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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 84, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2015/84\hfil Ground state solutions]
{Ground state solutions for semilinear elliptic equations with zero
mass in $\mathbb{R}^N$}
\author[J. Liu, J.-F. Liao, C.-L. Tang \hfil EJDE-2015/84\hfilneg]
{Jiu Liu, Jia-Feng Liao, Chun-Lei Tang}
\address{Jiu Liu \newline
School of Mathematics and Statistics, Southwest University,
Chongqing 400715, China}
\email{jiuliu2011@163.com}
\address{Jia-Feng Liao \newline
School of Mathematics and Statistics, Southwest University,
Chongqing 400715, China.\newline
School of Mathematics and Computational Science,
Zunyi Normal College, Zunyi, \newline
Guizhou 563002, China}
\email{liaojiafeng@163.com}
\address{Chun-Lei Tang (corresponding author)\newline
School of Mathematics and Statistics, Southwest University,
Chongqing 400715, China}
\email{tangcl@swu.edu.cn, Phone +86 23 68253135, Fax +86 23 68253135}
\thanks{Submitted January 17, 2015. Published April 7, 2015.}
\subjclass[2000]{35J20, 35J61}
\keywords{ Semilinear elliptic equation; zero mass; Nehari manifold;
\hfill\break\indent ground state solution}
\begin{abstract}
In this article, we study the semilinear elliptic equation
\begin{gather*}
-\Delta u=|u|^{p(x)-2}u, \quad x\in \mathbb{R}^N\\
u\in D^{1,2}(\mathbb{R}^N),
\end{gather*}
where $N\geq3$,
$p(x)=\begin{cases}
p, &x\in\Omega\\
2^*, &x\not\in\Omega,
\end{cases}$
with $2
0$
such that $t_uu\in\mathcal{N}$.
\end{lemma}
\begin{proof}
For any $u\in D^{1,2}(\mathbb{R}^N)\backslash\{0\}$, define
\[
f(t):=I(tu)=\frac{t^2}{2}\|u\|_{D^{1,2}(\mathbb{R}^N)}^2
-\frac{t^p}{p}\int_{\Omega}|u|^{p}\,dx-\frac{t^{2^*}}{2^*}
\int_{\mathbb{R}^N\backslash\Omega}|u|^{2^*}\,dx,\quad \forall t\in(0,+\infty).
\]
Then one has
\[
f'(t)t=\langle I'(tu),tu\rangle=t^2\|u\|_{D^{1,2}(\mathbb{R}^N)}^2
-t^{p}\int_{\Omega}|u|^{p}\,dx-t^{2^*}\int_{\mathbb{R}^N\backslash\Omega}|u|^{2^*}\,dx.
\]
Combining $2
0$ for $t>0$ small enough and $f'(t)t<0$
for $t>0$ large enough. Thus there exists $t_u>0$ such that
$f'(t_u)t_u=\langle I'(t_uu),t_uu\rangle=0$. That is $t_uu\in\mathcal{N}$.
The proof is complete.
\end{proof}
\begin{lemma} \label{lem3.3}
Assume that $N\geq3$, $2
0$.
\end{lemma}
\begin{proof}
For any $u\in\mathcal{N}$, one has
\begin{align*}
\|u\|_{D^{1,2}(\mathbb{R}^N)}^2
&= \int_{\Omega}|u|^{p}\,dx+\int_{\mathbb{R}^N\backslash\Omega}|u|^{2^*}\,dx\\
&\leq C\|u\|_{D^{1,2}(\mathbb{R}^N)}^p+C\|u\|_{D^{1,2}(\mathbb{R}^N)}^{2^*},
\end{align*}
which implies that there exists $\alpha>0$ such that
\begin{equation} \label{formula8}
\|u\|_{D^{1,2}(\mathbb{R}^N)}\geq\alpha,\quad \forall u\in\mathcal{N}.
\end{equation}
Thus for any $u\in\mathcal{N}$, we have
\begin{equation}\label{formula10}
\begin{aligned}
I(u)&= I(u)-\frac{1}{p}\langle I'(u),u\rangle\\
&= \big(\frac{1}{2}-\frac{1}{p}\big)\|u\|_{D^{1,2}(\mathbb{R}^N)}^2
+\big(\frac{1}{p}-\frac{1}{2^*}\big)\int_{\mathbb{R}^N\backslash\Omega}|u|^{2^*}\,dx\\
&\geq \big(\frac{1}{2}-\frac{1}{p}\big)\|u\|_{D^{1,2}(\mathbb{R}^N)}^2\\
&\geq \big(\frac{1}{2}-\frac{1}{p}\big)\alpha^2.
\end{aligned}
\end{equation}
Hence $m>0$. The proof is complete.
\end{proof}
\begin{lemma} \label{lem3.4}
Assume that $N\geq3$, $2
0$. Since $\Omega$ is bounded,
there exists $R>0$ such that $\Omega\subset B_R:=\{x\in\mathbb{R}^N:|x|0$ such that $B_r(z_0):=\{x\in\mathbb{R}^N:|x-z_0|0$.
This completes the proof.
\end{proof}
\subsection*{Acknowledgments}
The authors would like to thank Professor C. O. Alves for his suggestioins.
This research was supported by the National Natural Science Foundation of China
(No. 11471267), by the Fundamental Research Funds for
the Central Universities (No. XDJK2015D015), and by the
Natural Science Foundation of Education of Guizhou Province
(No. LKZS[2014]22)
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\end{document}