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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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. )

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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. )

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[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.

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