September  2018, 17(5): 1749-1764. doi: 10.3934/cpaa.2018083

On existence and nonexistence of positive solutions of an elliptic system with coupled terms

1. 

Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu, 210023, China

2. 

Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

Received  April 2017 Revised  December 2017 Published  April 2018

Fund Project: This research was supported by NSF (11471164, 11671209) of China

This paper is concerned with the elliptic system
$\begin{cases}-{\triangle u} = (q+1)u^qv^{p+1},~~ u>0~ in~ R^n,\\-{\triangle v} = (p+1)v^pu^{q+1},~~ v>0~in ~R^n,\end{cases}$
where
$ n ≥ 3 $
,
$ p,q>0 $
and
$ \max\{p,q\} ≥ 1 $
. We discuss the nonexistence of positive solutions in subcritical case and stable solutions in supercritical case, the necessary and sufficient conditions of classification in the critical case, and the Joseph-Lundgren-type condition for existence of local stable solutions.
Citation: Yayun Li, Yutian Lei. On existence and nonexistence of positive solutions of an elliptic system with coupled terms. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1749-1764. doi: 10.3934/cpaa.2018083
References:
[1]

M.-F. Bidaut-Véron and Th. Raoux, Asymptotics of solutions of some nonlinear elliptic systems, Comm. Partial Differential Equations, 21 (1996), 1035-1086. doi: 10.1080/03605309608821217. Google Scholar

[2]

L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. Google Scholar

[3]

G. CaristiL. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67. doi: 10.1007/s00032-008-0090-3. Google Scholar

[4]

W. ChenL. Dupaigne and M. Ghergu, A new critical curve for the Lane-Emden system, Discrete Contin. Dyn. Syst., 34 (2014), 2469-2479. doi: 10.3934/dcds.2014.34.2469. Google Scholar

[5]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar

[6]

W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Math. Sci., 29B (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5. Google Scholar

[7]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., Vol. 4, 2010. Google Scholar

[8]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497. Google Scholar

[9]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. Google Scholar

[10]

K. Chou and C. Chu, On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc., 2 (1993), 137-151. doi: 10.1112/jlms/s2-48.1.137. Google Scholar

[11]

J. DavilaL. DupaigneK. Wang and J. Wei, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258 (2014), 240-285. doi: 10.1016/j.aim.2014.02.034. Google Scholar

[12]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $ R^N $, J. Math. Pures Appl., 87 (2007), 537-561. doi: 10.1016/j.matpur.2007.03.001. Google Scholar

[13]

B. Gidas, W. -M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $ R^{n} $ (collected in the book Mathematical Analysis and Applications, which is vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981. ) Google Scholar

[14]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406. Google Scholar

[15]

D. Joseph and T. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Meth. Anal., 49 (1972/73), 241-269. doi: 10.1007/BF00250508. Google Scholar

[16]

Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905. doi: 10.1007/s00209-012-1036-6. Google Scholar

[17]

Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst., 36 (2016), 3277-3315. doi: 10.3934/dcds.2016.36.3277. Google Scholar

[18]

W.-M. Ni, On the elliptic equation $ Δ u+K(x)u^{(n+2)/(n-2)} = 0 $, its generalizations, and applications in geometry, Indiana Univ. Math. J., 31 (1982), 493-529. doi: 10.1512/iumj.1982.31.31040. Google Scholar

[19]

W.-M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations. The anomalous case, Accad. Naz. Lincei., 77 (1986), 231-257. Google Scholar

[20]

P. PolacikP. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part Ⅰ: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579. doi: 10.1215/S0012-7094-07-13935-8. Google Scholar

[21]

P. Quittner and Ph. Souplet, Symmetry of components for semilinear elliptic systems, SIAM J. Math. Anal., 44 (2012), 2545-2559. doi: 10.1137/11085428X. Google Scholar

[22]

J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Math., 113 (1965), 219-240. Google Scholar

[23]

X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590. doi: 10.2307/2154232. Google Scholar

[24]

Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system, Nonlinear Anal., 75 (2012), 1989-1999. doi: 10.1016/j.na.2011.09.051. Google Scholar

show all references

References:
[1]

M.-F. Bidaut-Véron and Th. Raoux, Asymptotics of solutions of some nonlinear elliptic systems, Comm. Partial Differential Equations, 21 (1996), 1035-1086. doi: 10.1080/03605309608821217. Google Scholar

[2]

L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. Google Scholar

[3]

G. CaristiL. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67. doi: 10.1007/s00032-008-0090-3. Google Scholar

[4]

W. ChenL. Dupaigne and M. Ghergu, A new critical curve for the Lane-Emden system, Discrete Contin. Dyn. Syst., 34 (2014), 2469-2479. doi: 10.3934/dcds.2014.34.2469. Google Scholar

[5]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar

[6]

W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Math. Sci., 29B (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5. Google Scholar

[7]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., Vol. 4, 2010. Google Scholar

[8]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497. Google Scholar

[9]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. Google Scholar

[10]

K. Chou and C. Chu, On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc., 2 (1993), 137-151. doi: 10.1112/jlms/s2-48.1.137. Google Scholar

[11]

J. DavilaL. DupaigneK. Wang and J. Wei, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258 (2014), 240-285. doi: 10.1016/j.aim.2014.02.034. Google Scholar

[12]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $ R^N $, J. Math. Pures Appl., 87 (2007), 537-561. doi: 10.1016/j.matpur.2007.03.001. Google Scholar

[13]

B. Gidas, W. -M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $ R^{n} $ (collected in the book Mathematical Analysis and Applications, which is vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981. ) Google Scholar

[14]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406. Google Scholar

[15]

D. Joseph and T. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Meth. Anal., 49 (1972/73), 241-269. doi: 10.1007/BF00250508. Google Scholar

[16]

Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905. doi: 10.1007/s00209-012-1036-6. Google Scholar

[17]

Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst., 36 (2016), 3277-3315. doi: 10.3934/dcds.2016.36.3277. Google Scholar

[18]

W.-M. Ni, On the elliptic equation $ Δ u+K(x)u^{(n+2)/(n-2)} = 0 $, its generalizations, and applications in geometry, Indiana Univ. Math. J., 31 (1982), 493-529. doi: 10.1512/iumj.1982.31.31040. Google Scholar

[19]

W.-M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations. The anomalous case, Accad. Naz. Lincei., 77 (1986), 231-257. Google Scholar

[20]

P. PolacikP. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part Ⅰ: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579. doi: 10.1215/S0012-7094-07-13935-8. Google Scholar

[21]

P. Quittner and Ph. Souplet, Symmetry of components for semilinear elliptic systems, SIAM J. Math. Anal., 44 (2012), 2545-2559. doi: 10.1137/11085428X. Google Scholar

[22]

J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Math., 113 (1965), 219-240. Google Scholar

[23]

X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590. doi: 10.2307/2154232. Google Scholar

[24]

Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system, Nonlinear Anal., 75 (2012), 1989-1999. doi: 10.1016/j.na.2011.09.051. Google Scholar

[1]

Shoichi Hasegawa. A critical exponent of Joseph-Lundgren type for an Hénon equation on the hyperbolic space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1189-1198. doi: 10.3934/cpaa.2017058

[2]

Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067

[3]

Phuong Le. Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 511-525. doi: 10.3934/cpaa.2020025

[4]

Yuxia Guo, Jianjun Nie. Classification for positive solutions of degenerate elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1457-1475. doi: 10.3934/dcds.2018130

[5]

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

[6]

Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855

[7]

Genggeng Huang. A Liouville theorem of degenerate elliptic equation and its application. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4549-4566. doi: 10.3934/dcds.2013.33.4549

[8]

Chang-Shou Lin, Lei Zhang. Classification of radial solutions to Liouville systems with singularities. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2617-2637. doi: 10.3934/dcds.2014.34.2617

[9]

Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155

[10]

Xian-gao Liu, Xiaotao Zhang. Liouville theorem for MHD system and its applications. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2329-2350. doi: 10.3934/cpaa.2018111

[11]

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

[12]

Piero Montecchiari, Paul H. Rabinowitz. A nondegeneracy condition for a semilinear elliptic system and the existence of 1- bump solutions. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-18. doi: 10.3934/dcds.2019241

[13]

M. Á. Burgos-Pérez, J. García-Melián, A. Quaas. Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4703-4721. doi: 10.3934/dcds.2016004

[14]

Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947

[15]

Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317

[16]

Olivier Goubet. Regularity of extremal solutions of a Liouville system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 339-345. doi: 10.3934/dcdss.2019023

[17]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511

[18]

Ovidiu Savin. A Liouville theorem for solutions to the linearized Monge-Ampere equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 865-873. doi: 10.3934/dcds.2010.28.865

[19]

Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248

[20]

Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (102)
  • HTML views (199)
  • Cited by (0)

Other articles
by authors

[Back to Top]