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

Abstract

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.

Some new Lyapunov-type inequalities for a class of (<i>n</i>+1)th order nonlinear differential equations

“…The relativistic operator and prescribed curvature operator play an important role in the study of the theory of classical relativity and the mean curvature hypersurfaces in differential geometry, see [2,3,4,5,6,7,8,9,12,13,14,15,19,20,21,22], and the references therein.…”