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Positive solutions of singular multiparameter p-Laplacian elliptic systems
School of Applied Science, Beijing Information Science & Technology University, Beijing, 100192, China |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
doi: 10.1007/978-3-662-00547-7. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
M. Ghergu,
Lane-Emden systems with negative exponents, J. Funct. Anal., 258 (2010), 3295-3318.
doi: 10.1016/j.jfa.2010.02.003. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[42] |
Q. A. Morris, Analysis of Classes of Superlinear Semipositone Problems with Nonlinear Boundary Conditions, Dissertation of University of North Carolina at Greensboro, 2017. |
[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. |
[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. |
[45] |
J. Serrin and H. Zou,
Nonexistence of positive solutions of Lane-Emden systems, Differ. Integral Equ., 9 (1996), 635-653.
|
[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. |
[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. |
[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. |
[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. |
[50] |
Z. Zhang and Z. Qi,
On a power-type coupled system of Monge-Ampère equations, Topol. Method. Nonl. Anal., 46 (2015), 717-729.
|
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
doi: 10.1007/978-3-662-00547-7. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
M. Ghergu,
Lane-Emden systems with negative exponents, J. Funct. Anal., 258 (2010), 3295-3318.
doi: 10.1016/j.jfa.2010.02.003. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[42] |
Q. A. Morris, Analysis of Classes of Superlinear Semipositone Problems with Nonlinear Boundary Conditions, Dissertation of University of North Carolina at Greensboro, 2017. |
[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. |
[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. |
[45] |
J. Serrin and H. Zou,
Nonexistence of positive solutions of Lane-Emden systems, Differ. Integral Equ., 9 (1996), 635-653.
|
[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. |
[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. |
[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. |
[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. |
[50] |
Z. Zhang and Z. Qi,
On a power-type coupled system of Monge-Ampère equations, Topol. Method. Nonl. Anal., 46 (2015), 717-729.
|
[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. |
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