Multiple solutions to Bahri-Coron problem involving fractional $p$-Laplacian in some domain with nontrivial topology

Abstract

In this article, we establish the existence of positive and multiple sign-changing solutions to the fractional $p$-Laplacian equation with purely critical nonlinearity \begin{equation} \label{Ppomegas-a}\tag{P$_{p,\Omega}^{s}$} \begin{cases} (-\Delta)_{p}^s u =|u|^{p_s^*-2} u& \text{in }\Omega, \\ u =0 & \text{on }\Omega^{c}, \end{cases} \end{equation} in a bounded domain $\Omega\subset \mathbb{R}^{N}$ for $s\in (0,1)$, $p\in (1,\infty)$, and the fractional critical Sobolev exponent $p^{*}_{s}={Np}/({N-sp})$ under some symmetry assumptions. We study Struwe's type global compactness results for the Palais-Smale sequence in the presence of symmetries.