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

Author(s)

Biliotti, Leonardo ^[1] ; Siciliano, Gaetano ^[2]

Affiliation

^[1] Univ Parma, Dipartimento Sci Matemat Fis & Informat, Parco Area Sci 53-A, I-43124 Parma - Italy

^[2] Univ Sao Paulo, Dept Matemat, Inst Matemat & Estat, Rua Matao 1010, BR-05508090 Sao Paulo - Brazil

• Document type: Journal article
• Source: Annali di Matematica Pura ed Applicata; v. 200, n. 2 JUL 2020.
• Web of Science Citations: 0

Abstract

Symmetry plays a basic role in variational problems (settled, e.g., in R-n or in a more general manifold), for example, to deal with the lack of compactness which naturally appears when the problem is invariant under the action of a noncompact group. In R-n, 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 R-n is compatible, see Willem (Minimax theorem. Progress in nonlinear differential equations and their applications, vol 24, Birkhauser Boston Inc., Boston, 1996). 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 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 noncompact type. (AU)

• FAPESP's process: 18/17264-4 - Existence of solutions for nonlinear elliptic equations
• Grantee: Gaetano Siciliano
• Support Opportunities: Regular Research Grants