# Concentration phenomena for a class of fractional Kirchhoff equations in $\mathbb{R}^{N}$ with general nonlinearities

## TL;DR

This paper investigates the concentration behavior of positive solutions to fractional Kirchhoff equations with general nonlinearities in b^N, showing solutions localize at minima of the potential as a small parameter tends to zero.

## Contribution

It establishes the existence of solutions that concentrate at potential minima for a broad class of fractional Kirchhoff problems with general nonlinearities.

## Key findings

- Solutions concentrate at local minima of V as ps .
- Existence of positive solutions proven using variational methods.
- Results hold for subcritical and critical nonlinearities.

## Abstract

In this paper we study the following class of fractional Kirchhoff problems: \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}M(\varepsilon^{2s-N}[u]^{2}_{s})(-\Delta)^{s}u + V(x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $N\geq 2$, $(-\Delta)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a positive continuous function, $M: [0, \infty)\rightarrow \mathbb{R}$ is a Kirchhoff function satisfying suitable conditions and $f:\mathbb{R}\rightarrow \mathbb{R}$ fulfills Berestycki-Lions type assumptions of subcritical or critical type. Using suitable variational arguments, we prove the existence of a family of positive solutions $(u_{\varepsilon})$ which concentrates at a local minimum of $V$ as $\varepsilon\rightarrow 0$.