Existence and concentration of positive solutions for nonlinear Kirchhoff type problems with a general critical nonlinearity

Abstract

We are concerned with the following Kirchhoff type equation $$-{\epsilon }^{2}M\left({\epsilon }^{2-N}{\int }_{{\mathbb{R}}^N}|\nabla u{|}^2\mathrm{d}x\right)\Delta u+V(x)u=f(u),x\in {\mathbb{R}}^N,N\ge 2,$$ where $M\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$, $V\in C({\mathbb{R}}^N,{\mathbb{R}}^{+})$ and $f(s)$ is of critical growth. In this paper, we construct a localized bound state solution concentrating at a local minimum of $V$ as $\epsilon \to 0$ under certain conditions on $f(s)$, $M$ and $V$. In particular, the monotonicity of $f(s)/s$ and the Ambrosetti-Rabinowitz condition are not required.