Symmetry properties of positive solutions for fully nonlinear elliptic systems

Abstract

We investigate symmetry properties of positive solutions for fully nonlinear uniformly elliptic systems, such as $$F_i(x,D{u}_i,D^2{u}_i)+f_i(x,u_1,\dots ,u_n,D{u}_i)=0,1\le i\le n,$$ in a bounded domain $\Omega$ in ${\mathbb{R}}^N$ with Dirichlet boundary condition $u_1=\dots ,u_n=0$ on $\partial \Omega$. Here, $f_i$'s are nonincreasing with the radius $r=|x|$, and satisfy a cooperativity assumption. In addition, each $f_i$ is the sum of a locally Lipschitz with a nondecreasing function in the variable $u_i$, and may have superlinear gradient growth. We show that symmetry occurs for systems with nondifferentiable $f_i$'s by developing a unified treatment of the classical moving planes method in the spirit of Gidas-Ni-Nirenberg. We also present different applications of our results, including uniqueness of positive solutions for Lane-Emden systems in the subcritical case in a ball, and symmetry for a class of systems with natural growth in the gradient.