The multiplicity of solutions for the critical problem involving the fracional p-Laplacian operator

Abstract

This paper deals with the existence of multiple solutions

for the following critical fractional $p$-Laplacian problem

\begin{equation*}

\left\{

\begin{array}{l}

\mathbf{(-}\Delta \mathbf{)}_{p}^{s}u(x)=\lambda \left\vert u\right\vert

^{p-2}u+f(x,u)+\mu g(x,u)\ \text{in }\Omega ,u>0, \\

\\

u=0\text{ on}\ \mathbb{R}^{n}\setminus \Omega ,%

\end{array}%

\right.

\end{equation*}%

where $p>1$, $s\in (0,1)$, $\Omega \subset \mathbb{R}^{n}(n>ps),$ be a bounded smooth domain, $\lambda $, $\mu $ are positive parameters and the functions $f,g:\overline{%

\Omega }\times \lbrack 0,\infty )\longrightarrow [0,\infty),$ are continuous and differentiable with respect to the second variable. Our main tools are based on variational methods combined with a classical concentration

compacteness method.