Positive solutions of singular multiparameter p-Laplacian elliptic systems

Meiqiang Feng; Yichen Zhang

doi:10.3934/dcdsb.2021083

Article Contents

2022, Volume 27, Issue 2: 1121-1147. Doi: 10.3934/dcdsb.2021083

This issue Previous Article Discrete-time dynamics of structured populations via Feller kernels Next Article Mixed Hegselmann-Krause dynamics

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
^* Corresponding author: Meiqiang Feng

Received: August 31, 2020

Revised: October 31, 2020

Early access: March 2021

Published: February 2022

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J60, 35J66; Secondary: 35J92.

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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.
[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.
[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.