CLC number: O175.29

Existence of Entire Radial Solutions to Monge-Ampère Type Systems

Monge-Ampère型方程组整体镜像对称解的存在性

School of Mathematics, Jilin University, Changchun 130012, Jilin, China

Received: 15 October 2024

Abstract

This paper mainly studies the following Monge-Ampère type systems $\{\begin{array}{l}\mathrm{d}\mathrm{e}{\mathrm{t}}^{\mathrm{1}/n}(\mathrm{\Delta }u\mathit{I}-D^{\mathrm{2}}u)=p(|\mathit{x}|)f(v),\mathit{x}\in {\mathbb{R}}^n,\\ \mathrm{d}\mathrm{e}{\mathrm{t}}^{\mathrm{1}/n}(\mathrm{\Delta }v\mathit{I}-D^{\mathrm{2}}v)=q(|\mathit{x}|)g(u),\mathit{x}\in {\mathbb{R}}^n.\end{array}$ The existence of entire radial solutions is obtained by using monotone iteration method and Arzelà-Ascoli theorem. These results generalize the classical Keller-Osserman condition to fully nonlinear systems.

摘要

本文主要研究了下面的Monge-Ampère型方程组：$\{\begin{array}{l}\mathrm{d}\mathrm{e}{\mathrm{t}}^{\mathrm{1}/n}(\mathrm{\Delta }u\mathit{I}-D^{\mathrm{2}}u)=p(|\mathit{x}|)f(v),\mathit{x}\in {\mathbb{R}}^n,\\ \mathrm{d}\mathrm{e}{\mathrm{t}}^{\mathrm{1}/n}(\mathrm{\Delta }v\mathit{I}-D^{\mathrm{2}}v)=q(|\mathit{x}|)g(u),\mathit{x}\in {\mathbb{R}}^n.\end{array}$利用单调迭代法和Arzelà-Ascoli定理，得到了整体镜像对称解的存在性。这些结果把经典的Keller-Osserman条件推广到了完全非线性方程组中。