Positive solution curves of semipositone problems with concave nonlinearities

Abstract

SynopsisWe consider the positive solutions to the semilinear equation:where Ω denotes a smooth bounded region in ℝN(N> 1) and ℷ 0. Heref:[0, ∞)→ℝ is assumed to be monotonically increasing, concave and such thatf(0)<0 (semipositone). Assuming thatf′(∞)≡limt→∞f(t)> 0, we establish the stability and uniqueness of large positive solutions in terms of (f(t)/t)′ When Ω is a ball, we determine the exact number of positive solutions for each λ > 0. We also obtain the geometry of the branches of positive solutions completely and establish how they evolve. This work extends and complements that of [3, 7] wheref′(∞)≦0.