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  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:
[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.  Google Scholar

[2]

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

[3]

W. ChenC. 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

[4]

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

[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.  Google Scholar

[6]

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

[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.   Google Scholar

[8]

C. LiZ. 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

[9]

C. MaW. 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

[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.  Google Scholar

[11]

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

[12]

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

[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.   Google Scholar

[14]

P. PoláčikP. 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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[2]

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

[3]

W. ChenC. 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

[4]

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

[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.  Google Scholar

[6]

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

[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.   Google Scholar

[8]

C. LiZ. 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

[9]

C. MaW. 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

[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.  Google Scholar

[11]

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

[12]

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

[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.   Google Scholar

[14]

P. PoláčikP. 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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.   Google Scholar

Figure 1.  Required initial integrability
[1]

Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951

[2]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987

[3]

Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935

[4]

Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057

[5]

Wenjing Chen, Louis Dupaigne, Marius Ghergu. A new critical curve for the Lane-Emden system. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2469-2479. doi: 10.3934/dcds.2014.34.2469

[6]

Ze Cheng, Genggeng Huang. A Liouville theorem for the subcritical Lane-Emden system. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1359-1377. doi: 10.3934/dcds.2019058

[7]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291

[8]

Wenxiong Chen, Congming Li. An integral system and the Lane-Emden conjecture. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1167-1184. doi: 10.3934/dcds.2009.24.1167

[9]

Ze Cheng, Genggeng Huang, Congming Li. On the Hardy-Littlewood-Sobolev type systems. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2059-2074. doi: 10.3934/cpaa.2016027

[10]

Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171

[11]

Hatem Hajlaoui, Abdellaziz Harrabi, Foued Mtiri. Liouville theorems for stable solutions of the weighted Lane-Emden system. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 265-279. doi: 10.3934/dcds.2017011

[12]

Yu Zheng, Carlos A. Santos, Zifei Shen, Minbo Yang. Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents. Communications on Pure & Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018

[13]

Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791

[14]

Philip Korman, Junping Shi. On lane-emden type systems. Conference Publications, 2005, 2005 (Special) : 510-517. doi: 10.3934/proc.2005.2005.510

[15]

Lorenzo D'Ambrosio, Enzo Mitidieri. Hardy-Littlewood-Sobolev systems and related Liouville theorems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 653-671. doi: 10.3934/dcdss.2014.7.653

[16]

Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164

[17]

Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1319-1345. doi: 10.3934/cpaa.2021022

[18]

Jingbo Dou, Fangfang Ren, John Villavert. Classification of positive solutions to a Lane-Emden type integral system with negative exponents. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6767-6780. doi: 10.3934/dcds.2016094

[19]

Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure & Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015

[20]

Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (42)
  • HTML views (59)
  • Cited by (0)

Other articles
by authors

[Back to Top]