We consider the positive solutions to the semilinear equation:
-Δu(x) = λf(u(x)) for x ∈ Ω
u(x) = 0 for x ∈ ∂Ω
where Ω denotes a smooth bounded region in RN (N > 1) and λ > 0. Here f :[0, ∞)→R is assumed to be monotonically increasing, concave and such that f(0) < 0 (semipositone). Assuming that f'(∞) ≡ lim t→∞ 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] where f'(∞) ≤ 0.
© 1997 Royal Society of Edinburgh
A. Castro, S. Gadam and R. Shivaji. “Positive solutions curves of semipositone problems with concave nonlinearities”, Proc. Royal Society of Edinburgh, Vol. 127A (1997), pp. 921-934.