A theorem of Radò’s type for the solutions of a quasi-linear equation

In this note, we prove a Radò type theorem for the solutions of the so-called $p$-harmonic equation $$\div(|\nabla u|^{p-2}\nabla u)=0$$ in any dimension. We show that a function $u\in C^1(\Omega)$ which is $p$-harmonic in $\Omega\setminus\{x\:u(x)=0\}$ is indeed $p$-harmonic in the whole domain $\Omega$. Our proof relies on a viscosity solution characterization of $p$-harmonic functions obtained in \cite{JLM}.

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Published 1 January 2004