# Concentration phenomena for a fractional Schr\"odinger-Kirchhoff type equation

## TL;DR

This paper studies the existence, multiplicity, and concentration of positive solutions for a fractional Schr"odinger-Kirchhoff equation, revealing how solutions cluster around the potential's minimum points as a small parameter approaches zero.

## Contribution

It introduces a novel approach combining penalization techniques and Ljusternik-Schnirelmann theory to analyze solution multiplicity related to the potential's topology.

## Key findings

- Multiple positive solutions concentrate near potential minima.
- Solution count relates to the topological complexity of the minimum set.
- The methods handle fractional operators with superlinear nonlinearities.

## Abstract

In this paper we deal with the multiplicity and concentration of positive solutions for the following fractional Schr\"odinger-Kirchhoff type equation \begin{equation*} M\left(\frac{1}{\varepsilon^{3-2s}} \iint_{\mathbb{R}^{6}}\frac{|u(x)- u(y)|^{2}}{|x-y|^{3+2s}} dxdy + \frac{1}{\varepsilon^{3}} \int_{\mathbb{R}^{3}} V(x)u^{2} dx\right)[\varepsilon^{2s} (-\Delta)^{s}u+ V(x)u]= f(u) \, \mbox{in} \mathbb{R}^{3} \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (\frac{3}{4}, 1)$, $(-\Delta)^{s}$ is the fractional Laplacian, $M$ is a Kirchhoff function, $V$ is a continuous positive potential and $f$ is a superlinear continuous function with subcritical growth. By using penalization techniques and Ljusternik-Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum.