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

TL;DR

This paper proves the existence of multiple sign-changing solutions to a critical p-Laplacian problem in higher dimensions, advancing understanding of nonlinear PDEs with critical exponents.

Contribution

It establishes the existence of entire nodal solutions for the critical p-Laplacian problem in  dimensions, a significant extension of previous results.

Findings

Multiple sign-changing solutions exist for the critical p-Laplacian problem.

The solutions are entire and nodal, changing sign across the domain.

The results apply for dimensions Na0a0a4 with 1<p<N.

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