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Article Contents

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

• * Corresponding author: Petru Jebelean
• 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.

Mathematics Subject Classification: Primary: 35J66, 34B16; Secondary: 34B18.

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