Exact multiplicity results and bifurcation for nonhomogeneous elliptic problems

In this paper we consider the following problem: (⋆) { − Δ u ( x ) + u ( x ) = λ ( f ( x , u ) + h ( x ) ) in R N , u ∈ H 1 ( R N ) , u > 0 in R N , where λ > 0 is a parameter. We assume lim | x | → ∞ f ( x , u ) = f ( u ) uniformly on any compact subset of [ 0 , ∞ ) , but we do not require f ( x , u ) ≥ f ( u ) for all x ∈ R N . We prove that there exists + ∞ > λ ∗ > 0 such that (⋆) has exactly two positive solutions for λ ∈ ( 0 , λ ∗ ) , no solution for λ > λ ∗ , a unique positive solution u ∗ for λ = λ ∗ , and ( λ ∗ , u ∗ ) is a bifurcation point in C 2 , α ( R N ) ∩ W 2 , 2 ( R N ) .