Existence and concentration of ground state solution to a nonlocal Schrödinger equation

Abstract

We study a class of Schrödinger-Kirchhoff system involving critical exponent. We aim to find suitable conditions to assure the existence of a positive ground state solution of Nehari-Pohouzaev type $u_{\epsilon }$ with exponential decay at infinity for $\epsilon$ and $u_{\epsilon }$ concentrates around a global minimum point of $V$ as $\epsilon \to 0^{+}.$ The nonlinear term includes the nonlinearity $f(u)\sim |u{|}^{p-1}u$ for the well-studied case $p\in [3,5)$, and the less-studied case $p\in (2,3)$.