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 $\mathcal{X}=\mathrm{\Omega }$, with $\mathrm{\Omega }\subset {\mathbb{R}}^n$ being a bounded smooth domain. In particular we consider integral functionals of the form $$\mathbb{F}[u]:={\int }_{\mathrm{\Omega }}F(x,u(x),\mathrm{\nabla }u(x))dx,$$ 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 $\mathbb{F}[\cdot ]$.