Symmetry and Preservation of Nodal Structure in Elliptic Equations Satisfying Fully Nonlinear Neumann Boundary Condtions

Abstract

In this paper we study a general class of quasi-linear elliptic equations of the form

$$ a_{ij} (\nabla u,u)u_{x_i x_j } + g(\lambda \nabla u,u) = 0{\text{ }}in{\text{ }}\Omega $$

((1.1))

, where Ω ⊂ ℝ² is a bounded domain, u: Ω → ℝ, “∇(†)” denotes the gradient, and λ ε ℝ. On the boundary aΩ we impose fully nonlinear Neumann conditions:

$$ q(\nabla u,u) \cdot n|_{\partial \Omega } = 0 $$

((1.2))

, where q(∇u,u) ε ℝ², n denotes the outward unit normal to ∂Ω, and “•” is the usual inner (dot) product on R². Such problems arise naturally in continuum physics; (1.1) models a general class of nonlinear diffusion processes, and (1.2) is a zero-flux condition. In a recent work [5] we studied global bifurcation problems for (1.1) defined on all of Rⁿ in the presence of lattice symmetry. In particular, (given certain hypotheses, which we recapitulate in Section 2) we showed that the precise nodal configuration of the eigenfunction (of the linearized problem at the trivial solution) is preserved along the associated global bifurcating solution branch.