Liouville theorems for stable solutions to the mean curvature equation

Abstract

We study stable solutions of the mean curvature equation $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) =|u|^{p-1}u $$ in domains of $\mathbb R^n$. We establish Liouville-type results in three settings. First, for globally stable solutions in $\mathbb R^n$, we prove that $u\equiv0$ if $1 2$ and $p=p_S$, where $p_S$ is the Sobolev critical exponent, then $u\equiv0$. Third, for solutions of the Dirichlet problem on connected unbounded domains with smooth boundary satisfying an epigraph-type geometric condition, we prove that if $n>2$, $1<p\le p_S$, and the solution is stable outside a compact set, then $u\equiv0$. The proofs are based on a gradient estimate for stable solutions, an integrability result, conservation laws and Pohozaev-type identities, and a unique continuation argument. These results extend Liouville-type theorems for stable solutions from the semilinear Lane--Emden equation to the mean curvature equation.