Positive solutions of nonlinear elliptic boundary value problems

Abstract

This paper considers the problem of finding positive vector-valued solutions U of the nonlinear elliptic boundary value problem L( U) + f( x, U) = 0 on a bounded region Ω, U + ∂U ∂v = 0 on ∂Ω. The operator L is uniformly elliptic and in divergence form, and f is, roughly speaking, superlinear; by the positivity of U is meant the positivity of each component of U on Ω. Under certain growth conditions on f and some further technical assumptions, the existence of a positive solution is proved, an a priori bound on all positive solutions is obtained, and a certain fixed point index is proved equal to − 1. As an example, information about fixed point indices is used to allow perturbations of the form ϵh( x, U, DU). In the final section, an essentially best possible theorem is given for Ω a ball and for radially symmetric solutions of the Laplacian with Dirichlet boundary conditions.