Positive solutions for classes of multiparameter elliptic semipositone problems

Abstract

We study positive solutions to multiparameter boundary-value problems of the form -Δu = λg(u) + μƒ(u) in Ω u = 0 on ∂Ω, where λ > 0, μ > 0, Ω ⊆ Rn; n ≥ 2 is a smooth bounded domain with ∂Ω in class C2 and Δ is the Laplacian operator. In particular, we assume g(0) > 0 and superlinear while ƒ(0) < 0, sublinear, and eventually strictly positive. For fixed μ, we establish existence and multiplicity for λ large. Our proofs are based on variational methods, the Mountain Pass Lemma, and sub-super solutions.