Three Topological Problems about Integral Functionals on Sobolev Spaces

In this paper, I propose some problems, of topological nature, on the energy functional associated to the Dirichlet problem −∆u = f(x, u) in Ω, u|∂Ω = 0. Positive answers to these problems would produce innovative multiplicity results on this Dirichlet problem. In the present very short paper, I wish to propose some problems, of topological nature, on the energy functional associated to the Dirichlet problem (Pf ) −∆u = f(x, u) in Ω, u|∂Ω = 0. and explain their motivations as well. So, let Ω ⊂ R (n ≥ 3) be an open bounded set. Put X = W 1,2 0 (Ω). For q > 0, denote by Aq the class of all Carathéodory functions f : Ω× R → R such that sup (x,ξ)∈Ω×R |f(x, ξ)| 1 + |ξ|q < +∞. For 0 < q ≤ n+2 n−2 and f ∈ Aq, put