Existence of entire positive radial large solutions to the Monge-Amp\`{e}re type equations and systems
首发时间:2018-05-30
Abstract:Under simple conditions on $f$ and $f+g$, weshow that entire positive radial large solutions exist for theMonge-Amp\`{e}re type equations $${\rm det} D^2 u(x)+p(|x|)|\nabla u|^N=\alpha \Delta u+a(|x|)f(u)+\alpha N|x|^{N-1}p(|x|)|\nabla u|,\ x\in \mathbb R^N, $$ and systems\begin{eqnarray*}&&{\rm det} D^2 u(x)+p(|x|)|\nabla u|^N =\alpha \Deltau+a(|x|)f(v)+\alpha N |x|^{N-1}p(|x|)|\nabla u|, \ x\in \mathbb R^N,\\%&&{\rm det} D^2 v(x)+q(|x|)|\nabla v|^N =\beta \Deltav+b(|x|)g(u)+\beta N |x|^{N-1}q(|x|)|\nabla v|,\ x\in \mathbb R^N,\end{eqnarray*}where ${\rm det}D^2 u$ is the so-called Monge-Amp\`{e}re operator, $\triangle$ is the classicalLaplace operator, $N\geq 2$, $\alpha, \beta$ are positive constantsand $a,b, p, q:\mathbb R^N\rightarrow [0,\infty)$ are continuous.
keywords: the Monge-Amp\`{e}re type equations systems entire radial large solutions existence
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Monge-Ampere型方程和方程组正的整体径向大解的存在性
摘要:本文应用单调迭代方法和Arzela-Ascoli 定理, 结合精巧的估计, 证明了在$f$和 $f+g$满足适当条件下, Monge-Ampere型方程 \\$${\rm det} D^2 u(x)+p(|x|)|\nabla u|^N=\alpha \Delta u+a(|x|)f(u)+\alpha N|x|^{N-1}p(|x|)|\nabla u|,\ x\in \mathbb R^N$$和方程组\begin{eqnarray*}&&{\rm det} D^2 u(x)+p(|x|)|\nabla u|^N =\alpha \Deltau+a(|x|)f(v)+\alpha N |x|^{N-1}p(|x|)|\nabla u|, \ x\in \mathbb R^N,\\%&&{\rm det} D^2 v(x)+q(|x|)|\nabla v|^N =\beta \Deltav+b(|x|)g(u)+\beta N |x|^{N-1}q(|x|)|\nabla v|,\ x\in \mathbb R^N,\end{eqnarray*}正的整体径向大解的存在性.其中${\rm det}D^2 u$被称为Monge-Ampere算子, $\triangle$ 是经典的拉普拉斯算子, $N\geq2$, $\alpha,\beta$ 是正常数, $a, b, p, q:\mathbb R^N\rightarrow[0,\infty)$ 是连续的.
关键词: Monge-Amp\`{e}re方程 方程组 整体径向大解 存在性
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Monge-Ampere型方程和方程组正的整体径向大解的存在性
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