Entire large solutions to the k-Hessian equations with weights: Existence, uniqueness and asymptotic behavior

Abstract

This paper is concerned with the k-Hessian equation Sk(D2u)=b(x)f(u) in RN, where b∈C∞(RN) is positive in RN, f∈C∞(β,∞) is positive and increasing on (β,∞) with β∈R∪{−∞} and satisfies the so-called Keller-Osserman condition. We first establish the existence of entire strictly k-convex large solutions to the equation. And then by constructing a new class of Karamata functions, we investigate the exact asymptotic behavior of entire strictly k-convex large solutions at infinity. Finally, we prove the uniqueness of solutions.