Symmetry and nonuniformly elliptic operators

Abstract

The goal of this paper is to study the symmetry properties of nonnegative solutions of elliptic equations involving a nonuniformly elliptic operator. We consider on a ball the solutions of \[ \Delta_pu+f(u)=0 \] with zero Dirichlet boundary conditions, for $p>2$, where $\Delta_p$ is the $p$-Laplace operator and $f$ a continuous nonlinearity. The main tools are a comparison result for weak solutions and a local moving-plane method which has been previously used in the $p=2$ case. We prove local and global symmetry results when $u$ is of class $C^{1,\gamma}$ for $\gamma$ large enough, under some additional technical assumptions.