In this paper, we use the critical point theory for convex, lower semicontinuous perturbations of {C^{1}}-functionals to obtain the existence of multiple nontrivial solutions for one parameter potential systems involving the operator {u\mapsto\operatorname{div}(\frac{\nabla u}{\sqrt{1-|\nabla u|^{2}}})}. The solvability of a general non-potential system is also established.
We deal with a multiparameter Dirichlet system having the formwhere M stands for the mean curvature operator in Minkowski spaceΩ is a general bounded regular domain in 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 O 0 and a closed one F , such that the system has zero or at least one strictly positive solution, according to (λ 1 , λ 2 ) ∈ O 0 or (λ 1 , λ 2 ) ∈ F . Moreover, we show that inside of 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.2010 Mathematics Subject Classification. Primary: 35J66, 34B16; Secondary: 34B18.
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