Standing waves for nonlinear Schrödinger-Poisson equation with high frequency

Abstract

We study the existence of ground state andbound state for the following Schrödinger-Poisson equation\begin{equation}\nonumber\begin{cases} -\Delta u + V(x) u+ \lambda\phi (x) u =\mu u+|u|^{p-1}u, & x\in \mathbb{R}^3, \\ -\Delta\phi = u^2, \quad \lim\limits_{|x|\to +\infty}\phi (x)=0,\end{cases}\leqno{(\mathrm{P})}\end{equation}where $p\in(3,5)$, $\lambda > 0$, $V\inC(\mathbb{R}^3,\mathbb{R}^+)$ and $\lim\limits_{|x|\to+\infty}V(x)=\infty$. By using variational method, we prove thatfor any $\lambda > 0$, there exists $\delta_1(\lambda) > 0$ such thatfor $\mu_1 < \mu < \mu_1 + \delta_1(\lambda)$, problem (P) has a nonnegativeground state with negative energy, which bifurcates from zero solution; problem (P) has a nonnegative bound state withpositive energy, which can not bifurcate from zero solution. Here $\mu_1$ is the first eigenvalue of $-\Delta+V$. Infinitely many nontrivial bound states are also obtained withthe help of a generalized version of symmetric mountain passtheorem.