Entire positive k-convex solutions to k-Hessian type equations and systems

Abstract

In this paper, we study the existence of entire positive solutions for the $ k $-Hessian type equation $ {\rm S}_k(D^2u+\alpha I) = p(|x|)f^k(u), \ \ x\in \mathbb{R}^n $ and system $ \begin{cases} {\rm S}_k(D^2u+\alpha I) = p(|x|)f^k(v), \ \ x\in \mathbb{R}^n, \\ {\rm S}_k(D^2v+\alpha I) = q(|x|)g^k(u), \ \ x\in \mathbb{R}^n, \end{cases} $ where $ D^2u $ is the Hessian of $ u $ and $ I $ denotes unit matrix. The arguments are based upon a new monotone iteration scheme.

Keywords

• k-hessian type equation and system
• entire positive k-convex solution
• monotone iterative
• existence