Positive Solutions of Semilinear Elliptic Equations on General Domains

Abstract

The existence problem for positive solutions to the equation

$$ \Delta U\left( x \right) + {\lambda ^{2}}f\left( {U\left( x \right)} \right) = 0$$

((1))

with homogeneous Dirichlet boundary conditions on a smooth, bounded domain, Ω, has recently been of much interest; cf. the survey article by Lions [L]. All the theorems cited therein have required f(0) > 0 or f(0) = 0 and f′(0) > 0. However, it was shown in [SW1] that a necessary condition for symmetry breaking of positive solutions of (1) on a disk is that f(0) < 0. In this note we outline a general procedure for constructing positive solutions to (1) on Ω which do not require any assumptions on the behavior of f near 0, nor do we restrict ourselves to superlinear ((f(U)/U)′ > 0) or sublinear ((f(U)/U)′ < 0) functions; furthermore, we make no assumptions about the sectional curvature of boundary Ω; (cf. [L]).