Stable solutions to $p$-Laplacian type equations with variable coefficients

Abstract

We study stable regular solutions of the variable-coefficient equation $-\operatorname{div}((A(x)\nabla u\cdot\nabla u)^{p/2-1}A(x)\nabla u)=f(u)$ in $B_1\subset\mathbb{R}^n$, where $p>1$, $f\in C^1(\mathbb{R})$ is nonnegative, and $A$ is a locally Lipschitz, symmetric, uniformly elliptic matrix field. We prove quantitative interior Hölder estimates for $n<p+4p/(p-1)$ when $p\ge2$, and for $n<5p$ when $1<p<2$. We also obtain interior energy estimates and higher integrability of the gradient in every dimension. These estimates are given in terms of $\|u\|_{L^1(B_1)}$, with constants depending only on $n,p$, the ellipticity bounds, and a local $L^\infty$ bound for $DA$, but not on $f$ or higher derivatives of $A$. These results extend the corresponding interior estimates of Cabré, Miraglio, and Sanchón for the $p$-Laplace equation to this variable-coefficient setting. The proof uses weighted Hessian estimates to control coefficient errors and a comparison between radial and full gradient energies.