Gradient-Type Systems on Unbounded Domains of the Heisenberg Group

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.