    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  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, doi: 10.3934/cpaa.2021036
##### References:
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##### References:
  W. Chen and C. Li, An integral system and the LaneEmden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar  W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, American Institute of Mathematical Sciences, Springfield, MO, 2010. Google Scholar  W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar  W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co, 2019. Google Scholar  Z. Cheng and G. Huang, A Liouville theorem for the subcritical Lane-Emden system, Discrete Contin. Dyn. Syst., 39 (2019), 1359-1377.  doi: 10.3934/dcds.2019058.  Google Scholar  L. Evans, Partial Differential Equations, Wadsworth and Brooks/cole Mathematics, 2010. Google Scholar  D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397. Google Scholar  C. Li, Z. Wu and H. Xu, Maximum principles and Bocher type theorems, Proc. Natl. Acad. Sci., 115 (2018), 6976-6979.  doi: 10.1073/pnas.1804225115.  Google Scholar  C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.  doi: 10.1016/j.aim.2010.07.020.  Google Scholar  L. Ma and D. Chen, A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859.  doi: 10.3934/cpaa.2006.5.855.  Google Scholar  E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in ${\textbf{R}}^N$, Differ. Integral Equ., 9 (1996), 465-479. Google Scholar  E. Mitidieri, A Rellich type identity and applications, Commun. Partial Differ. Equ., 18 (1993), 125-151.  doi: 10.1080/03605309308820923.  Google Scholar  E. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-384. Google Scholar  P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar  W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differ. Equ., 161 (2000), 219-243.  doi: 10.1006/jdeq.1999.3700.  Google Scholar  J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 369-380. Google Scholar  E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.  doi: 10.1512/iumj.1958.7.57030.  Google Scholar  J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.  doi: 10.1007/BF02392645.  Google Scholar  J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equ., 9 (1996), 635-653. Google Scholar  P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar  M. A. S. Souto, A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differ. Integral Equ., 8 (1995), 1245-1258. Google Scholar
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