Symmetry properties of positive solutions for fully nonlinear elliptic systems

2018/12/18 by Santos, Ederson Moreira dos, Nornberg, Gabrielle
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1812.07161

Abstract

We investigate symmetry properties of positive solutions for fully nonlinear uniformly elliptic systems, such as F_i (x,Du_i,D²u_i) +f_i (x,u₁, … , u_n,Du_i)=0, 1 ≤ i ≤ n, in a bounded domain Ω in ℝ^N with Dirichlet boundary condition u₁=…,u_n=0 on ∂Ω. 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.

Related

• Symmetry and monotonicity properties of singular solutions to some\n cooperative semilinear elliptic systems involving critical nonlinearities
• Symmetry of Convex Solutions to Fully Nonlinear Elliptic Systems: Bounded Domains
• Symmetry results for cooperative elliptic systems in unbounded domains
• Symmetry of solutions of some semilinear elliptic equations with singular nonlinearities
• Symmetry of Convex Solutions to Fully Nonlinear Elliptic Systems: Unbounded Domains