EXISTENCE OF A POSITIVE SOLUTION FOR A p-LAPLACIAN SEMIPOSITONE PROBLEM

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.