# American Institute of Mathematical Sciences

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. 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. [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. [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. [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. 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. [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. 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. [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. 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. [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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##### 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. 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. [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. [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. [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. 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. [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. 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. [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. 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. [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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