Symmetry of Mountain Pass Solutions of Some Vector Field Equations |
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Authors: | Orlando Lopes Marcelo Montenegro |
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Institution: | (1) Departamento de Matemática, Universidade Estadual de Campinas, IMECC, Caixa Postal 6065, CEP 13083-970 Campinas, SP, Brasil |
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Abstract: | We prove radial symmetry (or axial symmetry) of the mountain pass solution of variational elliptic systems − AΔu(x) + ∇ F(u(x)) = 0 (or − ∇.(A(r) ∇ u(x)) + ∇ F(r,u(x)) = 0,) u(x) = (u
1(x),...,u
N
(x)), where A (or A(r)) is a symmetric positive definite matrix. The solutions are defined in a domain Ω which can be
, a ball, an annulus or the exterior of a ball. The boundary conditions are either Dirichlet or Neumann (or any one which is invariant under rotation). The mountain pass solutions studied here are given by constrained minimization on the Nehari manifold. We prove symmetry using the reflection method introduced in Lopes (1996), J. Diff. Eq. 124, 378–388; (1996), Eletron. J. Diff. Eq. 3, 1–14]. |
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Keywords: | Vector field equations mountain pass radial symmetry axial symmetry |
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