Abstract

We investigate symmetry properties of positive solutions for fully nonlinear uniformly elliptic systems, such as F-i (x, Du(i), D(2)u(i)) + f(i) (x, u(1),..., u(n), Du(i)) = 0, 1 = i = n, in a bounded domain Omega in R-N with Dirichlet boundary condition u(1)=..., u(n)= 0 on partial derivative Omega. Here, f(i)'s are nonincreasing with the radius r=vertical bar x vertical bar, and satisfy a cooperativity assumption. In addition, each f(i) is the sum of a locally Lipschitz with a nondecreasing function in the variable ui, 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. (c) 2020 Elsevier Inc. All rights reserved.