Subsolutions: a Journey from Positone to Infinite Semipositone Problems

We discuss the existence of positive solutions to −∆u = λf(u) in Ω, with u = 0 on the boundary, where λ is a positive parameter, Ω is a bounded domain with smooth boundary ∆ is the Laplacian operator, and f : (0,∞) → R is a continuous function. We first discuss the cases when f(0) > 0 (positone), f(0) = 0 and f(0) < 0 (semipositone). In particular, we will review the existence of non-negative strict subsolutions. Along with these subsolutions and appropriate assumptions on f(s) for s 1 (which will lead to large supersolutions) we discuss the existence of positive solutions. Finally, we obtain new results on the case of infinite semipositone problems (lims→0+ f(s) = −∞).