一类具有强奇性的矩阵型偏微分方程的正解的存在性
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摘要
研究矩阵型强奇异偏微分方程■其中,Ω?R~n是有界开集,M(x)是定义在Ω上的实对称矩阵,-p<-1, 00是参数,f(x)∈L~1(Ω),f(x)>0 a.e. inΩ。证明,如果存在u_0∈H■(Ω)满足∫_Ωf(x)|u_0|~(1-p)dx<+∞,则对任意的λ>0上述方程都有正H■-解,即慢速解。我们注意到,对于奇异方程,古典解即■解不一定是H■(Ω)解。
We investigate the strongly singular partial differential equations of matrix-type, ■where Ω is a bound and open set in R~n, M(x) is a real symmetric matrix on Ω,-p<-1, 00 are parameters, f(x)∈L~1(Ω), f(x)>0 a.e. in Ω. We prove that the above-mentioned equation admits at least one positive H■-solution when λ>0 if there exists u_0 ∈H■(Ω) such that ∫_Ωf(x)|u_0|~(1-p)dx<+∞. It should be noted that a classical solution, namely, the ■, is not necessarily a H■(Ω)-solution for singular equations.
We investigate the strongly singular partial differential equations of matrix-type, ■where Ω is a bound and open set in R~n, M(x) is a real symmetric matrix on Ω,-p<-1, 0
引文
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[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):7 121-7 128.
[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):2 771-2 795.
[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):4 436-4 440.
[7] Boccardo L,Orsina L.A variational semilinear singular system[J].Nonlinear Analysis Theory Methods & Applications,2011,74 (12):3 849-3 860.
[8] Sun Y J.Compatible phenomena in singular problems[J].Proceedings of the Royal Society of Edinburgh,2013,143(A):1 321-1 330.
[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):1 257-1 284.
[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.