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 ▫$\mathbb{H}^n = \mathbb{C}^n \times \mathbb{R} \, (n \ge 1)$▫ 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 non-trivial 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.
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