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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

Yayun Li 1, and  Yutian Lei 2,

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.
Keywords: Elliptic system, Liouville theorem, classification, stable solutions, Joseph-Lundgren condition.
Mathematics Subject Classification: Primary: 35J47; Secondary: 45G15.
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. Caffarelli, B. 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. Caristi, L. 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. Chen, L. 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. Chen, C. 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. Davila, L. Dupaigne, K. 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. Polacik, P. 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. Caffarelli, B. 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. Caristi, L. 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. Chen, L. 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. Chen, C. 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. Davila, L. Dupaigne, K. 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. Polacik, P. 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

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