Radial symmetry and applications for a problem involving the $-\Delta_p(\cdot)$ operator and critical nonlinearity in~$\mathbb{R}^N$

TL;DR

This paper proves that all solutions to a critical p-Laplace equation in ^N are radially symmetric and decreasing when the nonlinearity is locally Lipschitz continuous, extending symmetry results to a singular case.

Contribution

It establishes radial symmetry of solutions for the critical p-Laplace equation in the singular case where 1<p<2, under Lipschitz continuity of the nonlinearity.

Findings

Solutions are radially symmetric and decreasing.

Symmetry holds for locally Lipschitz continuous nonlinearities.

Results extend symmetry understanding to singular p-Laplace equations.

Abstract

We consider weak non-negative solutions to the critical $p$-Laplace equation in $\mathbb{R}^N$, $-\Delta_p u =u^{p^*-1}$ in the singular case $1<p<2$. We prove that if the nonlinearity is locally Lipschitz continuous, namely $p^*\geqslant2$ then all the solutions in ${\mathcal D}^{1,p}(\R^N)$ are radial (and radially decreasing) about some point.