Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity

Abstract

We study the existence, nonexistence and multiplicity of positive solutions for the family of problems-Δu = fλ(x,u),u ∈01(Ω), where Ω is a bounded domain in ℝN, N ≥ 3 and λ > 0 is a parameter. The results include the well-known nonlinearities of the Ambrosetti-Brezis-Cerami type in a more general form, namely λa(x)uq + b(x)up, where 0 ≤ q < 1 < p ≤ 2* - 1. The coefficient a(x) is assumed to be nonnegative but b(x) is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in [9] are essential in this more general framework. The techniques used in the proofs are lower and upper solutions and variational methods. © European Mathematical Society 2006.