Liouville theorems for stable solutions to the mean curvature equation

We study stable solutions of the mean curvature equation $$-\mathrm{div}\left(\frac{\mathrm{\nabla }u}{\sqrt{1+\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2}}\right)=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$$ 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 $u\equiv 0$ if $1<p<p_{JL}$, where $p_{JL}$ is the Joseph–Lundgren exponent. Second, for solutions in ${\mathbb{R}}^n$ that are stable outside a compact set, we show that if $n>2$ and $p=p_S$, where $p_S$ is the Sobolev critical exponent, then $u\equiv 0$. 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 $u\equiv 0$. 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.