MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS SCHRÖDINGER-POISSON SYSTEMS WITH THE ASYMPTOTICAL NONLINEARITY IN $\mathbb{R}^3$

Abstract

In this paper, we study nonhomogeneous Schrödinger-Poisson systems \[ \begin{cases} -\Delta u + u + K(x) \phi(x) u = a(x) f(u) + h(x), & x \in \mathbb{R}^3, \\ -\Delta \phi = K(x) u^2, & x \in \mathbb{R}^3, \end{cases}\] where $f(t)$ is either asymptotically linear or asymptotically 3-linear with respect to $t$ at infinity. Under appropriate assumptions on $K, a, f$ and $ h$, the existence of two positive solutions of the above system is obtained by using the Ekeland's variational principle and the MountainPass Theorem in critical point theory.