Standing waves for nonlinear Schrödinger-Poisson equation with high frequency

Abstract

We study the existence of ground state and
bound state for the following Schrödinger-Poisson equation
where $p\in(3,5)$, $\lambda > 0$, $V\in
C(\mathbb{R}^3,\mathbb{R}^+)$ and $\lim\limits_{|x|\to
+\infty}V(x)=\infty$. By using  variational method, we prove that
for any $\lambda > 0$, there exists $\delta_1(\lambda) > 0$ such that
for $\mu_1 < \mu < \mu_1 + \delta_1(\lambda)$, problem (P) has  a nonnegative
ground state with negative energy, which bifurcates from zero solution; problem (P) has a nonnegative bound state with
positive 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 with
the help of a generalized version of symmetric mountain pass
theorem.