Topological results on a certain class of functionals and application

We study in this paper problems of the type $\text{Δu + ¦u¦}{}^{\mathrm{p\; ?\; 1}}\text{u = ?(x)}$, Ω bounded ? $\text{R}$^N, $\text{u = 0¦}{}_{\mathrm{?\Omega }}$, (I) where $\text{?(x)}$ is given and where $\text{p ? (1, (}\text{N + 2}\text{N ? 2}\text{)) (p ? (1, + ∞)}\text{if}\text{N ? 2)}$. Our main result is that (I) has an infinite number of solutions for a residual set of ? in $\text{H}{}^{\mathrm{?1}}\text{(Ω)}$. In particular, for many n ∈ $\text{N}$ there exists an open and dense subset of ? in $\text{H}{}^{\mathrm{?1}}\text{(Ω)}$ such that (I) has n distinct solutions for such an ? This result is to be related to the conjecture developed in 1] of the existence of an infinite number of solutions to (I). The proof relies on a general characterization of level sets for a certain class of functionals, when there are no critical value in a large enough interval. In addition to the study of problem (I), we apply this characterization to give another proof (using, e.g., Brouwer's fixed point theorem) for some classical results about even functionals and saddle points.