Inner regularity and Liouville theorems for stable solutions to the mean curvature equation

Abstract

Let $f\in C^1(\mathbb{R})$. We study stable solutions $u$ of the mean curvature equation \[ \operatorname{div}\left( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}} \right) = -f(u) \qquad \text{in}\ \Omega \subset \mathbb{R}^n. \] In the local setting we prove that $\nabla u$ satisfies inner Morrey regularity $M^{p_n}$, where \[ p_n := \left\{ \begin{array}{ll} n,\qquad & \text{if}\ 2\leq n\leq 5, \\ \frac{n}{n-4\sqrt{n-1}+4},\qquad & \text{if}\ n\geq 6, \end{array} \right. \] together with the estimate \[ \|\nabla u\|_{M^{p_n}(B_1)} \leq C \left( 1+\|\nabla u\|_{L^1(B_2)} \right). \] The exponent $p_n$ is optimal for $n\leq5$, as shown by an explicit one-dimensional example. For radial solutions we show that the symmetry center is at most a removable singularity. Globally, we establish Liouville-type theorem: any stable solution satisfying the growth condition \[ |\nabla u(x)| = \left\{ \begin{array}{lll} o(|x|^{-1}) \ & \text{as}\ |x|\rightarrow +\infty& \text{when}\ 2\leq n\leq 10, \\ o(|x|^{-n/2+\sqrt{n-1}+1}) \ & \text{as}\ |x|\rightarrow +\infty& \text{when}\ n\geq 11, \end{array} \right. \] must be constant. In particular, no nonconstant radial stable solution exists in dimensions \(2\leq n\leq6\), which highlights a global rigidity of stable radial solutions in low dimensions and extend the classical Liouville theorem of Farina and Navarro. Several exponents appearing in our results are new for mean curvature equations, showing both similarities and differences with the corresponding theorems for semilinear equations.