Entire nodal solutions to the pure critical exponent problem for the $p$-Laplacian

Abstract

We establish the existence of multiple sign-changing solutions to the quasilinear critical problem $$-{\Delta }_{p}u=|u{|}^{p^{\ast }-2}u,u\in D^{1,p}({\mathbb{R}}^N),$$ for $N\ge 4$, where ${\Delta }_{p}u:=\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)$ is the $p$-Laplace operator, $1<p<N$ and $p^{\ast }:=\frac{Np}{N-p}$ is the critical Sobolev exponent