2020/02/04 by Biliotti, Leonardo, Siciliano, Gaetano · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2002.01333
Symmetry plays a basic role in variational problems (settled e.g. in \mathbb Rn 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 Rn, a compactness result for invariant functions with respect to a subgroup G of O(n) has been proved under the condition that the G action on \mathbb Rn is compatible, see \citewillem. 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 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 Rn 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.