含非齐次项椭圆方程组的边界爆破解

王志敏, 马飞遥

含非齐次项椭圆方程组的边界爆破解

王志敏, 马飞遥*

（宁波大学 数学与统计学院, 浙江 宁波 315211）

利用上下解方法和比较原理研究了含非齐次项椭圆方程组边界爆破解的存在性问题. 首先证得包含非齐次项的加奇性权单个椭圆方程边界爆破解的存在性, 进一步得到方程组在边界爆破条件下解的存在性.

椭圆方程组; 边界爆破解; 存在性; 非齐次项

本文研究如下带有非齐次项的椭圆方程组的边界爆破问题, 并证明其解的存在性:

近些年大量国内外学者研究了单个方程边界爆破问题[1-3]. 文献[4]考虑了带权函数和非齐次项的单个方程边界爆破解的存在性以及解渐近行为:

García-Melián[5]研究了如下单个方程边界爆破解的存在性、唯一性以及解的不存在性:

椭圆方程组的边界爆破解也有许多研究[7-10]. 文献[11]考虑了如下方程组在边界爆破条件下解的存在性、唯一性及边界渐近行为:

本文考虑含非齐次项椭圆方程组边界爆破问题,利用文献[5-6]的方法得出单个方程存在唯一正解, 进而根据上下解方法得出该问题解的存在性.

1 单个方程解的存在性

为了证明方程组解的存在性, 先证明如下单个方程解的存在性:

通过引理2的证明过程可知:

2 方程组解的存在性

根据文献[1]中的标准方法, 可证得.

根据引理3、4可以得出问题(1)存在一个正解.

[1] Pao C V. Nonlinear parabolic and elliptic equations[M]. Berlin: Springer Science & Business Media, 2012.

[2] 黄水波, 田巧玉, 田双亮. 半线性椭圆方程(组)边界爆破解的研究[M]. 北京: 中国水利水电出版社, 2016.

[3] Keller J B. On solutions of Δ=()[J]. Communications on Pure and Applied Mathematics, 1957, 10(4):503-510.

[4] Zhang Z. A boundary blow-up elliptic problem with an inhomogeneous term[J]. Nonlinear Analysis: Theory, Methods & Applications, 2008, 68(11):3428-3438.

[5] García-Melián J. Large solutions for an elliptic equation with a nonhomogeneous term[J]. Journal of Mathematical Analysis and Applications, 2016, 434(1):872-881.

[6] Chuaqui M, Cortázar C, Elgueta M, et al. Uniqueness and boundary behaviour of large solutions to elliptic problems with singular weights[J]. Communications on Pure and Applied Analysis, 2004, 3(4):653-662.

[7] García-Melián J, de Lis J S, Letelier-Albornoz R. The solvability of an elliptic system under a singular boundary condition[J]. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 2006, 136(3):509-546.

[8] García-Melián J. A remark on uniqueness of large solutions for elliptic systems of competitive type[J]. Journal of Mathematical Analysis and Applications, 2007, 331(1):608-616.

[9] Mu C, Huang S, Tian Q, et al. Large solutions for an elliptic system of competitive type: Existence, uniqueness and asymptotic behavior[J]. Nonlinear Analysis: Theory, Methods & Applications, 2009, 71(10):4544-4552.

[10] García-Melián J. Large solutions for an elliptic system of quasilinear equations[J]. Journal of Differential Equations, 2008, 245(12):3735-3752.

[11] García-Melián J, Rossi J D. Boundary blow-up solutions to elliptic systems of competitive type[J]. Journal of Differential Equations, 2004, 206(1):156-181.

[12] Osserman R. On the inequality Δ≥()[J]. Pacific Journal of Mathematics, 1957, 7(4):1641-1647.

Boundary blow-up solutions for elliptic systems with nonhomogeneous term

WANG Zhimin, MA Feiyao*

( School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China )

Using the method of sub and super solutions and the comparison principle, we study the boundary blow-up problem for an elliptic system with non-homogeneous terms. Firstly, the existence of boundary blow-up solutions for a single equation with singular weights and nonhomogeneous term is derived. Further on, the existence of solutions for the system under the condition of boundary blow-up is proven.

elliptic equations; boundary blow-up solutions; existence; non-homogeneous term

O175.25

A

1001-5132（2020）01-0069-03

2019−07−16.

宁波大学学报（理工版）网址: http://journallg.nbu.edu.cn/

国家自然科学基金（11471174）; 浙江省自然科学基金（LY20A010010, LY20A010011）.

王志敏（1994－）, 男, 安徽滁州人, 在读硕士研究生, 主要研究方向: 偏微分方程. E-mail: 1459947773@qq.com

马飞遥（1979－）, 男, 湖南衡阳人, 博士/副教授, 主要研究方向: 偏微分方程. E-mail: mafeiyao@nbu.edu.cn

（责任编辑 韩 超）