Constant sign solutions for variable exponent system Neumann boundary value problems with singular coefficient (Q1724029)

Summary: We deal with the existence of constant sign solutions for the following variable exponent system Neumann boundary value problem: \(- \text{div}(| \nabla u |^{p(x) - 2} \nabla u) + \lambda | u |^{p(x) - 2} u = F_u(x, u, v)\) in \(\Omega, - \text{div}(| \nabla v |^{q(x) - 2} \nabla v) + \lambda | v |^{q(x) - 2} v = F_v(x, u, v)\) in \(\Omega, \partial u / \partial \gamma = 0 = \partial v / \partial \gamma\) on \(\partial \Omega\). We give several sufficient conditions for the existence of the constant sign solutions, when \(F(x, \cdot, \cdot)\) satisfies neither sub-(\(p(x), q(x)\)) growth condition, nor Ambrosetti-Rabinowitz condition (subcritical). In particular, we obtain the existence of eight constant sign solutions.