Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space

We deal with a multiparameter Dirichlet system having the form

$ \begin{equation*} \left\{ \begin{array}{ll} -\mathcal M(u) = \lambda_1f_1(u,v), & \hbox{in $\Omega$},\\ -\mathcal M(v) = \lambda_2f_2(u,v), & \hbox{in $\Omega$},\\ u|_{\partial\Omega} = 0 = v|_{\partial\Omega}, \end{array} \right. \end{equation*} $

where $ \mathcal M $ stands for the mean curvature operator in Minkowski space

$ \mathcal M(u) = \mbox{div} \left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right), $

$ \Omega $ is a general bounded regular domain in $ \mathbb{R}^N $ and the continuous functions $ f_1,f_2 $ satisfy some sign and quasi-monotonicity conditions. Among others, these type of nonlinearities, include the Lane-Emden ones. For such a system we show the existence of a hyperbola like curve which separates the first quadrant in two disjoint sets, an open one $ \mathcal{O}_0 $ and a closed one $ \mathcal{F} $, such that the system has zero or at least one strictly positive solution, according to $ (\lambda_1, \lambda_2)\in \mathcal{O}_0 $ or $ (\lambda_1, \lambda_2)\in \mathcal{F} $. Moreover, we show that inside of $ \mathcal{F} $ there exists an infinite rectangle in which the parameters being, the system has at least two strictly positive solutions. Our approach relies on a lower and upper solutions method - which we develop here, together with topological degree type arguments. In a sense, our results extend to non-radial systems some recent existence/non-existence and multiplicity results obtained in the radial case.