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 $$-\Delta_{p} u=|u|^{p^*-2}u, \qquad u\in D^{1,p}(\mathbb{R}^{N}),$$ for $N\geq4$, where $\Delta_{p}u:=\mathrm{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplace operator, $1<p<N$ and $p^*:=\frac{Np}{N-p}$ is the critical Sobolev exponent