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Multiplicity of solutions for a class of critical Schrödinger-Poisson system with two parameters (2007.13830v1)
Published 27 Jul 2020 in math.AP
Abstract: We study a class of critical Schr\"odinger-Poisson system of the form \begin{equation*} \begin{cases} -\Delta u+\lambda V(x)u+\phi u=\mu |u|{p-2}u+|u|{4}u& \quad x\in \mathbb{R}3,\ -\Delta \phi=u2&\quad x\in \mathbb{R}3,\ \end{cases} \end{equation*} where $\lambda, \mu>0$ are two parameters, $p\in(4,6)$ and $V$ satisfies some potential well conditions. By using the variational arguments, we prove the existence of positive ground state solutions for $\lambda$ large enough and $\mu>0$, and their asymptotical behavior as $\lambda\to\infty$. Moreover, by using Ljusternik-Schnirelmann theory, we obtain the existence of multiple positive solutions if $\lambda$ is large and $\mu$ is small.