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

摘要

We are concerned with the following Kirchhoff type equation $$-\varepsilon^2M \left(\varepsilon^{2-N} \int_{\mathbb{R}^N} | \nabla u|^2\, \mathrm{d}x\right) \Delta u+V(x)u = f(u),\ x \in \mathbb{R}^N,\ \ N\ge2, $$ where $M \inC(\mathbb{R}^+,\mathbb{R}^+)$, $V\in C(\mathbb{R}^N,\mathbb{R}^+)$ and $f(s)$is of critical growth. In this paper, we construct a localized bound statesolution concentrating at a local minimum of $V$ as $\varepsilon\to 0$ undercertain conditions on $f(s)$, $M$ and $V$. In particular, the monotonicity of$f(s)/s$ and the Ambrosetti-Rabinowitz condition are not required.