On a power-type coupled system of k-Hessian equations

Abstract

We deal with a coupled system of k-Hessian equations: where k = 1, 2, · · · , N , B is a unit ball in ℝ N , N ≥ 2, α and β are positive constants. By using the fixed-point index theory in cone, we obtain the existence, uniqueness and nonexistence of radial convex solutions for some suitable constants α and β. Furthermore, by using a generalized Krein-Rutman theorem, we also obtain a necessary and sufficient existence condition of the convex solutions to a nonlinear eigenvalue problem.