Multiplicity and concentration of solutions for a fractional $p$-Kirchhoff type equation

TL;DR

This paper investigates the existence, multiplicity, and concentration of solutions for a fractional p-Kirchhoff equation with critical Sobolev exponent, employing penalization and topological methods for small parameters.

Contribution

It introduces new results on solutions to a fractional p-Kirchhoff problem with critical growth, using variational and topological techniques.

Findings

Existence of solutions for small epsilon

Multiple solutions under certain conditions

Solutions concentrate as epsilon approaches zero

Abstract

This paper is concerned with the following fractional $p$-Kirchhoff equation \begin{eqnarray*} \varepsilon ^{sp}M\left( {\varepsilon ^{sp - N}}\iint_{\mathbb{R}^{2N}}\frac{{{{\left| {u(x) - u(y)} \right|}^p}}}{{{{\left| {x - y} \right|}^{N + sp}}}}dxdy\right)(-\Delta)_p^su + V(x){u^{p - 1}} = {u^{p_s^* - 1}}+f(u),\ \ u>0, \ \mbox{in}\ {\mathbb{R}^N}, %u \in {W^{s,p}}(\mathbb{R}^N), \end{eqnarray*} where $\varepsilon>0$ is a parameter, $M(t)=a+bt^{\theta-1}$ with $a>0$, $b>0$, $\theta>1$, $(-\Delta)_p^s$ denotes the fractional $p$-Laplacian operator with $0<s<1$ and $1<p<\infty$, $N>sp$, $\theta p<p_s^*$ with $p_s^*=\frac{Np}{N-sp}$ is the fractional critical Sobolev exponent, $f$ is a superlinear continuous function with subcritical growth and $V$ is a positive continuous potential. Using penalization method and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of nontrivial solutions for $\varepsilon>0$ small enough.