Schrödinger-Poisson system without growth and the Ambrosetti-Rabinowitz conditions

Abstract

We consider the following Schrodinger-Poisson system $ \left\{ \begin{array}{l} {\rm{ - }}\Delta u + V\left(x \right)u + \phi u = \lambda f\left(u \right)\; \; \; \; \; {\rm{in}}\; {\mathbb{R}^3}, \\ - \Delta \phi = {u^2}, \mathop {\lim }\limits_{|x| \to + \infty } \phi = 0, \; \; \; \; \; \; \; \; \; \; \; \; {\rm{in}}\; {\mathbb{R}^3}. \end{array} \right. $ Unlike most other papers on this problem, the Schrodinger-Poisson system without any growth and Ambrosetti-Rabinowitz condition is considered in this paper. Firstly, by Jeanjean's monotonicity trick and the mountain pass theorem, we prove that the problem possesses a positive solution for large value of $\lambda$. Secondly, we establish the multiplicity of solutions via the symmetric mountain pass theorem.