On the solutions to weakly coupled system of ${\mathbf{k}}_{\mathbf{i}}$-Hessian equations

Abstract

In this paper, the existence and multiplicity of nontrivial radial convex solutions to general coupled system of $k_i$-Hessian equations in a unit ball are studied via a fixed-point theorem. In particular, we obtain the uniqueness of nontrivial radial convex solution and nonexistence of nontrivial radial $\mathbf{k}$-admissible solution to a power-type system coupled by $k_i$-Hessian equations in a unit ball. Moreover, using a generalized Krein-Rutman theorem, the existence of $\mathbf{k}$-admissible solutions to an eigenvalue problem in a general strictly $(k-1)$-convex domain is also obtained.