Three Topological Problems about Integral Functionals on Sobolev Spaces

Abstract

In this paper, I propose some problems, of topological nature, on the energy functional associated to the Dirichlet problem

$$\left\{ \begin{gathered} - \Delta {\kern 1pt} {\kern 1pt} u = f\left( {x,u} \right){\text{in}}\Omega \hfill \\ u_{\left| {\wp \Omega } \right.} = 0 \hfill \\ \end{gathered} \right.$$

Positive answers to these problems would produce innovative multiplicity results on problem (P_f).