Three nontrivial solutions of a nonlocal problem involving critical exponent

Abstract

In this paper we will prove the existence of three nontrivial weak solutions of the following problem involving a nonlinear integro-differential operator and a term with critical exponent. \begin{align*} \begin{split} -\mathscr{L}_\Phi u & = |u|^{{p_{s}^{\ast}}-2}u+\lambda f(x,u)\,\,\mbox{in}\,\,\Omega,\\ u & = 0\,\, \mbox{in}\,\, \mathbb{R}^N\setminus \Omega, \end{split} \end{align*} Here $q\in(p, p_s^*)$, where $p_s^*$ is the fractional Sobolev conjugate of $p$ and $-\mathscr{L}_\Phi $ represents a general nonlocal integro-differential operator of order $s\in(0,1)$. This operator is possibly degenerate and covers the case of fractional $p$-Laplacian operator.