一类带有退化强制项的奇异椭圆方程解的存在性  

Existence of Solutions to Some Singular Elliptic Problems with Degenerate Coercivity

作　　者：李清微 高文杰[2] 韩玉柱[2] 

机构地区：[1]大连海事大学数学系,辽宁大连116000 [2]吉林大学数学学院,长春130012

出　　处：《数学物理学报（A辑）》2017年第5期877-894,共18页Acta Mathematica Scientia

基　　金：国家自然科学基金(11401252);吉林省科技厅青年基金(20160520103JH);吉林省教育厅科研基金(2015-463)~~

摘　　要：该文研究了一类具退化强制项和自然增长条件梯度项的奇异椭圆方程的Dirichlet边值问题{-div((∣▽u∣^(p-2)▽u)/((1+∣u∣)~r)+B(∣▽u∣~p)/(∣u∣~θ)=f,x∈Ω,u>0,x∈Ω,u=0,x∈аΩ,其中,ΩR^N(N≥p)是一有界区域,B,γ,θ>0,P>1,f是某一Lebesgue空间L^m(Ω)(m≥1)中的非负函数.利用截断技术并结合选取适当的检验函数,证明了该问题非负解的存在性以及正则性等结果.该文结果表明尽管低阶梯度项是奇异的,它的存在对解的正则性有"好"的影响.In this article, the authors consider the existence of solutions for the following elliptic boundary value problem with degenerate coercivity and a singular lower order term with natural growth with respect to the gradient of the following form {-div（（∣▽u∣^（p-2）▽u）/（（1＋∣u∣）~r）＋B（∣▽u∣~p）/（∣u∣~θ）=f,x∈Ω,u〉0,x∈Ω,u=0,x∈аΩ where ΩR^N（N≥p） is a bounded domain, B,γ,θ〉 0 and p 〉 1, and f is a non-negative function belonging to some Lebesgue space Lm（Ω） with m ≥ 1. By combining the truncation methods with several delicate test functions, the existence and regularity of solution is proved. The results show that the lower order term has some regularizing effects on the solution, even if it is singular.

关 键 词：退化强制 奇异性 低阶梯度项 存在 正则性 

分 类 号：O175.2[理学—数学]