论文标题
具有单调非线性的椭圆系统的爆破径向溶液
Blow-up radial solutions for elliptic systems with monotonic non-linearities
论文作者
论文摘要
我们关注系统\ begin {equation*} \ left \ {\ oken {Aligned}ΔU＆= g（| x |，v（x））&& \ quad \ quad \ mbox {in} && \ quad \ mbox {in} \ω，\ end {aligned} \ right。 \end{equation*} where $Ω\subset \mathbb{R}^N$ is either a ball centered at the origin or the whole space $\mathbb{R}^N$, and $f,g\in C^{1}([0,\infty)\times [0,\infty))$, are non-negative, and increasing.首先，我们研究了在系统中呈现与它们在边界行为相对应的球中的阳性径向溶液的存在。接下来，如果$ g（| x |，v（x））= | x |^{a} v^p $和$ f（| x |，| \ nabla u（x）|）= | x | x |^{b}最后，我们采用$ h（t）= t^s $，$ s> 1 $，$ω= \ mathbb {r}^n $，并且通过使用动力学系统技术，我们能够描述这种正径向解决方案的无穷大行为。
We are concerned with the existence and boundary behaviour of positive radial solutions for the system \begin{equation*} \left\{ \begin{aligned} Δu&=g(|x|,v(x)) &&\quad\mbox{in}\ Ω, \\ Δv&=f(|x|,|\nabla u(x)|) &&\quad\mbox{in}\ Ω, \end{aligned} \right. \end{equation*} where $Ω\subset \mathbb{R}^N$ is either a ball centered at the origin or the whole space $\mathbb{R}^N$, and $f,g\in C^{1}([0,\infty)\times [0,\infty))$, are non-negative, and increasing. Firstly, we study the existence of positive radial solutions in the case when the system is posed in a ball corresponding to their behaviour at the boundary. Next, we discuss the existence of positive radial solutions in case when $g(|x|,v(x)) = |x|^{a} v^p$ and $f(|x|, |\nabla u (x)|) = |x|^{b} h(|\nabla u|)$. Finally, we take $h(t) = t^s$, $s> 1$, $Ω= \mathbb{R}^N$ and by the use of dynamical system techniques we are able to describe the behaviour at infinity of such positive radial solutions.
