Branches of Radial Solutions for Semipositone Problems

Abstract

We consider the radially symmetric solutions to the equation −Δu(x) = λƒ(u(x)) for x ∈ Ω, u(x) = 0 for x ∈ ∂Ω, where Ω denotes the unit ball in R N ( N > 1), centered at the origin and λ > 0. Here ƒ: R → R is assumed to be semipositone (ƒ(0) < 0), monotonically increasing, superlinear with subcritical growth on [0, ∞). We establish the structure of radial solution branches for the above problem. We also prove that if ƒ is convex and ƒ( t)/( tƒ′( t)−ƒ( t)) is a nondecreasing function then for each λ > 0 there exists at most one positive solution u such that (λ, u) belongs to the unbounded branch of positive solutions. Further when ƒ( t) = t p − k, k > 0 and 1 < p < ( N + 2)/( N − 2), we prove that the set of positive solutions is connected. Our results are motivated by and extend the developments in [4].