MiS Preprint Repository

The aim of this paper is to discuss the question of existence and multiplicity of local minimizers for a relatively large class of functionals ${\mathbb F}[\cdot]: W^{1,p}({\mathcal X}, {\mathcal Y}) \to {\mathbb R}$ from a purely topological point of view. The basic assumptions on ${\mathbb F}[\cdot]$ are sequential lower semicontinuity with respect to $W^{1,p}$-weak convergence and $W ^{1,p}$-weak coercivity and the target is a multiplicity bound on the number of such minimizers in terms of convenient topological invariants of the manifolds ${\mathcal X}$ and ${\mathcal Y}$. In the first part of the paper, we focus on the case where ${\mathcal Y}$ is non contractible and proceed by establishing a link between the latter problem and the question of enumeration of homotopy classes of continuous maps from various skeleta of ${\mathcal X}$ into ${\mathcal Y}$. Naturally enough, our results in this direction are of a cohomological nature.

We devote the second part to the case where ${\mathcal Y}$ is the Euclidean space ${\mathbb R}^N$ and ${\mathcal X}= \Omega$, with $\Omega \subset {\mathbb R}^n$ being a bounded smooth domain. In particular we consider integral functionals of the form $$ {\mathbb F} [u] := \int_{\Omega } F(x, u(x), \nabla u(x)) \, dx, $$ where the above assumptions on ${\mathbb F}[\cdot]$ can be verified when the integrand $F$ is appropriately quasiconvex and pointwise $p$-coercive with respect to the gradient argument. We introduce the notion of a topologically {\it non trivial} domain and under this hypothesis establish the required multiplicity result for strong local minimizers of ${\mathbb F}[\cdot]$.