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Multiplicity results for nonlinear Schr\"{o}dinger-Poisson systems with subcritical or critical growth

Shangjiang Guo|Zhisu Liu
대한수학회 (The Korean Mathematical Society)|2016.03|대한수학회지 (Journal of the KMS)|제53권 제2호|pp. 247-262

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키워드

variational methodsSchr\"{o}dinger-Poisson systemcritical growthsubcritical growth
주제분류수학

초록

In this paper, we consider the following Schr\"{o}dinger-Poisson system: $$ \left\{ \begin{array}{ll} -\triangle {u}+ u+\lambda\phi u =\mu f(u)+|u|^{p-2}u, & \mbox{in}\,\,\Omega, \\ -\triangle {\phi}=u^2, & \mbox{in}\,\, \Omega,\\ \phi=u=0, & \mbox{on}\,\, \partial\Omega, \end{array} \right. $$ where $\Omega$ is a smooth and bounded domain in $\R^3$, $p\in (1,6]$, $\lambda,\mu$ are two parameters and $f:\R\rightarrow\R$ is a continuous function. Using some critical point theorems and truncation technique, we obtain three multiplicity results for such a problem with subcritical or critical growth.

Abstract

In this paper, we consider the following Schr\"{o}dinger-Poisson system: $$ \left\{ \begin{array}{ll} -\triangle {u}+ u+\lambda\phi u =\mu f(u)+|u|^{p-2}u, & \mbox{in}\,\,\Omega, \\ -\triangle {\phi}=u^2, & \mbox{in}\,\, \Omega,\\ \phi=u=0, & \mbox{on}\,\, \partial\Omega, \end{array} \right. $$ where $\Omega$ is a smooth and bounded domain in $\R^3$, $p\in (1,6]$, $\lambda,\mu$ are two parameters and $f:\R\rightarrow\R$ is a continuous function. Using some critical point theorems and truncation technique, we obtain three multiplicity results for such a problem with subcritical or critical growth.

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