American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2021083

Positive solutions of singular multiparameter p-Laplacian elliptic systems

Meiqiang Feng , and  Yichen Zhang

School of Applied Science, Beijing Information Science & Technology University, Beijing, 100192, China

* Corresponding author: Meiqiang Feng

Received  September 2020 Revised  November 2020 Early access  March 2021

Fund Project: The first author is supported by the Beijing Natural Science Foundation of China (1212003)

In this paper, by using the eigenvalue theory, the sub-supersolution method and the fixed point theory, we prove the existence, multiplicity, uniqueness, asymptotic behavior and approximation of positive solutions for singular multiparameter p-Laplacian elliptic systems on nonlinearities with separate variables or without separate variables. Various nonexistence results of positive solutions are also studied.

Keywords: Singular p-Laplacian systems, positive solutions, uniqueness and approximation, existence and asymptotic behavior, sub-supersolution method.
Mathematics Subject Classification: Primary: 35J60, 35J66; Secondary: 35J92.
Citation: Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021083
References:
[1]

A. Ahammou, Positive radial solutions of nonlinear elliptic systems, New York J. Math., 7 (2001), 267–280. http://nyjm.albany.edu/j/2001/7_267.html.  Google Scholar

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C. O. Alves and D. G. de Figueiredo, Nonvariational elliptic systems, Discrete Contin. Dynam. Systems, 8 (2002), 289-302.  doi: 10.3934/dcds.2002.8.289.  Google Scholar

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M. Benrhouma, Existence of solutions for a semilinear elliptic system, ESAIM Cont. Opt. Cal. Var., 19 (2013), 574-586.  doi: 10.1051/cocv/2012022.  Google Scholar

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M. Benrhouma, Existence and uniqueness of solutions for a singular semilinear elliptic system, Nonlinear Anal., 107 (2014), 134-146.  doi: 10.1016/j.na.2014.05.002.  Google Scholar

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I. Birindelli and E. Mitidieri, Liouville theorems for elliptic inequalities and applications, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1217-1247.  doi: 10.1017/S0308210500027293.  Google Scholar

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D. Bonheure, E. M. dos Santos and H. Tavares, Hamiltonian elliptic systems: A guide to variational frameworks, Port. Math., 71 (2014), 301-395.  doi: 10.4171/PM/1954.  Google Scholar

[7]

Y. Bozhkov and E. Mitidieri, Existence of multiple solutions for quasilinear systems via fibering method, J. Differential Equations, 190 (2003), 239-267.  doi: 10.1016/S0022-0396(02)00112-2.  Google Scholar

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J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56.  doi: 10.1006/jdeq.1999.3701.  Google Scholar

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J. Busca and R. Man$\acute{a}$sevich, A Liouville-type theorem for Lane-Emden systems, Indiana Univ. Math. J., 51 (2002), 37-51.  doi: 10.1512/iumj.2002.51.2160.  Google Scholar

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C. Cosner, Positive solutions for superlinear elliptic systems, without variational structure, Nonlinear Anal., 8 (1984), 1427-1436.  doi: 10.1016/0362-546X(84)90053-1.  Google Scholar

[11]

D. Cao, S. Peng and S. Yan, Infinitely many solutions for $p$-Laplacian equation involving critical Sobolev growth, J. Funct. Anal., 262 (2012), 2861-2902.  doi: 10.1016/j.jfa.2012.01.006.  Google Scholar

[12]

M. Chhetri, R. Shivaji, B. Son and L. Sankar, An existence result for superlinear semipositone $p$-Laplacian systems on the exterior of a ball, Differ. Integral Equ., 31 (2018), 643–656, https://mathscinet.ams.org/leavingmsn?url=https://projecteuclid.org/euclid.die/1526004034.  Google Scholar

[13]

M. Chhetri and P. Girg, Existence of positive solutions for a class of superlinear semipositone systems, J. Math. Anal. Appl., 408 (2013) 781–788. doi: 10.1016/j.jmaa.2013.06.041.  Google Scholar

[14]

F.-C. Şt. Cȋrstea and V. D. R$\breve{a}$dulescu, Entire solutions blowing up at infinity for semilinear elliptic systems, J. Math. Pures Appl., 81 (2002), 827-846.  doi: 10.1016/S0021-7824(02)01265-5.  Google Scholar

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Ph. Clément, D. G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations, 17 (1992), 923-940.  doi: 10.1080/03605309208820869.  Google Scholar

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R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems, Nonlinear Anal., 39 (2000), 559-568.  doi: 10.1016/S0362-546X(98)00221-1.  Google Scholar

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L. D'Ambrosio and E. Mitidieri, Quasilinear elliptic systems in divergence form associated to general nonlinearities, Adv. Nonlinear Anal., 7 (2018), 425-447.  doi: 10.1515/anona-2018-0171.  Google Scholar

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J. M. do Ó, S. Lorca, J. S$\acute{a}$nchez and P. Ubilla, Positive solutions for a class of multiparameter ordinary elliptic systems, J. Math. Anal. Appl., 332 (2007), 1249-1266.  doi: 10.1016/j.jmaa.2006.10.063.  Google Scholar

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D. R. Dunninger and H. Wang, Multiplicity of positive radial solutions for an elliptic system on an annulus, Nonlinear Anal., 42 (2000), 803-811.  doi: 10.1016/S0362-546X(99)00125-X.  Google Scholar

[21]

P. Felmer, R. F. Manásevich and F. de Thélin, Existence and uniqueness of positive solutions for certain quasilinear elliptic systems, Comm. Partial Differential Equations, 17 (1992), 2013-2029.  doi: 10.1080/03605309208820912.  Google Scholar

[22]

M. Feng, Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior, Adv. Nonlinear Anal., 10 (2021), 371-399.  doi: 10.1515/anona-2020-0139.  Google Scholar

[23]

M. Feng, B. Du and W. Ge, Impulsive boundary value problems with integral boundary conditions and one-dimensional $p$-Laplacian, Nonlinear Anal., 70 (2009), 3119-3126.  doi: 10.1016/j.na.2008.04.015.  Google Scholar

[24]

G. Galise, On positive solutions of fully nonlinear degenerate Lane-Emden type equations, J. Differential Equations, 266 (2019), 1675-169.  doi: 10.1016/j.jde.2018.08.014.  Google Scholar

[25]

M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal., 258 (2010), 3295-3318.  doi: 10.1016/j.jfa.2010.02.003.  Google Scholar

[26]

M. Ghergu and V. R$\breve{a}$dulescu, Explosive solutions of semilinear elliptic systems with gradient term, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 97 (2003), 437-445.   Google Scholar

[27]

D. Guo, Eigenvalue and eigenvectors of nonlinear operators, Chin. Ann. Math., 2 (Eng. Issue) (1981), 65-80. Google Scholar

[28]

D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988.  Google Scholar

[29]

D. D. Hai, Existence and uniqueness of solutions for quasilinear elliptic systems, Proc. Amer. Math. Soc., 133 (2005), 223-228.  doi: 10.1090/S0002-9939-04-07602-6.  Google Scholar

[30]

D. D. Hai, On a class of semilinear elliptic systems, J. Math. Anal. Appl., 285 (2003), 477-486.  doi: 10.1016/S0022-247X(03)00413-X.  Google Scholar

[31]

D. D. Hai, Uniqueness of positive solutions for a class of semilinear elliptic systems, Nonlinear Anal., 52 (2003), 595-603.  doi: 10.1016/S0362-546X(02)00125-6.  Google Scholar

[32]

D. D. Hai and R. Shivaji, Positive radial solutions for a class of singular superlinear problems on the exterior of a ball with nonlinear boundary conditions, J. Math. Anal. Appl., 456 (2017), 872-881.  doi: 10.1016/j.jmaa.2017.06.088.  Google Scholar

[33]

D. D. Hai and R. Shivaji, Existence and multiplicity of positive radial solutions for singular superlinear elliptic systems in the exterior of a ball, J. Differential Equations, 266 (2019), 2232-2243.  doi: 10.1016/j.jde.2018.08.027.  Google Scholar

[34]

S. Hu and H. Wang, Convex solutions of boundary value problems arising from Monge-Ampère equations, Discrete Contin. Dynam. Systems, 16 (2006), 705-720.  doi: 10.3934/dcds.2006.16.705.  Google Scholar

[35]

N. Kawano and T. Kusano, On positive entire solutions of a class of second order semilinear elliptic systems, Math. Zeitschrift, 186 (1984), 287-297.  doi: 10.1007/BF01174883.  Google Scholar

[36]

M. A. Krasnosel'skii, Positive Solutions of Operators Equations, Noordhoff, Groningen, 1964. Google Scholar

[37]

A. V. Lair and A. W. Wood, Existence of entire large positive solutions of semilinear elliptic systems, J. Differential Equations, 164 (2000), 380-394.  doi: 10.1006/jdeq.2000.3768.  Google Scholar

[38]

K. Q. Lan and Z. Zhang, Nonzero positive weak solutions of systems of $p$-Laplace equations, J. Math. Anal. Appl., 394 (2012), 581-591.  doi: 10.1016/j.jmaa.2012.04.061.  Google Scholar

[39]

Y.-H. Lee, Multiplicity of positive radial solutions for multiparameter semilinear elliptic systems on an annulus, J. Differential Equations, 174 (2001), 420-441.  doi: 10.1006/jdeq.2000.3915.  Google Scholar

[40]

M. Maniwa, Uniqueness and existence of positive solutions for some semilinear elliptic systems, Nonlinear Anal., 59 (2004), 993-999.  doi: 10.1016/j.na.2004.08.006.  Google Scholar

[41]

N. Mavinga and R. Pardo, A priori bounds and existence of positive solutions for semilinear elliptic systems, J. Math. Anal. Appl., 449 (2017), 1172-1188.  doi: 10.1016/j.jmaa.2016.12.058.  Google Scholar

[42]

Q. A. Morris, Analysis of Classes of Superlinear Semipositone Problems with Nonlinear Boundary Conditions, Dissertation of University of North Carolina at Greensboro, 2017.  Google Scholar

[43]

R. Precup, Existence, localization and multiplicity results for positive radial solutions of semilinear elliptic systems, J. Math. Anal. Appl., 352 (2009), 48-56.  doi: 10.1016/j.jmaa.2008.01.097.  Google Scholar

[44]

P. Quittner and Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal., 174 (2004), 49-81.  doi: 10.1007/s00205-004-0323-8.  Google Scholar

[45]

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

[46]

B. Son and P. Wang, Analysis of positive radial solutions for singular superlinear $p$-Laplacian systems on the exterior of a ball, Nonlinear Anal., 192 (2020), 111657, 15 pp. doi: 10.1016/j.na.2019.111657.  Google Scholar

[47]

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

[48]

M. Xiang, B. Zhang and V. D. R$\breve{a}$dulescu, Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 3186-3205.  doi: 10.1088/0951-7715/29/10/3186.  Google Scholar

[49]

Y. Zhang and M. Feng, A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior, Electron. Res. Arch., 28 (2020), 1419-1438.  doi: 10.3934/era.2020075.  Google Scholar

[50]

Z. Zhang and Z. Qi, On a power-type coupled system of Monge-Ampère equations, Topol. Method. Nonl. Anal., 46 (2015), 717-729.   Google Scholar

[51]

H. Zou, A priori estimates for a semilinear elliptic system without variational structure and their applications, Math. Ann., 323 (2002), 713-735.  doi: 10.1007/s002080200324.  Google Scholar

show all references

References:
[1]

A. Ahammou, Positive radial solutions of nonlinear elliptic systems, New York J. Math., 7 (2001), 267–280. http://nyjm.albany.edu/j/2001/7_267.html.  Google Scholar

[2]

C. O. Alves and D. G. de Figueiredo, Nonvariational elliptic systems, Discrete Contin. Dynam. Systems, 8 (2002), 289-302.  doi: 10.3934/dcds.2002.8.289.  Google Scholar

[3]

M. Benrhouma, Existence of solutions for a semilinear elliptic system, ESAIM Cont. Opt. Cal. Var., 19 (2013), 574-586.  doi: 10.1051/cocv/2012022.  Google Scholar

[4]

M. Benrhouma, Existence and uniqueness of solutions for a singular semilinear elliptic system, Nonlinear Anal., 107 (2014), 134-146.  doi: 10.1016/j.na.2014.05.002.  Google Scholar

[5]

I. Birindelli and E. Mitidieri, Liouville theorems for elliptic inequalities and applications, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1217-1247.  doi: 10.1017/S0308210500027293.  Google Scholar

[6]

D. Bonheure, E. M. dos Santos and H. Tavares, Hamiltonian elliptic systems: A guide to variational frameworks, Port. Math., 71 (2014), 301-395.  doi: 10.4171/PM/1954.  Google Scholar

[7]

Y. Bozhkov and E. Mitidieri, Existence of multiple solutions for quasilinear systems via fibering method, J. Differential Equations, 190 (2003), 239-267.  doi: 10.1016/S0022-0396(02)00112-2.  Google Scholar

[8]

J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56.  doi: 10.1006/jdeq.1999.3701.  Google Scholar

[9]

J. Busca and R. Man$\acute{a}$sevich, A Liouville-type theorem for Lane-Emden systems, Indiana Univ. Math. J., 51 (2002), 37-51.  doi: 10.1512/iumj.2002.51.2160.  Google Scholar

[10]

C. Cosner, Positive solutions for superlinear elliptic systems, without variational structure, Nonlinear Anal., 8 (1984), 1427-1436.  doi: 10.1016/0362-546X(84)90053-1.  Google Scholar

[11]

D. Cao, S. Peng and S. Yan, Infinitely many solutions for $p$-Laplacian equation involving critical Sobolev growth, J. Funct. Anal., 262 (2012), 2861-2902.  doi: 10.1016/j.jfa.2012.01.006.  Google Scholar

[12]

M. Chhetri, R. Shivaji, B. Son and L. Sankar, An existence result for superlinear semipositone $p$-Laplacian systems on the exterior of a ball, Differ. Integral Equ., 31 (2018), 643–656, https://mathscinet.ams.org/leavingmsn?url=https://projecteuclid.org/euclid.die/1526004034.  Google Scholar

[13]

M. Chhetri and P. Girg, Existence of positive solutions for a class of superlinear semipositone systems, J. Math. Anal. Appl., 408 (2013) 781–788. doi: 10.1016/j.jmaa.2013.06.041.  Google Scholar

[14]

F.-C. Şt. Cȋrstea and V. D. R$\breve{a}$dulescu, Entire solutions blowing up at infinity for semilinear elliptic systems, J. Math. Pures Appl., 81 (2002), 827-846.  doi: 10.1016/S0021-7824(02)01265-5.  Google Scholar

[15]

Ph. Clément, D. G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations, 17 (1992), 923-940.  doi: 10.1080/03605309208820869.  Google Scholar

[16]

R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems, Nonlinear Anal., 39 (2000), 559-568.  doi: 10.1016/S0362-546X(98)00221-1.  Google Scholar

[17]

L. D'Ambrosio and E. Mitidieri, Quasilinear elliptic systems in divergence form associated to general nonlinearities, Adv. Nonlinear Anal., 7 (2018), 425-447.  doi: 10.1515/anona-2018-0171.  Google Scholar

[18]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[19]

J. M. do Ó, S. Lorca, J. S$\acute{a}$nchez and P. Ubilla, Positive solutions for a class of multiparameter ordinary elliptic systems, J. Math. Anal. Appl., 332 (2007), 1249-1266.  doi: 10.1016/j.jmaa.2006.10.063.  Google Scholar

[20]

D. R. Dunninger and H. Wang, Multiplicity of positive radial solutions for an elliptic system on an annulus, Nonlinear Anal., 42 (2000), 803-811.  doi: 10.1016/S0362-546X(99)00125-X.  Google Scholar

[21]

P. Felmer, R. F. Manásevich and F. de Thélin, Existence and uniqueness of positive solutions for certain quasilinear elliptic systems, Comm. Partial Differential Equations, 17 (1992), 2013-2029.  doi: 10.1080/03605309208820912.  Google Scholar

[22]

M. Feng, Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior, Adv. Nonlinear Anal., 10 (2021), 371-399.  doi: 10.1515/anona-2020-0139.  Google Scholar

[23]

M. Feng, B. Du and W. Ge, Impulsive boundary value problems with integral boundary conditions and one-dimensional $p$-Laplacian, Nonlinear Anal., 70 (2009), 3119-3126.  doi: 10.1016/j.na.2008.04.015.  Google Scholar

[24]

G. Galise, On positive solutions of fully nonlinear degenerate Lane-Emden type equations, J. Differential Equations, 266 (2019), 1675-169.  doi: 10.1016/j.jde.2018.08.014.  Google Scholar

[25]

M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal., 258 (2010), 3295-3318.  doi: 10.1016/j.jfa.2010.02.003.  Google Scholar

[26]

M. Ghergu and V. R$\breve{a}$dulescu, Explosive solutions of semilinear elliptic systems with gradient term, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 97 (2003), 437-445.   Google Scholar

[27]

D. Guo, Eigenvalue and eigenvectors of nonlinear operators, Chin. Ann. Math., 2 (Eng. Issue) (1981), 65-80. Google Scholar

[28]

D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988.  Google Scholar

[29]

D. D. Hai, Existence and uniqueness of solutions for quasilinear elliptic systems, Proc. Amer. Math. Soc., 133 (2005), 223-228.  doi: 10.1090/S0002-9939-04-07602-6.  Google Scholar

[30]

D. D. Hai, On a class of semilinear elliptic systems, J. Math. Anal. Appl., 285 (2003), 477-486.  doi: 10.1016/S0022-247X(03)00413-X.  Google Scholar

[31]

D. D. Hai, Uniqueness of positive solutions for a class of semilinear elliptic systems, Nonlinear Anal., 52 (2003), 595-603.  doi: 10.1016/S0362-546X(02)00125-6.  Google Scholar

[32]

D. D. Hai and R. Shivaji, Positive radial solutions for a class of singular superlinear problems on the exterior of a ball with nonlinear boundary conditions, J. Math. Anal. Appl., 456 (2017), 872-881.  doi: 10.1016/j.jmaa.2017.06.088.  Google Scholar

[33]

D. D. Hai and R. Shivaji, Existence and multiplicity of positive radial solutions for singular superlinear elliptic systems in the exterior of a ball, J. Differential Equations, 266 (2019), 2232-2243.  doi: 10.1016/j.jde.2018.08.027.  Google Scholar

[34]

S. Hu and H. Wang, Convex solutions of boundary value problems arising from Monge-Ampère equations, Discrete Contin. Dynam. Systems, 16 (2006), 705-720.  doi: 10.3934/dcds.2006.16.705.  Google Scholar

[35]

N. Kawano and T. Kusano, On positive entire solutions of a class of second order semilinear elliptic systems, Math. Zeitschrift, 186 (1984), 287-297.  doi: 10.1007/BF01174883.  Google Scholar

[36]

M. A. Krasnosel'skii, Positive Solutions of Operators Equations, Noordhoff, Groningen, 1964. Google Scholar

[37]

A. V. Lair and A. W. Wood, Existence of entire large positive solutions of semilinear elliptic systems, J. Differential Equations, 164 (2000), 380-394.  doi: 10.1006/jdeq.2000.3768.  Google Scholar

[38]

K. Q. Lan and Z. Zhang, Nonzero positive weak solutions of systems of $p$-Laplace equations, J. Math. Anal. Appl., 394 (2012), 581-591.  doi: 10.1016/j.jmaa.2012.04.061.  Google Scholar

[39]

Y.-H. Lee, Multiplicity of positive radial solutions for multiparameter semilinear elliptic systems on an annulus, J. Differential Equations, 174 (2001), 420-441.  doi: 10.1006/jdeq.2000.3915.  Google Scholar

[40]

M. Maniwa, Uniqueness and existence of positive solutions for some semilinear elliptic systems, Nonlinear Anal., 59 (2004), 993-999.  doi: 10.1016/j.na.2004.08.006.  Google Scholar

[41]

N. Mavinga and R. Pardo, A priori bounds and existence of positive solutions for semilinear elliptic systems, J. Math. Anal. Appl., 449 (2017), 1172-1188.  doi: 10.1016/j.jmaa.2016.12.058.  Google Scholar

[42]

Q. A. Morris, Analysis of Classes of Superlinear Semipositone Problems with Nonlinear Boundary Conditions, Dissertation of University of North Carolina at Greensboro, 2017.  Google Scholar

[43]

R. Precup, Existence, localization and multiplicity results for positive radial solutions of semilinear elliptic systems, J. Math. Anal. Appl., 352 (2009), 48-56.  doi: 10.1016/j.jmaa.2008.01.097.  Google Scholar

[44]

P. Quittner and Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal., 174 (2004), 49-81.  doi: 10.1007/s00205-004-0323-8.  Google Scholar

[45]

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

[46]

B. Son and P. Wang, Analysis of positive radial solutions for singular superlinear $p$-Laplacian systems on the exterior of a ball, Nonlinear Anal., 192 (2020), 111657, 15 pp. doi: 10.1016/j.na.2019.111657.  Google Scholar

[47]

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

[48]

M. Xiang, B. Zhang and V. D. R$\breve{a}$dulescu, Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 3186-3205.  doi: 10.1088/0951-7715/29/10/3186.  Google Scholar

[49]

Y. Zhang and M. Feng, A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior, Electron. Res. Arch., 28 (2020), 1419-1438.  doi: 10.3934/era.2020075.  Google Scholar

[50]

Z. Zhang and Z. Qi, On a power-type coupled system of Monge-Ampère equations, Topol. Method. Nonl. Anal., 46 (2015), 717-729.   Google Scholar

[51]

H. Zou, A priori estimates for a semilinear elliptic system without variational structure and their applications, Math. Ann., 323 (2002), 713-735.  doi: 10.1007/s002080200324.  Google Scholar

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