BOUNDARY BLOW-UP RATE OF THE LARGE SOLUTION FOR AN ELLIPTIC COOPERATIVE SYSTEM

doi:10.1007/s10473-019-0514-x

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (5): 1363-1379.doi: 10.1007/s10473-019-0514-x

BOUNDARY BLOW-UP RATE OF THE LARGE SOLUTION FOR AN ELLIPTIC COOPERATIVE SYSTEM

王影¹, 王明新²

1. School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China;
2. Natural Science Research Center, Harbin Institute of Technology, Harbin 150080, China

收稿日期:2017-12-12 修回日期:2018-11-30 出版日期:2019-10-25 发布日期:2019-11-11
通讯作者: Ying WANG E-mail:wy@ncwu.edu.cn
作者简介:Mingxin WANG,E-mail:mxwang@hit.edu.cn
基金资助:
This work was supported by the National Natural Science Foundation of China (11501199).

BOUNDARY BLOW-UP RATE OF THE LARGE SOLUTION FOR AN ELLIPTIC COOPERATIVE SYSTEM

Ying WANG¹, Mingxin WANG²

1. School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China;
2. Natural Science Research Center, Harbin Institute of Technology, Harbin 150080, China

Received:2017-12-12 Revised:2018-11-30 Online:2019-10-25 Published:2019-11-11
Contact: Ying WANG E-mail:wy@ncwu.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (11501199).

摘要/Abstract

摘要： In this article we consider positive large solution of cooperative systems of the form -∆u₁=λ₁u₁ + a₁u₁u₂^q₁ -b₁(x)u₁^p₁+1, -∆u₂=λ₂u₂ + a₂u₁^q₂u₂ -b₂(x)u₂^p₂+1 in a bounded smooth domain Ω ⊂ R^N(λ_i ∈ R, a_i, b_i>0, 0 < q_i < p_j, i, j ∈ {1, 2}, i ≠ j), Based on the construction of certain sup and sub-solution, we show existence, uniqueness and blow-up rate of the large solution.

关键词: cooperative system, blow-up rate, uniqueness

Abstract: In this article we consider positive large solution of cooperative systems of the form -∆u₁=λ₁u₁ + a₁u₁u₂^q₁ -b₁(x)u₁^p₁+1, -∆u₂=λ₂u₂ + a₂u₁^q₂u₂ -b₂(x)u₂^p₂+1 in a bounded smooth domain Ω ⊂ R^N(λ_i ∈ R, a_i, b_i>0, 0 < q_i < p_j, i, j ∈ {1, 2}, i ≠ j), Based on the construction of certain sup and sub-solution, we show existence, uniqueness and blow-up rate of the large solution.

Key words: cooperative system, blow-up rate, uniqueness

中图分类号:

35J55

王影, 王明新. BOUNDARY BLOW-UP RATE OF THE LARGE SOLUTION FOR AN ELLIPTIC COOPERATIVE SYSTEM[J]. 数学物理学报(英文版), 2019, 39(5): 1363-1379.

Ying WANG, Mingxin WANG. BOUNDARY BLOW-UP RATE OF THE LARGE SOLUTION FOR AN ELLIPTIC COOPERATIVE SYSTEM[J]. Acta mathematica scientia,Series B, 2019, 39(5): 1363-1379.

参考文献

[1] Bieberbach L. ∆u=e^u und die automorrphen Funktionen. Math Ann, 1916, 77:173-212
[2] Keller J B.On solutions of ∆u=f(u). Comm Pure Appl Math, 1957, 10:503-510
[3] Osserman R. On the inequality ∆u ≥ f(u). Pacific J Math, 1957, 7:1641-1647
[4] Du Y, Huang Q. Blow-up solutions for a class of semilinear elliptic and parabolic equations. SIAM J Math Anal, 1999, 31:1-18
[5] García-Melián J, Gómez-Reñasco R, López-Gómez J, Sabina de Lis J. Point-wise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs. Arch Ration Mech Anal, 1998, 145(3):261-289
[6] López-Gómez J. Large solutions, metasolution, and asymptotic behavior of the regular positive solutions of sublinear parabolic problems. Electron J Differential Equations, 2000, 5:135-171
[7] Wang Y, Wang M X. The blow-up rate and uniqueness of large solutions for a porous media logsitic equation. Nonlinear Anal, 2010, 11:1572-1580
[8] García-Melián J, Rossi J D. Boundary blow-up solutions to elliptic systems of competitive type. J Differential Equations, 2004, 206:156-181
[9] García-Melián J, Suárez A. Existence and uniqueness of positive large solutions to some cooperative elliptic systems. Adv Nonlinear Stud, 2003, 3:193-206
[10] Li H L, Wang M X. Existence and uniqueness of positive solutions to the boundary blow-up problem for an elliptic system. J Differential Equations, 2007, 234:246-266
[11] López-Gómez J, Maire L. Boundary blow-up rate and uniqueness of the large solution for an elliptic cooperative system of logistic type. Nonlinar Anal:RWA, 2017, 33:208-316
[12] López-Gómez J. The boundary blow-up rate of large solution. J Differential Equations, 2003, 195:25-45
[13] Delgado M, López-Gómez J, Suárez A. On the symbiotic Lotka-Volterra model with diffusion and transport effects. J Differential Equations, 2000, 160:175-262
[14] López-Gómez J, Maire L. Coupled versus uncoupled blow-up rates in cooperative n-species logistic systems. Adv Nonl Studies, 2017, 17:411-428
[15] García-Melián J, Rossi J D, Sabina J C. Elliptic systems with boundary blow-up:Existence, uniqueness and application to removability of singularities. Comm Pure Appl Anal, 2016, 15:549-562

BOUNDARY BLOW-UP RATE OF THE LARGE SOLUTION FOR AN ELLIPTIC COOPERATIVE SYSTEM

BOUNDARY BLOW-UP RATE OF THE LARGE SOLUTION FOR AN ELLIPTIC COOPERATIVE SYSTEM

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