STANDING WAVES FOR NONLINEAR SCHRODINGER-POISSON EQUATION WITH HIGH FREQUENCY

We study the existence of ground state and bound state for the following Schrodinger-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.