Existence of multiple solutions for a nonhomogeneoussemilinear elliptic equation involving critical exponent

Yinbin Deng; Shuangjie Peng; Li Wang

doi:10.3934/dcds.2012.32.795

Article Contents

2012, Volume 32, Issue 3: 795-826. Doi: 10.3934/dcds.2012.32.795

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Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent

1.
Department of Mathematics, Huazhong Normal University, Wuhan 430079
2.
Department of Mathematics, Central China Normal University, Wuhan 430079

Received: March 2010

Revised: August 2011

Published: March 2012

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Abstract

In this paper, we consider the following problem $$ \left\{ \begin{array}{ll} -\Delta u+u=u^{2^{*}-1}+\lambda(f(x,u)+h(x))\ \ \hbox{in}\ \mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),\ \ u>0 \ \hbox{in}\ \mathbb{R}^{N}, \end{array} \right. (\star) $$ where $\lambda>0$ is a parameter, $2^* =\frac {2N}{N-2}$ is the critical Sobolev exponent and $N>4$, $f(x,t)$ and $h(x)$ are some given functions. We prove that there exists $0<\lambda^{*}<+\infty$ such that $(\star)$ has exactly two positive solutions for $\lambda\in(0,\lambda^{*})$ by Barrier method and Mountain Pass Lemma and no positive solutions for $\lambda >\lambda^*$. Moreover, if $\lambda=\lambda^*$, $(\star)$ has a unique solution $(\lambda^{*}, u_{\lambda^{*}})$, which means that $(\lambda^{*}, u_{\lambda^{*}})$ is a turning point in $H^{1}(\mathbb{R}^{N})$ for problem $(\star)$.

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Mathematics Subject Classification: 35J65, 35J40.

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