### Standing waves for nonlinear Schrödinger-Poisson equation with high frequency

DOI: http://dx.doi.org/10.12775/TMNA.2015.028

#### 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.

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.

#### Keywords

Schrödinger-Poisson equation; ground states; bound states; variational method

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