Multiplicity of positive solutions for an indefinite superlinear elliptic problem on R^n

Abstract

We consider the elliptic problem -Δu − λu = a(x)u^p, with p >1 and (x) sign-changing. Under suitable conditions on p and a(x), we extend the multiplicity, existence and nonexistence results known to hold for this equation on a bounded domain with standard homogeneous boundary conditions) to the case that the bounded domain is replaced by the entire space R^N. More precisely, we show that there exists ∧>0 such that this equation on R^N has no positive solution for λ>∧, at least two positive solutions for λ ∈ (0,Λ), and at least one positive solution for λ ∊ (−∞, 0] ∪ {Λ}.