一类具有强奇性的矩阵型偏微分方程的正解的存在性

双震; 孙义静

doi:10.7523/j.issn.2095-6134.2019.03.003

中国科学院大学学报 >

2019 , Vol. 36 >Issue 3: 311 - 319

DOI: https://doi.org/10.7523/j.issn.2095-6134.2019.03.003

数学与物理学

一类具有强奇性的矩阵型偏微分方程的正解的存在性

双震 ,
孙义静

展开

中国科学院大学数学科学学院, 北京 100049

收稿日期: 2018-01-22

修回日期: 2018-04-13

网络出版日期: 2019-05-15

基金资助

国家自然科学基金（11571339，11771468）资助

收起

Exsitence of positive solutions for matrix-type partial differential equations with strongly singular nonlinearities

SHUANG Zhen ,
SUN Yijing

Expand

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received date: 2018-01-22

Revised date: 2018-04-13

Online published: 2019-05-15

Fold

摘要

研究矩阵型强奇异偏微分方程

其中，Ω⊂Rⁿ是有界开集，M（x）是定义在Ω上的实对称矩阵，-p<-1，0 < q < 1，λ>0是参数，f（x）∈L¹（Ω），f（x）>0 a.e.in Ω。证明，如果存在u₀ ∈H₀¹（Ω）满足∫_Ωf（x）|u₀|^1-pdx <+∞，则对任意的λ>0上述方程都有正H₀¹-解，即慢速解。我们注意到，对于奇异方程，古典解即C²（Ω）∩C（Ω）解不一定是H₀¹（Ω）解。

关键词： H₀¹-解; 实对称矩阵; 强奇性

本文引用格式

双震 , 孙义静 . 一类具有强奇性的矩阵型偏微分方程的正解的存在性[J]. 中国科学院大学学报, 2019 , 36(3) : 311 -319 . DOI: 10.7523/j.issn.2095-6134.2019.03.003

Abstract

We investigate the strongly singular partial differential equations of matrix-type,

where Ω is a bound and open set in Rⁿ, M(x) is a real symmetric matrix on Ω, -p<-1,0 < q < 1,λ>0 are parameters, f(x)∈L¹(Ω),f(x)>0 a.e. in Ω. We prove that the above-mentioned equation admits at least one positive H₀¹-solution when λ>0 if there exists u₀ ∈H₀¹(Ω) such that ∫_Ωf(x)|u₀|^1-pdx <+∞. It should be noted that a classical solution, namely, the C²(Ω)∩C(Ω)-solution, is not necessarily a H₀¹(Ω)-solution for singular equations.

Key words： H₀¹-solution; real symmetric matrix; strong singularity

参考文献

[1] Lazer A C, McKenna P J. On a singular nonlinear elliptic boundaryvalue problem[J]. Proceeding of the American Mathemarical Socirty, 1991, 111(3):721-730.
[2] Boccardo L, Orsina L. Semilinear elliptic equations with singular nonlinearities[J]. Calculus of Variations and Partial Differential Equations, 2010, 37(3):363-380.
[3] Arcoya D, Boccardo L. Multiplicity of solutions for a Dirichlet problem with a singular and a supercritical nonlinearities[J]. Differential and Integral Equations, 2013, 26(26):7121-7128.
[4] Arcoya D, Boccardo L, Leonori T, et al. Some elliptic problems with singular natural growth lower order terms[J]. Journal of Differential Equations, 2010, 249(11):2771-2795.
[5] Boccardo L. Dirichlet problems with singular and gradient quadratic lower order terms[J]. Esaim Control Optimisation & Calculus of Variations, 2008, 14(3):411-426.
[6] Boccardo L. A Dirichlet problem with singular and supercritical nonlinearities[J]. Nonlinear Analysis Theory Methods & Application, 2012, 75(12):4436-4440.
[7] Boccardo L, Orsina L. A variational semilinear singular system[J]. Nonlinear Analysis Theory Methods & Applications, 2011, 74(12):3849-3860.
[8] Sun Y J. Compatible phenomena in singular problems[J]. Proceedings of the Royal Society of Edinburgh, 2013, 143(A):1321-1330.
[9] Sun Y J, Wu S P. An exact estimate result for a class of singular equations with critical exponents[J]. Journal of Functional Analysis, 2011, 260(5):1257-1284.
[10] Sun Y J, Wu S P, Long Y M. Combined effects of singular and superlinear nonlinearities in some singular boundary value problems[J]. Journal of Differential Equations, 2001, 176(2):511-531.
[11] Sun Y J, Zhang D Z. The role of the power 3 for elliptic equations with negative exponents[J]. Calculus of Variations & Partial Differential Equations, 2014, 49(3/4):909-922.
[12] Adams A R. Sobolev spaces[M]. New York:Academic Press, 1975:95-107.

Options

文章导航

摘要

本文引用格式

Abstract

参考文献

访问统计

联系我们

模态框（Modal）标题

摘要

本文引用格式

Abstract

参考文献

访问统计

联系我们