Perturbation from Dirichlet problem involving oscillating nonlinearities

In this paper we prove that if the potential F(x,t)=∫0tf(x,s)ds has a suitable oscillating behavior in any neighborhood of the origin (respectively +∞), then under very mild conditions on the perturbation term g  , for every k∈Nk∈N there exists bk>0bk>0 such that{−Δu=f(x,u)+λg(x,u)inΩ,u=0on∂Ω has at least k   distinct weak solutions in W01,2(Ω), for every λ∈Rλ∈R with |λ|⩽bk|λ|⩽bk. Moreover, information about the location of such solutions is also given. In fact, there exists a positive real number σ>0σ>0, which does not depend on λ  , such that the W01,2(Ω)-norm of each of those k solutions is not greater than σ.