Exact multiplicity results and bifurcation for nonhomogeneous elliptic problems

In this paper we consider the following problem: equation(⋆){−Δu(x)+u(x)=λ(f(x,u)+h(x))in RN,u∈H1(RN),u>0 in RN, where λ>0λ>0 is a parameter. We assume lim|x|→∞f(x,u)=f̄(u) uniformly on any compact subset of [0,∞)[0,∞), but we do not require f(x,u)≥f̄(u) for all x∈RNx∈RN. We prove that there exists +∞>λ∗>0+∞>λ∗>0 such that (⋆) has exactly two positive solutions for λ∈(0,λ∗)λ∈(0,λ∗), no solution for λ>λ∗λ>λ∗, a unique positive solution u∗u∗ for λ=λ∗λ=λ∗, and (λ∗,u∗)(λ∗,u∗) is a bifurcation point in C2,α(RN)∩W2,2(RN)C2,α(RN)∩W2,2(RN).