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

SHUANG Zhen; SUN Yijing

doi:10.7523/j.issn.2095-6134.2019.03.003

Journal of University of Chinese Academy of Sciences >

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

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

Research Articles

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

SHUANG Zhen ,
SUN Yijing

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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

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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

Cite this article

SHUANG Zhen , SUN Yijing . Exsitence of positive solutions for matrix-type partial differential equations with strongly singular nonlinearities[J]. Journal of University of Chinese Academy of Sciences, 2019 , 36(3) : 311 -319 . DOI: 10.7523/j.issn.2095-6134.2019.03.003

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