Ground state solutions for asymptotically periodic Schrodinger-Poisson systems in R^2

Abstract

This article concerns the planar Schrodinger-Poisson system $$\displaylines{ -\Delta u+V(x)u+\phi u=f(x,u), \quad x\in \mathbb{R}^2,\cr \Delta \phi= u^2, \quad x\in \mathbb{R}^2, } $$ where V(x) and f(x, u) are periodic or asymptotically periodic in x. By combining the variational approach, the non-Nehari manifold approach and new analytic techniques, we establish the existence of ground state solutions for the above problem in the periodic and asymptotically periodic cases. In particular, in our study, f is not required to satisfy the Ambrosetti-Rabinowitz type condition or the Nehari-type monotonic condition.

Keywords

• Planar Schrodinger-Poisson system
• ground state solution
• Logarithmic convolution potential