### Multiple solutions to a perturbed Neumann problem

We consider the perturbed Neumann problem ⎧ -Δu + α(x)u = α(x)f(u) + λg(x,u) a.e. in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where Ω is an open bounded set in ${\mathbb{R}}^{N}$ with boundary of class C², $\alpha \in {L}^{\infty}\left(\Omega \right)$ with $essin{f}_{\Omega}\alpha >0$, f: ℝ → ℝ is a continuous function and g: Ω × ℝ → ℝ, besides being a Carathéodory function, is such that, for some p > N, $su{p}_{\left|s\right|\le t}\left|g(\cdot ,s)\right|\in {L}^{p}\left(\Omega \right)$ and $g(\cdot ,t)\in {L}^{\infty}\left(\Omega \right)$ for all t ∈ ℝ. In this setting, supposing only that the set of global minima of the function $1/2\xi \xb2-{\int}_{0}^{\xi}f\left(t\right)dt$ has M ≥ 2 bounded connected components, we prove that, for all λ ∈ ℝ small enough, the above...