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

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

Tianyu Liao

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.
Keywords: Lane-Emden system, Integrability, regularity lifting, contraction mappings, Hardy-Littlewood-Sobolev inequality.
Mathematics Subject Classification: Primary: 35R11.
Citation: Tianyu Liao. The regularity lifting methods for nonnegative solutions of Lane-Emden system. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1681-1698. doi: 10.3934/cpaa.2021036
References:
[1]

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.

[2]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, American Institute of Mathematical Sciences, Springfield, MO, 2010.

[3]

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.

[4]

W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co, 2019.

[5]

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.

[6]

L. Evans, Partial Differential Equations, Wadsworth and Brooks/cole Mathematics, 2010.

[7]

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. 

[8]

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.

[9]

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.

[10]

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.

[11]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in ${\textbf{R}}^N$, Differ. Integral Equ., 9 (1996), 465-479. 

[12]

E. Mitidieri, A Rellich type identity and applications, Commun. Partial Differ. Equ., 18 (1993), 125-151.  doi: 10.1080/03605309308820923.

[13]

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. 

[14]

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.

[15]

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.

[16]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 369-380. 

[17]

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.

[18]

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.

[19]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equ., 9 (1996), 635-653. 

[20]

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.

[21]

M. A. S. Souto, A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differ. Integral Equ., 8 (1995), 1245-1258. 

show all references

References:
[1]

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.

[2]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, American Institute of Mathematical Sciences, Springfield, MO, 2010.

[3]

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.

[4]

W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co, 2019.

[5]

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.

[6]

L. Evans, Partial Differential Equations, Wadsworth and Brooks/cole Mathematics, 2010.

[7]

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. 

[8]

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.

[9]

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.

[10]

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.

[11]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in ${\textbf{R}}^N$, Differ. Integral Equ., 9 (1996), 465-479. 

[12]

E. Mitidieri, A Rellich type identity and applications, Commun. Partial Differ. Equ., 18 (1993), 125-151.  doi: 10.1080/03605309308820923.

[13]

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. 

[14]

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.

[15]

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.

[16]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 369-380. 

[17]

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.

[18]

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.

[19]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equ., 9 (1996), 635-653. 

[20]

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.

[21]

M. A. S. Souto, A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differ. Integral Equ., 8 (1995), 1245-1258. 

Figure 1.  Required initial integrability
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