Symmetry properties of positive solutions for fully nonlinear elliptic systems

We investigate symmetry properties of positive solutions for fully nonlinear uniformly elliptic systems, such asFi(x,Dui,D2ui)+fi(x,u1,…,un,Dui)=0,1≤i≤n, in a bounded domain Ω in RN with Dirichlet boundary condition u1=…,un=0 on ∂Ω. Here, fi's are nonincreasing with the radius r=|x|, and satisf...

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Bibliographic Details
Published in:Journal of Differential Equations 2020-08, Vol.269 (5), p.4175-4191
Main Authors: Moreira dos Santos, Ederson, Nornberg, Gabrielle
Format: Article
Language:English
Subjects:
Elliptic system of fully nonlinear equations
Moving planes
Positive solution
Symmetry
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Summary:We investigate symmetry properties of positive solutions for fully nonlinear uniformly elliptic systems, such asFi(x,Dui,D2ui)+fi(x,u1,…,un,Dui)=0,1≤i≤n, in a bounded domain Ω in RN with Dirichlet boundary condition u1=…,un=0 on ∂Ω. Here, fi's are nonincreasing with the radius r=|x|, and satisfy a cooperativity assumption. In addition, each fi 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 fi'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.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2020.03.023