A group theoretic proof of a compactness lemma and existence of nonradial solutions for semilinear elliptic equations

Abstract

Symmetry plays a basic role in variational problems (settled e.g. in $\mathbb R^{n}$ or in a more general manifold), for example to deal with the lack of compactness which naturally appear when the problem is invariant under the action of a noncompact group. In $\mathbb R^n$, a compactness result for invariant functions with respect to a subgroup $G$ of $\mathrm{O}(n)$ has been proved under the condition that the $G$ action on $\mathbb R^n$ is compatible, see \cite{willem}. As a first result we generalize this and show here that the compactness is recovered for particular subgroups of the isometry group of a Riemannian manifold. We investigate also isometric action on Hadamard manifold $(M,g)$ proving that a large class of subgroups of $\mathrm{Iso}(M,g)$ is compatible. As an application we get a compactness result for ``invariant'' functions which allows us to prove the existence of nonradial solutions for a classical scalar equation and for a nonlocal fractional equation on $\mathbb R^n$ for $n=3$ and $n=5$, improving some results known in the literature. Finally, we prove the existence of nonradial invariant functions such that a compactness result holds for some symmetric spaces of non compact type.