Abstract:

The purpose of this paper is to study the following Schrödinger-Poisson system

(P)$\begin{equation} \left\{\begin{array}{ll} -\Delta u + \Big(\omega-\sum\limits_{i=1}^m\frac{1}{|x-x_i|}\Big)u+\lambda\phi (x)u =f(u)\,\,\, &x\in \mathbb{R}^3, \\ -\Delta\phi = |u|^{2},\, \ &u\in H^1(\mathbb{R}^3), \end{array}\right. \end{equation}$

where $\omega>0$, $\lambda>0$, $x_i\in\mathbb{R}^3$, $m\in\mathbb{N}$, $f(u)\sim lu$ (as $u\rightarrow+\infty$) is the asymptotically linear term. We study the effect of values of parameters $\omega$, $\lambda$ and asymptotic coefficient $l$ on the existence of ground state, multiple solutions to system (P), by using the variational method.

Key words: Elliptic equation, Asymptotically linear, Variational method