Gradient-type systems on unbounded domains of the Heisenberg group

TL;DR

This paper establishes the existence of multiple symmetric weak solutions for certain gradient-type systems on unbounded domains of the Heisenberg group, using variational methods and symmetry group techniques.

Contribution

It introduces a novel group-theoretical approach combined with compactness results to prove solution existence for subelliptic systems with Sobolev-Hardy potentials.

Findings

Existence of at least two nontrivial symmetric weak solutions for parameter values in an open interval.

Solutions are uniformly bounded in the Sobolev $HW^{1,2}_0$-norm.

Existence result is stable under small subcritical perturbations of the nonlinear term.

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 $\Omega_\psi$ of the Heisenberg group $\mathbb{H}^n=\mathbb{C}^n\times \mathbb{R}$ ($n\geq 1$) whose geometrical profile is determined by two real positive functions $\psi_1$ and $\psi_2$ 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 $\Lambda\subset (0,\infty)$ such that, for every parameter $\lambda\in \Lambda$, the system has at least two nontrivial symmetric weak solutions that are uniformly bounded with respect to the Sobolev $HW^{1,2}_0$-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 $\mathbb{U}(n)=U(n)\times\{1\}$ and by exploiting some compactness embedding results into Lebesgue spaces, recently proved for suitable $\mathbb{U}(n)$-invariant subspaces of the Folland-Stein space $HW^{1,2}_0(\Omega_\psi)$. A key ingredient for our variational approach is a very general min-max argument valid for sufficiently smooth functionals defined on reflexive Banach spaces.