论文标题
由$ k $ -Hessian操作员驱动的椭圆系统的径向解决方案的分类
Classification of radial solutions for elliptic systems driven by the $k$-Hessian operator
论文作者
论文摘要
我们关注系统的非恒定正径向解决方案$$ \ weft \ webt {aLigned} s_k（d^2 u）＆= | \ nabla u |^{m} m} v^{p} v^{p} && \ quad \ quad \ quad \ quad \ mbox {in} v^{s} && \ quad \ mbox {in}ω，\ end {aligned} \ right。 $$ $ s_k（d^2u）$是$ k $ -Hessian运算符，$ u \ in C^2（ω）$（$ 1 \ leq k \ leq n $）和$ω\ subset \ subset \ subset \ mathbb {r}^n $ $（n $ geq 2）$要么是一个球或整个空间。指数满足$ q> 0 $，$ m，s \ geq 0 $，$ p \ geq s \ geq 0 $和$（k-m）（k-s）（k-s）\ neq pq $。如果$ω$是一个球，我们根据其在边界处的行为对所有正径向溶液进行了分类。此外，我们考虑了$ω= \ mathbb {r}^n $，并在且仅当$ 0 \ leq m <k $和$ pq <（k-m）（k-s）$时，发现上述系统允许非恒定正径向解决方案。使用来自三个组件合作和不可还原动力学系统的参数，我们推断出此类解决方案的无穷大行为。
We are concerned with non-constant positive radial solutions of the system $$ \left\{ \begin{aligned} S_k(D^2 u)&=|\nabla u|^{m} v^{p}&&\quad\mbox{ in }Ω,\\ S_k(D^2 v)&=|\nabla u|^{q} v^{s} &&\quad\mbox{ in }Ω, \end{aligned} \right. $$ where $S_k(D^2u)$ is the $k$-Hessian operator of $u\in C^2(Ω)$ ($1\leq k\leq N$) and $Ω\subset\mathbb{R}^N$ $(N\geq 2)$ is either a ball or the whole space. The exponents satisfy $q>0$, $m,s\geq 0$, $p\geq s\geq 0$ and $(k-m)(k-s)\neq pq$. In the case where $Ω$ is a ball, we classify all the positive radial solutions according to their behavior at the boundary. Further, we consider the case $Ω=\mathbb{R}^N$ and find that the above system admits non-constant positive radial solutions if and only if $0\leq m<k$ and $pq < (k-m)(k-s)$. Using arguments from three component cooperative and irreducible dynamical systems we deduce the behavior at infinity of such solutions.
