Existence of a positive solution for a p-Laplacian semipositone problem, Boundary Value Problems 3

We consider the boundary value problem −∆pu = λ f (u) in Ω satisfying u = 0 on ∂Ω, where u = 0 on ∂Ω, λ> 0 is a parameter,Ω is a bounded domain in Rn with C2 boundary ∂Ω, and ∆pu: = div(|∇u|p−2∇u) for p> 1. Here, f: [0,r] → R is a C1 nondecreasing function for some r> 0 satisfying f (0) < 0 (semipositone). We establish a range of λ for which the above problem has a positive solution when f satisfies certain additional conditions. We employ the method of subsuper solutions to obtain the result. 1