    April  2021, 20(4): 1681-1698. doi: 10.3934/cpaa.2021036

## The regularity lifting methods for nonnegative solutions of Lane-Emden system

 School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

Received  June 2020 Revised  January 2021 Published  April 2021 Early access  March 2021

Fund Project: The first author is partially supported by NSFC-12031012 and NSFC-11831003

In this paper, we focus on the regularity of nonnegative solutions of Lane-Emden system
 $\begin{equation*} \begin{cases} -\Delta u = v^p\\ -\Delta v = u^q \end{cases} \mbox{ in } \mathbb{R}^n. \end{equation*}$
By means of Kelvin transform, we turn this problem into estimating the local integrability of
 $(\bar{u},\bar{v})$
. Assume that
 $(\bar{u},\bar{v})$
possesses some initial local integrability beforehand.
 $(\bar{u},\bar{v})\in L_{loc}^{r_0}(\mathbb{R}^n)\times L_{loc}^{s_0}(\mathbb{R}^n)$
for any suitable
 $r_0$
and
 $s_0$
under specified conditions. Then through a regularity lifting method by contracting operators, we prove that
 $(\bar{u},\bar{v})\in L_{loc}^r(\mathbb{R}^n)\times L_{loc}^s(\mathbb{R}^n)$
for
 $r$
and
 $s$
sufficiently large under twice regularity lifting if needed. Furthermore, we lift the regularity of solutions to
 $L^\infty(\mathbb{R}^n)\times L^\infty(\mathbb{R}^n).$
We believe that these new methods employed in this paper can be widely applied to study a variety of other problems with different spaces and linear or nonlinear problems.
Citation: Tianyu Liao. The regularity lifting methods for nonnegative solutions of Lane-Emden system. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1681-1698. doi: 10.3934/cpaa.2021036
##### References:
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##### References:
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