Sublinear singular elliptic problems with two parameters

Abstract

We establish several existence and nonexistence results for the boundary value problem −Δ u+ K( x) g( u)= λf( x, u)+ μh( x) in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in R N , λ and μ are positive parameters, h is a positive function, while f has a sublinear growth. The main feature of this paper is that the nonlinearity g is assumed to be unbounded around the origin. Our analysis shows the importance of the role played by the decay rate of g combined with the signs of the extremal values of the potential K( x) on Ω ̄ . The proofs are based on various techniques related to the maximum principle for elliptic equations.