Existence and concentration result for a quasilinear Schrödinger equation with critical growth

Abstract

This paper is concerned with the concentration of positive ground states solutions for a modified Schrodinger equation $$\begin{aligned} -\varepsilon ^{2}\triangle u+V(x)u-\varepsilon ^{2}\triangle (u^{2})u=K(x)|u|^{p-2}u+|u|^{22^{*}-2}u,\quad \text{ in } \; \mathbb {R}^{N}, \end{aligned}$$ where $$4 0$$ is a parameter and $$2^{*}:=\frac{2N}{N-2}(N\ge 3)$$ is the critical Sobolev exponent. We prove the existence of a positive ground state solution $$v_{\varepsilon }$$ and $$\varepsilon $$ sufficiently small under some suitable conditions on the nonnegative functions V(x) and K(x). Moreover, $$v_{\varepsilon }$$ concentrates around a global minimum point of V as $$\varepsilon \rightarrow 0$$ . The proof of the main result is based on minimax theorems and concentration compact theory.