On positive solutions of semilinear elliptic equations

Abstract

This paper is concerned with necessary conditions for the existence of positive solutions of the semilinear problem Δ u + f ( u ) = 0 , x ∈ Ω , u = 0 , x ∈ ∂ Ω \Delta u + f(u) = 0,x \in \Omega ,u = 0,x \in \partial \Omega , whose supremum norm bears a certain relationship to zeros of the nonlinearity f f . We first discuss the smooth case (i.e., f f and ∂ Ω \partial \Omega smooth) and then show how to obtain similar results in the nonsmooth case.