Existence and concentration of positive ground states for Schrodinger-Poisson equations with competing potential functions

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.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/78/abstr.html