Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 78, pp. 1-19.

Title: Existence and concentration of positive ground states for Schrodinger-Poisson
       equations with competing potential functions
       
Authors: Wenbo Wang (Yunnan Univ., Kunming,650500, Yunnan, China)
         Quanqing Li (Honghe Univ., Mengzi, 661100, Yunnan, China)

Abstract:
 This article concerns the Schrodinger-Poisson equation
 $$\displaylines{
 -\varepsilon^2\Delta u+V(x)u+K(x)\phi u=P(x)|u|^{p-1}u+Q(x) |u|^{q-1}u,\quad
 x\in\mathbb{R}^3,\cr
 -\varepsilon^2\Delta \phi=K(x)u^2,\quad x\in\mathbb{R}^3,
 }$$
 where $3<q<p<5=2^{\ast}-1$. We prove that for all $\varepsilon>0$, the equation
 has a ground state solution. The methods used here are based on the
 Nehari manifold and the concentration-compactness principle.
 Furthermore, for 
\varepsilon>0
 small, these ground states concentrate at
 a global minimum point of the least energy function.
 
Submitted January 11, 2019. Published July 22, 2020.
Math Subject Classifications: 35J15, 35J20, 35J50.
Key Words: Schrodinger-Poisson equation; Nehari manifold; ground states;
           concentration-compactness; concentration.