含非齊次項(xiàng)橢圓方程組的邊界爆破解

王志敏, 馬飛遙

含非齊次項(xiàng)橢圓方程組的邊界爆破解

王志敏, 馬飛遙*

（寧波大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院, 浙江 寧波 315211）

利用上下解方法和比較原理研究了含非齊次項(xiàng)橢圓方程組邊界爆破解的存在性問題. 首先證得包含非齊次項(xiàng)的加奇性權(quán)單個(gè)橢圓方程邊界爆破解的存在性, 進(jìn)一步得到方程組在邊界爆破條件下解的存在性.

橢圓方程組; 邊界爆破解; 存在性; 非齊次項(xiàng)

本文研究如下帶有非齊次項(xiàng)的橢圓方程組的邊界爆破問題, 并證明其解的存在性:

近些年大量國(guó)內(nèi)外學(xué)者研究了單個(gè)方程邊界爆破問題[1-3]. 文獻(xiàn)[4]考慮了帶權(quán)函數(shù)和非齊次項(xiàng)的單個(gè)方程邊界爆破解的存在性以及解漸近行為:

García-Melián[5]研究了如下單個(gè)方程邊界爆破解的存在性、唯一性以及解的不存在性:

橢圓方程組的邊界爆破解也有許多研究[7-10]. 文獻(xiàn)[11]考慮了如下方程組在邊界爆破條件下解的存在性、唯一性及邊界漸近行為:

本文考慮含非齊次項(xiàng)橢圓方程組邊界爆破問題,利用文獻(xiàn)[5-6]的方法得出單個(gè)方程存在唯一正解, 進(jìn)而根據(jù)上下解方法得出該問題解的存在性.

1 單個(gè)方程解的存在性

為了證明方程組解的存在性, 先證明如下單個(gè)方程解的存在性:

通過引理2的證明過程可知:

2 方程組解的存在性

根據(jù)文獻(xiàn)[1]中的標(biāo)準(zhǔn)方法, 可證得.

根據(jù)引理3、4可以得出問題(1)存在一個(gè)正解.

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

[2] 黃水波, 田巧玉, 田雙亮. 半線性橢圓方程(組)邊界爆破解的研究[M]. 北京: 中國(guó)水利水電出版社, 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.

寧波大學(xué)學(xué)報(bào)（理工版）網(wǎng)址: http://journallg.nbu.edu.cn/

國(guó)家自然科學(xué)基金（11471174）; 浙江省自然科學(xué)基金（LY20A010010, LY20A010011）.

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

馬飛遙（1979－）, 男, 湖南衡陽(yáng)人, 博士/副教授, 主要研究方向: 偏微分方程. E-mail: mafeiyao@nbu.edu.cn

（責(zé)任編輯 韓 超）