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

## TL;DR

This paper establishes the existence and concentration of positive solutions for a class of nonlinear Kirchhoff equations with critical nonlinearity, without requiring the monotonicity of the nonlinearity or the Ambrosetti-Rabinowitz condition.

## Contribution

It constructs localized bound state solutions concentrating at minima of the potential without assuming monotonicity or Ambrosetti-Rabinowitz conditions.

## Key findings

- Solutions concentrate at local minima of V as epsilon approaches zero
- Existence of solutions under general conditions on f, M, and V
- No need for monotonicity of f(s)/s or Ambrosetti-Rabinowitz condition

## Abstract

We are concerned with the following Kirchhoff type equation $$-\varepsilon^2 M \left(\varepsilon^{2-N} \int_{\mathbb{R}^N} | \nabla u|^2\, \mathrm{d} x\right) \Delta u+V(x)u = f(u),\ x \in \mathbb{R}^N,\ \ N\ge2, $$ where $M \in C(\mathbb{R}^+,\mathbb{R}^+)$, $V\in C(\mathbb{R}^N,\mathbb{R}^+)$ 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 $\varepsilon\to 0$ 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.