Three Solutions for a Neumann Problem

In this paper we consider a Neumann problem of the type (Pλ) 8< : −∆u = α(x)(|u|q−2u− u) + λf(x, u) in Ω, ∂u ∂ν = 0 on ∂Ω. Applying the theory developed in [13], we establish, under suitable assumptions, the existence of an open interval Λ ⊆ R and of a positive real number %, such that, for each λ ∈ Λ, problem (Pλ) admits at least three weak solutions in W 1,2(Ω) whose norms are less than %. Let us recall that a Gâteaux differentiable functional J on a real Banach space X is said to satisfy the Palais–Smale condition if each sequence {xn} in X such that supn∈N |J(xn)| < ∞ and limn→∞ ‖J (xn)‖X∗ = 0 admits a strongly converging subsequence. In [13], we proved the following result: Theorem A ([13, Theorem 3]). Let X be a separable and reflexive real Banach space, I ⊆ R an interval, and g:X×I → R a continuous function satisfying the following conditions: (i) for each x ∈ X, the function g(x, · ) is concave, 2000 Mathematics Subject Classification. 35J20, 35J65.