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

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.