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- Title
Gradient-Type Systems on Unbounded Domains of the Heisenberg Group.
- Authors
Molica Bisci, Giovanni; Repovš, Dušan
- Abstract
The purpose of this paper is to study the existence of weak solutions for some classes of one-parameter subelliptic gradient-type systems involving a Sobolev–Hardy potential defined on an unbounded domain Ω ψ of the Heisenberg group H n = C n × R ( n ≥ 1 ) whose geometrical profile is determined by two real positive functions ψ 1 and ψ 2 that are bounded on bounded sets. The treated problems have a variational structure, and thanks to this, we are able to prove the existence of an open interval Λ ⊂ (0 , ∞) such that, for every parameter λ ∈ Λ , the system has at least two non-trivial symmetric weak solutions that are uniformly bounded with respect to the Sobolev H W 0 1 , 2 -norm. Moreover, the existence is stable under certain small subcritical perturbations of the nonlinear term. The main proof, crucially based on the Palais principle of symmetric criticality, is obtained by developing a group-theoretical procedure on the unitary group U (n) = U (n) × { 1 } and by exploiting some compactness embedding results into Lebesgue spaces, recently proved for suitable U (n) -invariant subspaces of the Folland–Stein space H W 0 1 , 2 (Ω ψ) . A key ingredient for our variational approach is a very general min–max argument valid for sufficiently smooth functionals defined on reflexive Banach spaces.
- Publication
Journal of Geometric Analysis, 2020, Vol 30, Issue 2, p1724
- ISSN
1050-6926
- Publication type
Article
- DOI
10.1007/s12220-019-00276-2