Integral functionals on Sobolev spaces having multiple local minima

THEOREM A. Let (X, τ) be a Hausdorff topological space and Ψ : X →]−∞,+∞], Φ : X → R two functions. Assume that there is r > infX Ψ such that the set Ψ (]−∞, r]) is compact and first-countable. Moreover, suppose that the function Φ is bounded below in Ψ(]−∞, r]) and that the function Ψ+ λΦ is sequentially lower semicontinuous for each λ ≥ 0 small enough. Finally, assume that the set of all global minima of Ψ has at least k connected components. Then, there exists λ > 0 such that, for each λ ∈]0, λ[, the function Ψ + λΦ has at least k τΨ-local minima lying in Ψ (]−∞, r[).