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http://dx.doi.org/10.4134/JKMS.2016.53.2.247

MULTIPLICITY RESULTS FOR NONLINEAR SCHRÖDINGER-POISSON SYSTEMS WITH SUBCRITICAL OR CRITICAL GROWTH  

Guo, Shangjiang (College of Mathematics and Econometrics Hunan University)
Liu, Zhisu (School of Mathematics and Physics University of South China)
Publication Information
Journal of the Korean Mathematical Society / v.53, no.2, 2016 , pp. 247-262 More about this Journal
Abstract
In this paper, we consider the following $Schr{\ddot{o}}dinger$-Poisson system: $$\{\begin{array}{lll}-{\Delta}u+u+{\lambda}{\phi}u={\mu}f(u)+{\mid}u{\mid}^{p-2}u,\;\text{ in }{\Omega},\\-{\Delta}{\phi}=u^2,\;\text{ in }{\Omega},\\{\phi}=u=0,\;\text{ on }{\partial}{\Omega},\end{array}$$ where ${\Omega}$ is a smooth and bounded domain in $\mathbb{R}^3$, $p{\in}(1,6]$, ${\lambda}$, ${\mu}$ are two parameters and $f:\mathbb{R}{\rightarrow}\mathbb{R}$ 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.
Keywords
$Schr{\ddot{o}}dinger$-Poisson system; subcritical growth; critical growth; variational methods;
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