On the solutions to weakly coupled system of $\boldsymbol{k_i}$-Hessian equations

TL;DR

This paper investigates the existence, multiplicity, and uniqueness of radial convex solutions to coupled $k_i$-Hessian equations in a unit ball, and explores eigenvalue problems in convex domains using fixed-point and Krein-Rutman theorems.

Contribution

It provides new results on the existence, uniqueness, and nonexistence of solutions to coupled $k_i$-Hessian systems, extending the understanding of such nonlinear PDEs.

Findings

Existence of radial convex solutions in a unit ball.

Uniqueness of nontrivial radial convex solutions.

Nonexistence of certain $k$-admissible solutions.

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 $\boldsymbol{k}$-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 $\boldsymbol{k}$-admissible solutions to an eigenvalue problem in a general strictly $(k-1)$-convex domain is also obtained.