Local conditions insuring bifurcation from the continuous spectrum

We consider a family of equations −∆u(x) + λu(x) = f(x, u(x)), λ > 0, x ∈ R , (1)λ where the nonlinearity f : RN ×R → R satisfiesf(x, 0) = 0, a.e.x ∈ RN . We say thatλ = 0 is a bifurcation point for(1)λ if there exists a sequence {(λn, un)} ⊂ R+ × H1(RN ) of nontrivial solutions of(1)λn with λn → 0 and ||un||H1(RN ) → 0. In this case{(λn, un)} is called a bifurcating sequence. The aim of the paper is to show that weak conditions on f(x, .) around zero suffice to guarantee that λ = 0 is a bifurcation point for(1)λ. More precisely suppose there is δ > 0 such that (H1) f : RN × [−δ, δ] → R is Caratheodory. (H2) lim |x|→∞ f(x, s) = 0 uniformly for s ∈ [−δ, δ].