Asymptotic behavior of solutions to the critical $p$-Laplace equation near isolated singularities

Abstract

In this paper, we study the asymptotic behavior of weak solutions to the following critical $p$-Laplace equation \[ -Δ_p u=u^{p^*-1}, \quad u\ge0 \quad\hbox{in }B_1\setminus\{0\}, \] where $Δ_pu=\diver(|\nabla u|^{p-2}\nabla u)$ is the classical $p$-Laplacian with $1<p<n$, $p^*=\frac{np}{n-p}$ is the Sobolev critical exponent and $B_1\setminus\{0\}\subset \mathbb{R}^n$ is the punctured unit ball. We develop a nonlinear potential-theoretic method to establish sharp upper bounds for solutions, without using the method of moving planes or moving spheres. Furthermore, we use Steiner symmetrization to study the radial symmetry of global singular solutions, and thereby classify all Fowler singular solutions. Finally, we prove that every local singular solution is asymptotic to a global Fowler singular solution.