Classification of radial solutions for elliptic systems driven by the $k$-Hessian operator

Abstract

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 $S_k(D^{2}u)$ is the $k$-Hessian operator of $u\in C^2(\Omega )$ ($1\le k\le N$) and $\Omega \subset {\mathbb{R}}^N$ $(N\ge 2)$ is either a ball or the whole space. The exponents satisfy $q>0$, $m,s\ge 0$, $p\ge s\ge 0$ and $(k-m)(k-s)\ne pq$. In the case where $\Omega$ is a ball, we classify all the positive radial solutions according to their behavior at the boundary. Further, we consider the case $\Omega ={\mathbb{R}}^N$ and find that the above system admits non-constant positive radial solutions if and only if $0\le m<k$ 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.

21 pages. arXiv admin note: text overlap with arXiv:1808.00407

References in corpus (1)

• Entire large solutions for semilinear elliptic equations