Three solutions for a perturbed Dirichlet problem

In this paper we prove the existence of at least three distinct solutions to the following perturbed Dirichlet problem: {−Δu=f(x,u)+λg(x,u)in Ωu=0on ∂Ω, where Ω⊂RN is an open bounded set with smooth boundary ∂Ω and λ∈R. Under very mild conditions on g and some assumptions on the behaviour of the potential of f at 0 and +∞, our result assures the existence of at least three distinct solutions to the above problem for λ small enough. Moreover such solutions belong to a ball of the space W01,2(Ω) centered in the origin and with radius not dependent on λ.