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Existence of Nontrivial Solutions for Schrödinger–Poisson Systems with Critical Exponent on Bounded Domains

Article

Abstract

In this paper, we concern with the following Schrödinger–Poisson system:
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\phi u = f(x,u)+u^5 ,\quad &{} x\in \Omega ,\\ -\Delta \phi =u^2,\quad &{} x\in \Omega ,\\ u=\phi =0, \quad &{} x \in \partial \Omega , \end{array}\right. } \end{aligned}$$
where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^{3}\) and \(u^5\) reaches the Sobolev critical exponent since \(2^*=6\) in dimension 3. Under some appropriate assumptions on f, a new result on the existence of nontrivial solutions is obtained via variational methods.

Keywords

Schrödinger–Poisson system Variational methods Nontrivial solution 

Mathematics Subject Classification

35J20 35J60 

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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaPeople’s Republic of China

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