On a weighted Lane–Emden system

ABSTRACT

Liouville type results for positive radial entire solutions $(u,v)\in \left[C^2\right({\mathbb{R}}^N\setminus \left\{0\right\})\cap C^0({\mathbb{R}}^N\left)\right]\times \left[C^2\right({\mathbb{R}}^N\setminus \left\{0\right\})\cap C^0({\mathbb{R}}^N\left)\right]$ $(N\ge 3)$ of the weighted Lane–Emden system $\begin{array}{rl}& \mathrm{div}\left(|x{|}^{{\theta }_1}\mathrm{\nabla }u\right)+|x{|}^{l_1}{v}^p=0\mathrm{in}{\mathbb{R}}^N\setminus \left\{0\right\},\\ & \mathrm{div}\left(|x{|}^{{\theta }_2}\mathrm{\nabla }v\right)+|x{|}^{l_2}{u}^q=0\mathrm{in}{\mathbb{R}}^N\setminus \left\{0\right\}\end{array}$ are established, where $(p,q)\in (0,\mathrm{\infty })\times (0,\mathrm{\infty })$, ${\theta }_1,{\theta }_2,l_1,l_2$ satisfy $N+{\theta }_i>2,{\tau }_i=l_i-{\theta }_i>-2$, i=1,2. Meanwhile, classification of singularities at the origin of positive radial solutions of this system on the punctured unit ball and singular positive radial solutions of the Dirichlet problem of this system in the unit ball are also obtained.

COMMUNICATED BY:

• Daomin Cao

KEYWORDS:

• Singular positive radial solutions
• existence and non-existence
• weighted Lane–Emden system

1991 MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary 35B45
• Secondary 35J40

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of the corresponding author is supported by National Natural Science Foundation of China [11771125, 11571193].