Systems of singular Poisson equations in unbounded domains (Q2504063)

The authors consider the systems \[ \begin{cases} -\text{div}(\nabla u|x|^{-2a}) = H_v(x,u,v), \\ -\text{div}(\nabla v|x|^{-2a}) = H_u(x,u,v); \end{cases} \] and \[ \begin{cases} -\text{div}(\nabla u|x|^{-2a}) = \beta |x|^{-2(1+a)}v+H_v(x,u,v), \\ -\text{div}(\nabla v|x|^{-2a}) = \eta |x|^{-2(1+a)}u+H_u(x,u,v); \end{cases} \] where \(N \geq 3\), \(a \in [0,(N-2)/2)\), and \(\beta, \eta \in [0,S(a,a+1))\), with \(S(a,a+1)\) being the best constant in the Caffarelli, Konh, and Nirenberg inequality. The \(C^1\)-potential \(H(x,u,v)\) satisfies some technical assumptions which characterize it as a singular subcritical nonlinearity. The authors prove the existence of one nontrivial solution for the first system in a domain \(\Omega \subset \mathbb{R}^N \setminus \mathbb{Z}^N\) invariant by \(\mathbb{Z}^N\) translations; and to the second one in an unbounded cylindrical domain and for \(\beta,\eta \in (0\,\lambda_1)\) with \(\lambda_1\) the best constant in the Poincaré inequality. The proofs rely on variational methods. Since the associated functional is indefinite, a generalized linking theorem due to \textit{W. Kryszewski} and \textit{A. Szulkin} [Adv. Differ. Equ. 3, No. 3, 441--472 (1998; Zbl 0947.35061)] is used.