Three solutions for a perturbed Dirichlet problem Original Research Article

In this paper we prove the existence of at least three distinct solutions to the following perturbed Dirichlet problem: View the MathML source{−Δu=f(x,u)+λg(x,u)in Ωu=0on ∂Ω, Turn MathJax on where Ω⊂RNΩ⊂RN is an open bounded set with smooth boundary ∂Ω∂Ω and λ∈Rλ∈R. Under very mild conditions on gg and some assumptions on the behaviour of the potential of ff at 0 and +∞+∞, our result assures the existence of at least three distinct solutions to the above problem for λλ small enough. Moreover such solutions belong to a ball of the space View the MathML sourceW01,2(Ω) centered in the origin and with radius not dependent on λλ.