EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A QUASILINEAR ELLIPTIC SYSTEM ON UNBOUNDED DOMAINS INVOLVING NONLINEAR BOUNDARY CONDITIONS

Somayeh Khademloo; Ghasem Alizadeh Afrouzi; Jiafa Xu

doi:10.11948/20190192

2020 Volume 10 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2020 > 10(3): 1094-1106

Next Article Previous Article

Somayeh Khademloo, Ghasem Alizadeh Afrouzi, Jiafa Xu. EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A QUASILINEAR ELLIPTIC SYSTEM ON UNBOUNDED DOMAINS INVOLVING NONLINEAR BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 1094-1106. doi: 10.11948/20190192

Citation:

Somayeh Khademloo, Ghasem Alizadeh Afrouzi, Jiafa Xu. EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A QUASILINEAR ELLIPTIC SYSTEM ON UNBOUNDED DOMAINS INVOLVING NONLINEAR BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 1094-1106. doi: 10.11948/20190192

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A QUASILINEAR ELLIPTIC SYSTEM ON UNBOUNDED DOMAINS INVOLVING NONLINEAR BOUNDARY CONDITIONS

1.
Department of Mathematics, Faculty of Basic Sciences, Babol (Noushirvani) University of Technology Babol, Iran
2.
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
3.
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China

Author Bio: Email address: s.khademloo@nit.ac.ir (S. Khademloo); Email address: afrouzi@umz.ac.ir (G. A. Afrouzi)
Corresponding author: Email address: xujiafa292@sina.com (J. Xu)
Fund Project: The authors were supported by Talent Project of Chongqing Normal University (Grant No. 02030307-0040), the National Natural Science Foundation of China(Grant No. 11601048), Natural Science Foundation of Chongqing Normal University (Grant No. 16XYY24)

Abstract

Abstract

We prove two existence results for the nonlinear elliptic boundary value system involving $p$-Laplacian over an unbounded domain in $R^N$ with noncompact boundary. The proofs are based on variational methods applied to weighted spaces.
- Quasilinear elliptic systems /
- nonlinear boundary conditions /
- variational methods /
- weighted function spaces
MSC: 35J20, 35J60, 35J70

FullText(HTML)

References

[1]	G. Aronsson and U. Janfalk, On Hele-Shaw flow of power-law fluids, Eur. J. Appl. Math., 1992, 3(4), 343-366. doi: 10.1017/S0956792500000905 CrossRef Google Scholar
[2]	Z. Bai, Z. Du and S. Zhang, Iterative method for a class of fourth-order p-Laplacian beam equation, Journal of Applied Analysis and Computation, 2019, 9(4), 1443-1453. Google Scholar
[3]	F. Browder, Existence theorems for nonlinear partial differential equations, Proc. Sympos. Pure Math., Vol. 16, Amer. Math. Sot., Providence, RI, 1970. Google Scholar
[4]	L. Chen, C. Chen, H. Yang and H. Song, Infinite radial solutions for the fractional Kirchhoff equation, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 2019,113(3), 2309-2318. Google Scholar
[5]	C. Chen and H. Yang, Multiple solutions for a class of quasilinear Schrödinger systems in $\mathbb R.N$, Bull. Malays. Math. Sci. Soc., 2019, 42(2), 611-636. doi: 10.1007/s40840-017-0502-z CrossRef $\mathbb R.N$" target="_blank">Google Scholar
[6]	L. Chen, C. Chen, H. Yang and H. Song, Nonexistence of stable solutions for quasilinear Schrödinger equation, Bound. Value Probl., 2018,168, 11. Google Scholar
[7]	C. Chen, H. Song and H. Yang, Liouville-type theorems for stable solutions of singular quasilinear elliptic equations in $\mathbb R.N$, Electron. J. Differ. Equ., 2018, 2018(81), 1-11. $\mathbb R.N$" target="_blank">Google Scholar
[8]	C. Chen, H. Song and H. Yang, Liouville type theorems for stable solutions of p-Laplace equation in $\mathbb R.N$, Nonlinear Anal., 2017,160, 44-52. doi: 10.1016/j.na.2017.05.004 CrossRef $\mathbb R.N$" target="_blank">Google Scholar
[9]	I. J. Díaz, Nonlinear partial differential equations and free boundaries, Vol. Ⅰ. Elliptic equations, Research Notes in Mathematics, 106. Pitman, Boston, MA, 1985. Google Scholar
[10]	X. Dong, Z. Bai and S. Zhang, Positive solutions to boundary value problems of p-Laplacian with fractional derivative, Bound. Value Probl., 2017, 5, 15. Google Scholar
[11]	R. Dautray and L. J. Lions, Mathematical analysis and numerical methods for science and technology, Vol. 1: Physical origins and classical methods, Springer-Verlag, Berlin, 1985. Google Scholar
[12]	L. W. Findley, S. Lai and K. Onaran, Creep and relaxation of nonlinear viscoelastic materials, North Holland Publ. House, Amsterdam-New York-Oxford, 1976. Google Scholar
[13]	X. Hao, H. Wang, L. Liu and Y. Cui, Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator, Bound. Value Probl., 2017,182, 18. Google Scholar
[14]	J. Jiang, D. O'Regan, J. Xu and Y. Cui, Positive solutions for a Hadamard fractional p-Laplacian three-point boundary value problem, Mathematics, 2019, 7(5), 439. doi: 10.3390/math7050439 CrossRef Google Scholar
[15]	M. L. Kachanov, The theory of creep, National Lending Liberary for science and technology, Boston Spa, Yorkshire, England, 1967. Google Scholar
[16]	M. L. Kachanov, Foundations of the theory of plasticity, North Holland Publ. House, Amsterdam-London, 1971. Google Scholar
[17]	H. Lian, D. Wang, Z. Bai and R. P. Agarwal, Periodic and subharmonic solutions for a class of second-order p-Laplacian Hamiltonian systems, Bound. Value Probl., 2014,260, 15. Google Scholar
[18]	E. Montefusco and V. Rădulescu, Nonlinear eigenvalue problems for quasilinear operators on unbounded domains, Nonlinear Differ. Eua. Appl., 2001, 8(4), 481-497. doi: 10.1007/PL00001460 CrossRef Google Scholar
[19]	V. Pao, Nonlinear parabolic and elliptic equations, Plenum press, New York, London, 1992. Google Scholar
[20]	K. Pfluger, Existence and multiplicity of solutions to a p-Laplacian equation with nonlinear boundary condition, Electron. J. Differ. Equ., 1998, 1998(10), 1-13. Google Scholar
[21]	K. Pfluger, Compact traces in weighted Sobolev spaces, Analysis, 1998, 18, 65-83. Google Scholar
[22]	T. Ren, S. Li, X. Zhang and L. Liu, Maximum and minimum solutions for a nonlocal p-Laplacian fractional differential system from eco-economical processes, Bound. Value Probl., 2017,118, 15. Google Scholar
[23]	H. P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CMBS Reg. Conf. Ser. Math., 1986, 65. Google Scholar
[24]	A. Szulkin, Ljusternik-Schnirelmann theory on $C.1$-manifold, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1988, 5(2), 119-139. doi: 10.1016/S0294-1449(16)30348-1 CrossRef $C.1$-manifold" target="_blank">Google Scholar
[25]	J. Sun, J. Chu and T. Wu, Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian, J. Differ. Equ., 2017,262(2), 945-977. Google Scholar
[26]	K. Sheng, W. Zhang and Z. Bai, Positive solutions to fractional boundary-value problems with p-Laplacian on time scales, Bound. Value Probl., 2018, 70, 15. Google Scholar
[27]	Y. Tian, S. Sun and Z. Bai, Positive solutions of fractional differential equations with p-Laplacian, J. Funct. Spaces., 2017, 3187492, 9. Google Scholar
[28]	J. Wu, X. Zhang, L. Liu, Y. Wu and Y. Cui, The convergence analysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity, Bound. Value Probl., 2018, 82, 15. Google Scholar
[29]	Y. Wang, Y. Liu and Y. Cui, Infinitely many solutions for impulsive fractional boundary value problem with p-Laplacian, Bound. Value Probl., 2018, 94, 16. Google Scholar
[30]	Y. Wei, C. Chen, Q. Chen and H. Yang, Liouville-type theorem for nonlinear elliptic equations involving p-Laplace-type Grushin operators, Math. MethodsAppl. Sci., https://doi.org/10.1002/mma.5886. Google Scholar
[31]	Y. Wei, C. Chen, H. Song and H. Yang, Liouville-type theorems for stable solutions of Kirchhoff equations with exponential and superlinear nonlinearities, Complex Var. Elliptic Equ., 2019, 64(8), 1297-1309. doi: 10.1080/17476933.2018.1514030 CrossRef Google Scholar
[32]	Y. Wei, C. Chen, H. Yang and H. Song, Multiplicity of solutions for a class of fractional p-Kirchhoff system with sign-changing weight functions, Bound. Value Probl., 2018, 78, 18. Google Scholar
[33]	Q. Yuan, C. Chen and H. Yang, Existence of positive solutions for a Schrödinger-Poisson system with bounded potential and weighted functions in $\mathbb R.3$, Bound. Value Probl., 2017,151, 17. $\mathbb R.3$" target="_blank">Google Scholar
[34]	X. Zhang, L. Liu, Y. Wu and Y. Cui, Entire blow-up solutions for a quasilinear p-Laplacian Schrödinger equation with a non-square diffusion term, Appl. Math. Lett., 2017, 74, 85-93. Google Scholar
[35]	X. Zhang, L. Liu, Y. Wu and Y. Cui, The existence and nonexistence of entire large solutions for a quasilinear Schrödinger elliptic system by dual approach, J. Math. Anal. Appl., 2018,464(2), 1089-1106. Google Scholar
[36]	X. Zhang, L. Liu, Y. Wu and Y. Cui, Existence of infinitely solutions for a modified nonlinear Schrödinger equation via dual approach, Electron. J. Differential Equ., 2018, 2018(147), 1-15. Google Scholar
[37]	X. Zhang, J. Jiang, Y. Wu and Y. Cui, Existence and asymptotic properties of solutions for a nonlinear Schrödinger elliptic equation from geophysical fluid flows, Appl. Math. Lett., 2019, 90,229-237. Google Scholar
[38]	X. Zhang, L. Liu, Y. Wu and Y. Cui, New result on the critical exponent for solution of an ordinary fractional differential problem, J. Funct. Spaces, 2017, 3976469, 4. Google Scholar
[39]	X. Zhang, L. Liu and Y. Wu, The entire large solutions for a quasilinear Schrödinger elliptic equation by the dual approach, Appl. Math. Lett., 2016, 55, 1-9. doi: 10.1016/j.aml.2015.11.005 CrossRef Google Scholar
[40]	X. Zhang, L. Liu, Y. Wu and L. Caccetta, Entire large solutions for a class of Schrödinger systems with a nonlinear random operator, J. Math. Anal. Appl., 2015,423(2), 1650-1659. doi: 10.1016/j.jmaa.2014.10.068 CrossRef Google Scholar
[41]	E. Zeidler, The Ljusternik-Schnirelmann theory for indefinite and not necessarily odd nonlinear operators and its applications, Nonlinear Anal. 1980, 4(3), 451-489. doi: 10.1016/0362-546X(80)90085-1 CrossRef Google Scholar