Symmetry of Ground States of Quasilinear Elliptic Equations

Abstract

. We consider the problem of radial symmetry for non‐negative continuously differentiable weak solutions of elliptic equations of the form ${\rm div}(A(\vert Du\vert)Du) + f(u) = 0,\quad x\in {\vec R}^n, \quad n\geq 2,\eqno(1)$ under the ground state condition $u(x)\to 0 \mbox{ as } \vert x\vert\to\infty. \eqno(2)$ Using the well‐known moving plane method of Alexandrov and Serrin, we show, under suitable conditions on A and f, that all ground states of (1) are radially symmetric about some origin O in ${\vec R}^n$. In particular, we obtain radial symmetry for compactly supported ground states and give sufficient conditions for the positivity of ground states in terms of the given operator A and the nonlinearity f.