Unique continuation, nonexistence and bubbling for the critical $p$-Laplace equation II: the degenerate elliptic case and higher dimensions

Abstract

For every $p>1$, we prove that a nontrivial solution of \begin{equation*} -Δ_pu=f(x,u,\nabla u) \end{equation*} in a planar domain, with $|f(x,s,ξ)|\leq C(|s|^{p-1}+|ξ|^{p-1})$, vanishes neither on an open set nor to infinite order at a point. This allows us to prove unique continuation properties of axially symmetric solutions to $p$-Laplace equations in higher dimensions $N\geq3$, with axially symmetric $p$-harmonic functions as a particular case. Unlike in our previous work, the analysis is based here on the regularity and quasiconformal properties of the whole stress field $|\nabla u|^{p-2}\nabla u$ in the plane, taking into account the more delicate regularity theory for $p>2$. This allows us to prove Liouville-type and bubble profile decomposition results for critical problems in dimensions $N\geq3$ and in the whole range $1<p<N$, under some symmetry conditions.