{equation}
To prove that, we employ the method used in \cite{su} and suppose
to the contrary that
\[
(2^{\ast}-2)\liminf_{n\to \infty}\|u_n^{+}\|^2
=(2^{\ast}-q)\int {K(x)a(x)|w_0^{+}|^{q}}.
\]
In view of (A1) and the fact $\|u_n^{+}\|\ge d>0$, we have
$\int {K(x)a(x)|w_0^{+}|^{q}}> 0$ and so
\begin{equation}\label{e3.52}
\liminf_{n\to \infty}\Big[{{(2^{\ast}-2)\|u_n^{+}\|^2}\over{(2^{\ast}-q)
\int K(x)a(x)|u_n^{+}|^{q}}}\Big]
= {{\liminf_{n\to \infty} \left[(2^{\ast}-2)\|u_n^{+}\|^2\right]}
\over{(2^{\ast}-q)\int K(x)a(x)|w_0^{+}|^{q}}}=1
\end{equation}
Notice that
\begin{equation}\label{e3.53}
{{(2^{\ast}-2)\|u_n^{+}\|^2}\over{(2^{\ast}-q)\int K(x)a(x)|u_n^{+}|^{q}}}>1,
\end{equation}
for $n=1,2,\dots$. Combining with \eqref{e3.52} and \eqref{e3.53},
we obtain that there exists a subsequence $\{u_{n_k}^+\}$ of $\{u_n^+\}$ such that
\[
{{(2^{\ast}-2)\|u_{n_k}^+\|^2}\over{(2^{\ast}-q)\int K(x)a(x)|u_{n_k}^+|^{q}}} \to 1
\]
as $k\to \infty$. Hence,
\begin{gather*}
\|u_{n_k}^+\|^2\to {{2^{\ast}-q}\over{2^{\ast}-2}}\int K(x)a(x)|w_0^+|^{q}, \\
\int {K(x)b(x)|u_{n_k}^+|^{2^{\ast}}}\to {{2-q}\over{2^{\ast}-2}}
\int K(x)a(x)|w_0^+|^{q}
\end{gather*}
and so we have that for $\|a\|_{\sigma_q}<\overline{M}$,
\begin{align*}
0&< {\Big[\Big({{2-q}\over{2^{\ast}-q}}\Big)
\Big({{2^{\ast}-2}\over{2^{\ast}-q}}\Big)^{{2^{\ast}-2}\over{2-q}}
\|a\|_{\sigma_q}^{-{{2^{\ast}-2}\over{2-q}}}S_{q\sigma_q'}^{{{q}
 \over{2}}{{2^{\ast}-2}\over{2-q}}}
-|b|_{\infty}S^{-{{2^{\ast}}\over{2}}}\Big]}\int K(x)|u_{n_k}^+|^{2^{\ast}} \\
&\leq {{2-q}\over{2^{\ast}-q}}\Big({{2^{\ast}-2}\over{2^{\ast}-q}}\Big)
 ^{{2^{\ast}-2}\over{2-q}} {{\|u_{n_k}^+\|^{2(2^{\ast}-q)\over{2-q}}}
\over{\big({\int  K(x)a(x)|u_{n_k}^+|^{q}}\big)^{{2^{\ast}-2}\over{2-q}} }}
 -\int K(x)b(x)|u_{n_k}^+|^{2^{\ast}}\\ 
&\to {{{2-q}\over{2^{\ast}-q}}\Big({{2^{\ast}-2}\over{2^{\ast}-q}}
 \Big)^{{2^{\ast}-2}\over{2-q}}
{{ \big[{{2^{\ast}-q}\over{2^{\ast}-2}}\int K(x)a(x)|w_{0}^+|^{q}\big]^{{2^{\ast}-q}
\over{2-q}} }
\over{\big({\int  K(x)a(x)|w_{0}^+|^{q}}\big)^{{2^{\ast}-2}\over{2-q}}}}}\\
&\quad -{{2-q}\over{2^{\ast}-2}}\int K(x)a(x)|w_0^+|^{q}
 =0,
\end{align*}
namely, $u_{n_k}^+\to 0$ in $L_K^{2^{\ast}}(\mathbb{R}^N)$, and consequently
 $w_0^+\equiv 0$, which leads to a contradiction.
Thus, \eqref{e3.51} follows. From \eqref{e3.51} we can further obtain
that there is a suitable positive constant $d$ for $n$ large enough
\[
(2^{\ast}-2)\|u_n^{+}\|^2-(2^{\ast}-q)\int K(x)a(x)|u_{n}^+|^{q}\ge d>0.
\]
Therefore, by Lemma \ref{lem33} and the boundness of $\{u_{n}^+\}$,
we conclude that $\|(g_n^{+})'(0)\| \le d_1$. Since $0<d_2\le \|u_n^-\| \le d_3$,
a similar argument can show $\|(g_n^{-})'(0)\| \le d_4$.
For fixed $n$, since
\begin{gather*}
(1-g_n^+(v_{\rho}))
=\rho \langle (g_n^+)'(0),v\rangle,\\
(1-g_n^-(v_{\rho}))=\rho \langle (g_n^-)'(0),v\rangle, \\
\|z_{\rho}-u_n\| \leq \rho+d(|1-g_n^+(v_{\rho})|+|1-g_n^-(v_{\rho})|),
\end{gather*}
$ \langle I'(u_n),u_n^{\pm}\rangle=0$ and $(u_n-v_{\rho})^{\pm} \to u_n^{\pm}$ as
 $\rho \to 0$, letting $\rho \to 0$ in \eqref{e3.4} we obtain
\[
\langle I'(u_n),v\rangle \leq {{d}\over n}.
\]

From the above discussion, we can conclude that $I'(u_n)\to 0$ in $H^{-1}$
as $n\to \infty$. By applying  \cite[Proposition 3.2]{es}, we obtain that the
sequence $\{u_n\}$ indeed satisfies the following
\begin{itemize}
\item[(i)]  $I(u_n) \to \beta_1 <c_1<c_0+{1\over N}
{{1}\over{|b|_{\infty}^{(N-2)/2}}}S^{N/2}$,

\item[(ii)]  $I'(u_n)\to 0$ in $H^{-1}$.
\end{itemize}
Then, we may use (i), (ii) and \cite[Lemma 3.1]{es} to guarantee a
convergent subsequence for $\{u_n\}$ whose strong limit will give the
desired minimizer.
\end{proof}

Clearly, Lemma \ref{lem35} would give the conclusion for Theorem \ref{thm11}
 only if the given relations $\beta_1 <c_1$ or $\beta_2 <c_1$ could be established.
 While it is not sure whether or not such inequalities should hold,
we shall use these values to compare with another minimization problem. Namely set
\[
\Lambda_{\ast}^-=\Lambda_1^-\cap \Lambda_2^-\subset \Lambda^-
\]
and then define
\begin{equation}
c_2=\inf_{u\in\Lambda_{\ast}^-}I(u).
\end{equation}
It is easy to see that $c_2\geq c_1$. Since $I$ satisfies $(PS)$ condition only
locally, we need the following upper bound for $c_2$.

\begin{lemma}\label{lem36}
(i) For any fixed $\varepsilon>0$, then there are $s>0$ and $t\in \mathbb{R}$ such that
$su_1-tU_{\varepsilon}\in \Lambda_{\ast}^-$.

(ii) For $\varepsilon>0$ sufficiently small , if $N\geq7$, ${(3N-2)/(2N-4)}<q<2$
and $\alpha>(N-2)/2$, then we have
\[
c_2\leq \sup_{s\geq0,t\in\mathbb{R}}I(su_1-tU_{\varepsilon})
<c_1+{1\over N}{{1}\over{|b|_{\infty}^{(N-2)/2}}}S^{N/2}.
\]
\end{lemma}

\begin{proof}
(i) It suffices to show that there are $s>0$ and $t\in\mathbb{R}$ so that
\[
s(u_1-tU_{\varepsilon})^+\in\Lambda^- \quad \text{and} \quad
-s(u_1-tU_{\varepsilon})^-\in\Lambda^-.
\]
To prove that, we set
\[
t_2=\max_{\mathbb{R}^N}{{u_1}\over{U_{\varepsilon}}} \quad \text{and} \quad
 t_1=\min_{\mathbb{R}^N}{{u_1}\over{U_{\varepsilon}}}.
\]
For each $t\in(t_1,t_2)$, we denote by $s^+(t)$ and $s^-(t)$
the positive values given by Lemma \ref{lem31}. Then one has
\[
s^+(t)(u_1-tU_{\varepsilon})^+\in\Lambda^- \quad \text{and} \quad
-s^-(t)(u_1-tU_{\varepsilon})^-\in\Lambda^-.
\]
Notice that $s^+(t)$ is continuous with respect to $t$ satisfying
\[
\lim_{t\to t_1^+}s^+(t)=t^+(u_1-t_1U_{\varepsilon})<+\infty \quad \text{and} \quad
\lim_{t\to t_2^-}s^+(t)=+\infty.
\]
Moreover, $s^-(t)$ is also continuous with respect to $t$ and
\[
\lim_{t\to t_1^+}s^-(t)=+\infty \quad \text{and} \quad
\lim_{t\to t_2^-}s^-(t)=t^+(t_2U_{\varepsilon}-u_1)<+\infty .
\]
By the continuity of $s^{\pm}(t)$, we conclude that there exists
$t_0\in(t_1,t_2)$ such that
$s^+(t_0)=s^-(t_0)=s_0>0$.
This proves (i) with $t=t_0$ and $s=s_0$.

(ii). Obviously, it suffices to estimate $I(su_1-tU_{\varepsilon})$ for
$s\geq 0$ and $t\in \mathbb{R}$. Since $\varepsilon$ can be now sufficiently small,
we let $U_{\varepsilon}=v_{\varepsilon}$. From the structure of $I$, we can
take $R_1>0$ possible large such that $I(su_1-tv_{\varepsilon}) \leq c_1$
for all $s^2+t^2\geq R_1^2$. Hence, we only need to estimate
$I(su_1-tv_{\varepsilon})$ for all $s^2+t^2\leq R_1^2$.
It follows from Lemma \ref{lem23} and the  elementary inequality
\[
|s+t|^m\geq|s|^m+|t|^m-d(|s|^{m-1}|t|+|s||t|^{m-1}),\quad \text{for any }
 s,\,t\in\mathbb{R},\,m>1
\]
that
\begin{align*}
&I(su_1-tv_{\varepsilon})\\
&\leq  I(su_1)+I(tv_{\varepsilon})-st\int K(x)a(x)|u_1|^{q-2}u_1v_{\varepsilon}
 -st\int K(x)b(x)|u_1|^{2^{\ast}-2}u_1v_{\varepsilon}\\
&\quad +d\Big( \int K(x)b(x)|su_1|^{2^{\ast}-1}|tv_{\varepsilon}|
 +\int K(x)b(x)|su_1||tv_{\varepsilon}|^{2^{\ast}-1}\Big)\\
&\quad +d\Big( \int K(x)a(x)|su_1|^{q-1}|tv_{\varepsilon}|
 +\int K(x)a(x)|su_1||tv_{\varepsilon}|^{q-1}\Big)\\
&\leq  I(su_1)+I(tv_{\varepsilon}) + O\big(\varepsilon^{{(N-2)(q-1)}\over4}\big)
 +O\big(\varepsilon^{{N-2}\over4}\big).
\end{align*}
Since $\int K(x)v_{\varepsilon}^{2^{\ast}}=1$, we have
\begin{align*}
I(tv_{\varepsilon})
&={{{t^2}\over2}\|v_{\varepsilon}\|^2-{{t^{2^{\ast}}}\over{2^{\ast}}}}
 \int K(x)b(x)v_{\varepsilon}^{2^{\ast}}-{{t^q}\over q}\int K(x)a(x)v_{\varepsilon}^q\\
&={\Big({{t^2}\over2}\|v_{\varepsilon}\|^2-{{t^{2^{\ast}}}\over{2^{\ast}}}
 |b|_{\infty}\Big)}
 +{{t^{2^{\ast} }}\over{2^{\ast}}}\int K(x)(|b|_{\infty}-b(x))v_{\varepsilon}^{2^{\ast}}\\
&\quad -{{t^q}\over q}\int K(x)a(x)v_{\varepsilon}^q.
\end{align*}
For any $\varepsilon >0$, it is easy to verify that the function
$t \to I(tv_{\varepsilon})$ attains its maximum at a point $t_{\varepsilon}>0$.
 Moreover, applying the arguments similar to that of
\cite[Proposition 3.2]{es} and \cite[Lemma 4.1]{dr}, we can conclude
that there are two positive constants $d_1$ and $d_2$ such that
$0<d_1 \le t_{\varepsilon}\le d_2$, independent of $\varepsilon$.

Let $h(t)={{t^2}\over2}\|v_{\varepsilon}\|^2-{{t^{2^{\ast}}}\over{2^{\ast}}}|b|_{\infty}$.
Clearly, $h(t)$ achieves its maximum at the point
$t_{\ast}={(\|v_{\varepsilon}\|^2/|b|_{\infty})^{(N-2)/4}}$.
In conclusion, we can deduce from
$\int K(x)(|b|_{\infty}-b(x))v_{\varepsilon}^{2^{\ast}}=O(\varepsilon^{N/2})$
(see \cite{fu2}),
$\|v_{\varepsilon}\|^N \leq S^{N/2}+O(\varepsilon^{\alpha/2})
+O(\varepsilon^{(N-2)/2})$ (see \cite{fu1,fu2}) and \eqref{e2.8} that
\begin{align*}
\max_{t>0}I(tv_{\varepsilon})
&\leq h(t_{\varepsilon})+{{{(t_{\varepsilon})^{2^{\ast}
 }}\over{2^{\ast}}}}\int K(x)(|b|_{\infty}
 -b(x))v_{\varepsilon}^{2^{\ast}}-{{{(t_{\varepsilon})^q}\over q}}
 \int K(x)a(x)v_{\varepsilon}^q\\
&\leq h(t_{\ast})+{{d_2^{2^{\ast}
 }}\over{2^{\ast}}}\int K(x)(|b|_{\infty}-b(x))v_{\varepsilon}^{2^{\ast}}
 -{{d_1^q}\over{q}}\int K(x)a(x)v_{\varepsilon}^q\\
&\leq {1\over N}{{1}\over{|b|_{\infty}^{(N-2)/2}}}S^{N/2}+O(\varepsilon^{\alpha
 /2})+O(\varepsilon^{(N-2)/2})+O(\varepsilon^{N/2})\\
&\quad -d\varepsilon^{\frac{N}{2}-{{N-2}\over4}q}
 +O\big(\varepsilon^{{(N-2)q}\over4}\big).
\end{align*}
Furthermore, we can obtain that for $\varepsilon >0$ small enough,
\begin{align*}
&\max_{s>0,\,t\in\mathbb{R}} I(su_1-tU_{\varepsilon}) \\
&\leq \max_{s>0} I(su_1)+\max_{t\in\mathbb{R}}I(tv_{\varepsilon})
+O\big(\varepsilon^ {{(N-2)(q-1)}\over4}\big)+O\big(\varepsilon^{{N-2}\over4}\big)\\
 &\leq c_1+{1\over N} {{1}\over{|b|_{\infty}^{(N-2)/2}}}S^{N/2}
 +O(\varepsilon^{\alpha/2})+ O\big(\varepsilon^{(N-2)/2}\big)+O(\varepsilon^{N/2})\\
 &\quad -d\varepsilon^{\frac{N}{2}-{{N-2}\over4}q}
+O\big(\varepsilon^{{(N-2)q}\over4}\big) +O\big(\varepsilon^{{(N-2)(q-1)}\over4}\big)
 +O(\varepsilon^{{N-2}\over4})\\
 &<c_1+{1\over N}{{1}\over{|b|_{\infty}^{(N-2)/2}}}S^{N/2},
\end{align*}
if $N\geq 7$, ${{3N-2}\over{2N-4}}<q<2$ and $\alpha>(N-2)/2$.
This completes the proof.
\end{proof}

\begin{lemma}\label{lem37}
Assume $\beta_1\geq c_1$ and $\beta_2\geq c_1$. The minimization problem
\begin{equation} \label{e3.70}
c_2=\inf_{u\in\Lambda_{\ast}^-}I(u)
\end{equation}
achieves its infimum at $u_2\in\Lambda_{\ast}^-$ which defines a
sign-changing critical point for $I$, provided $\|a\|_{\sigma_q}<M_2$
with some $M_2>0$.
\end{lemma}

\begin{proof}
Set $M_2=\min\{M_1,\overline{M}\}$. As in the proof of Lemma \ref{lem35},
we can construct a minimizing sequence $\{u_n\}\subset\Lambda_{\ast}^-$
for \eqref{e3.70} such that
$I(u_n)\to c_2$ and $I'(u_n)\to 0$.
Noting that $\{u_n\}\subset\Lambda_{\ast}^-$, we have
\begin{equation}\label{e3.71}
0<d_{1}\leq \|u_n^{\pm}\|\leq d_{2}
\end{equation}
for some positive constants $d_{1}$ and $d_{2}$.
Thus, we may assume that $u_n^{\pm}\rightharpoonup u_2^{\pm}$ in $H$.
\smallskip

\noindent\textbf{Claim.} $u_2^{\pm}\neq 0$. Suppose to the contrary,
we assume first that $u_2^+=0$, then we infer from
$u_n^+\in \Lambda^-\subset \Lambda$ and
$\lim_{n\to\infty}\int K(x)a(x)|u_n^+|^q= \int K(x)a(x)|w_0^+|^q$ that
\[
\|u_n\|^2-\int K(x)b(x)|u_n^+|^{2^{\ast}}=o(1).
\]
Combining this with \eqref{e2.1} and \eqref{e3.71}, we can obtain that for
$n$ large enough
\[
\int K(x)b(x)|u_n^+|^{2^{\ast}}\geq{{1}\over{|b|_{\infty}^{(N-2)/2}}}S^{N/2}+o(1),
\]
and so
\begin{equation} \label{e3.72}
I(u_n^+)={1\over2}\|u_n^+\|^2-{1\over{2^{\ast}}}\int K(x)b(x)|u_n^+|^{2^{\ast}}+o(1)
\geq {1\over N}{{1}\over{|b|_{\infty}^{(N-2)/2}}}S^{N/2}+o(1).
\end{equation}
On the other hand, from the upper bound of $c_2$ and $I(-u_n^-)\geq c_1$, we have
\[
I(u_n^+)\leq c_2-c_1+o(1)< {1\over N} {{1}\over{|b|_{\infty}^{(N-2)/2}}}S^{N/2},
\]
which is a contradiction to \eqref{e3.72}. Hence, $u_2^+\neq 0$.
Similarly, we can prove that $u_2^-\neq 0$.

Let $u_2=u_2^+-u_2^-$. Obviously, $u_2$ is sign-changing and
$u_n\rightharpoonup u_2$ in $H$. Since for any $\phi\in H$ there holds
$\langle I'(u_2),\phi \rangle=0$, $u_2$ is a weak solution of \eqref{e1.1}.
Now, to complete the proof of Theorem \ref{thm11}, we only need to show
that $u_n\to u_2$ in $H$. Define $u_n^+\,=u_2^++v_n^+$ and
$u_n^-\,=u_2^-+v_n^-$, then we have $v_n^{\pm} \rightharpoonup 0$ in $H$.
Combining this with $u_n^{\pm}\in \Lambda$ and
$\langle I'(u_2^+),u_2^+ \rangle=\langle I'(u_2^-),u_2^- \rangle=0$, we can use the
Brezis-Lieb Lemma \cite{wi} to obtain
\begin{equation} \label{e3.73}
\|v_n^{\pm}\|^2-\int K(x)b(x)|v_n^{\pm}|^{2^{\ast}}=o(1).
\end{equation}
Because the fact $c_1<c_0+{1\over N}{{1}\over{|b|_{\infty}^{(N-2)/2}}}S^{N/2}$,
it follows from Lemma \ref{lem36} that
\begin{align*}
\lim_{n\to \infty} (I(v_n^+)+I(-v_n^-))
&=\lim_{n\to \infty} I(u_n)-I(u_2)
\leq c_2-c_0\\
 &< {1\over N}{{1}\over{|b|_{\infty}^{(N-2)/2}}}S^{N/2}+c_1-c_0\\
 &< {2\over N} {{1}\over{|b|_{\infty}^{(N-2)/2}}}S^{N/2}.
\end{align*}
Therefore, we must have
 \[
\lim_{n\to \infty} \min\{I(v_n^+),\,I(-v_n^-)\}
<{1\over N}{{1}\over{|b|_{\infty}^{(N-2)/2}}}S^{N/2}.
\]
This and \eqref{e3.73} imply
\[
\|v_n^+\| \to 0 \quad \text{or} \quad \|v_n^-\| \to 0,
\]
that is, $u_2=u_2^+-u_2^-\in \Lambda_1^-$ or $u_2=u_2^+-u_2^-\in \Lambda_2^-$.
Thus, under the assumption $\beta_1\geq c_1$ and $\beta_2\geq c_1$, we get
$ I(u_2)\geq c_1$.
Hence, if writing $u_n=u_2+w_n$, we have $w_n\rightharpoonup 0$ in $H$.
 According to Brezis-Lieb Lemma, one has
\begin{equation}\label{e3.74}
I(u_n)=I(u_2+w_n)
=I(u_2)+o(1)+{1\over2}\|w_n\|^2-{1\over{2^{\ast}}}\int K(x)b(x)|w_n|^{2^{\ast}}.
\end{equation}
Since $u_2$ is a weak solution of \eqref{e1.1}, it follows from $u_n\in \Lambda$ that
\begin{equation} \label{e3.75}
\|w_n\|^2-\int K(x)b(x)|w_n|^{2^{\ast}}=o(1).
\end{equation}
Now assume
\[
\|w_n\|^2\to l\geq 0, \quad \int K(x)b(x)|w_n|^{2^{\ast}} \to l\geq 0.
\]
If $l\neq0$, then \eqref{e2.1} and \eqref{e3.75} yield that
$l\geq {{1}\over{|b|_{\infty}^{(N-2)/2}}}S^{N/2}$.
Using \eqref{e3.74}, $I(u_2)\geq c_1$ and Lemma \ref{lem36}, we obtain that
\[
c_1+o(1)+{1\over N}{{1}\over{|b|_{\infty}^{(N-2)/2}}}S^{N/2}
\leq I(u_n)=c_2+o(1)<c_1+{1\over N}{{1}\over{|b|_{\infty}^{(N-2)/2}}}S^{N/2}
\]
which is a contradiction. Therefore, $l=0$, that is, $u_n\to u_2$ in $H$
which defines a sign-changing solution of \eqref{e1.1}.
\end{proof}

The proof of Theorem \ref{thm11}  follows from Lemmas \ref{lem35} and \ref{lem37}
and  the symmetry of the functional $I$.

\subsection*{Acknowledgments}
This research was supported by National Natural Science Foundation of China
(No. 11371091, 11501107) and the innovation group of ``Nonlinear analysis and
its applications" (No. IRTL1206).

\begin{thebibliography}{99}

\bibitem{al} S. Alama and G. Tarantello;
\emph{On semilinear elliptic equations with indefinite nonlinearities},
 Calculus of Variations, 1 (1993), 439-475.

\bibitem{am} A. Ambrosetti, H. Brezis, G. Cerami;
\emph{Combined effects of concave and convex nonlinearities in some
elliptic problems}, J. Functional Anal., 122 (1994), 519-543.

\bibitem{ca} F. Catrina, M. Furtado, M. Montenegro;
\emph{Positive solutions for nonlinear
elliptic equations with fast increasing weight},
Proc. Royal Soc. Edinburgh, 137 (2007), 1157-1178.

\bibitem{ch2} J. Chen;
\emph{Multiple positive solutions for a class of nonlinear elliptic equations},
 J. Math. Anal. Apll., 295 (2004), 341-354.

\bibitem{ch} J. Chen;
\emph{Some further results on a semilinear equation with concave-convex nonlinearity},
 Nonlinear Anal., 62 (2005), 71-87.

\bibitem{dg} D. G. de Figueiredo, J. P. Gossez, P. Ubilla;
\emph{Local superlinearity and sublinearity for indefinite semilinear
elliptic problems}, J. Functional Anal., 199 (2003), 452-467.

\bibitem{dr} P. Dr\'abek, Y. X. Huang;
\emph{Multiplicity of positive solutions for some quasilinear elliptic equation
 in $R^N$ with critical Sobolev exponent},
J. Differential Equation, 140 (1997), 106-132.

\bibitem{es} M. Escobedo, O. Kavian;
\emph{Variational problems related to self-similar solutions of the heat equation},
 Nonlinear Anal., 11 (1987), 1103-1133.

\bibitem{fu1} M. Furtado, O. Myiagaki, J. P. Silva;
\emph{On a class of nonlinear elliptic
equations with fast increasing weight and critical growth},
J. Differential Equation, 249 (2010), 1035-1055.

\bibitem{fu2} M. Furtado, R. Ruviaro, J. P. Silva;
\emph{Two solutions for an elliptic equation with fast increasing
weight and concave-convex nonlinearties}, J. Math. Anal. Apll., 416 (2014), 698-709.

\bibitem{ha} A. Haraux, F. Weissler;
\emph{Nonunqueness for a semilinear initial value problem},
Indiana Univ. Math. J., 31 (1982), 167-189.

\bibitem{he} L. Herraiz;
\emph{Asymptotic behavior of solutions of some semilinear parabolic problems},
Ann. Inst. H. Poincar\'e Anal. Nonlin\'eaire, 16 (1999), 49-105.

\bibitem{na1} Y. Naito;
\emph{Self-similar solutions for a semilinear heat equation with critical
Sobolev exponent}, Indiana Univ. Math. J., 57 (2008), 1283-1315.

\bibitem{na2} Y. Naito, T. Suziki;
\emph{Radial symmetry of self-similar solutions for semilinear heat equation},
J. Differential Equation, 163 (2000), 407-428.

\bibitem{qi} Y. Qi;
\emph{The existence of ground states to a weakly coupled elliptic system},
Nonlinear Anal., 48 (2002), 905-925.

\bibitem{su} Y. Sun, S. Li;
\emph{A nonlinear elliptic equation with critical exponent:
Estimates for extremal values}, Nonlinear Anal., 69 (2008), 1856-1869.

\bibitem{ta1} G. Tarantello;
\emph{On nonhomogeneous elliptic equations
involving critical sobolev exponent}, Ann. Inst. H. Poincar\'e
Anal. Non Lin\'eaire, 9 (1992), 281-304.

\bibitem{ta2} G. Tarantello;
\emph{Multiplicity results for an inhomogeneous Neumann problem with
critical exponent},  Manu. Math., 81 (1993), 51-78.

\bibitem{wi} M. Willem;
\emph{Minimax Theorems},  Birkh\"auser, Boston, 1996.

\bibitem{wu} T. F. Wu;
\emph{Three positive solutions for Dirichlet problems involing critical
Sobolev exponent and sign-changing weight}, J. Differential Equations,
249 (2010), 1549-1578.

\end{thebibliography}

\end{document}