Positive radial solutions for multiparameter Dirichlet systems with mean curvature operator in Minkowski space and Lane–Emden type nonlinearities

Abstract

We deal with Dirichlet systems involving the mean curvature operator in Minkowski spaceM(w)=div(∇w1−|∇w|2) in a ball in RN. Using the fixed point index and the lower and upper solutions method, we first obtain the existence of positive solutions for a class of differential systems with a singular φ-Laplacian, subjected to homogeneous mixed boundary conditions. The main application of the general results concerns Dirichlet systems with Lane–Emden type nonlinearities in a superlinear case, depending on two parameters λ1,λ2. So, we prove that for such a system there exists a continuous curve Γ which separates the first quadrant in two disjoint unbounded open sets O1 and O2 such that the system has zero, at least one or at least two radial positive solutions, according to (λ1,λ2)∈O1, (λ1,λ2)∈Γ or (λ1,λ2)∈O2.