一类带非线性对数项的拟线性椭圆方程解的存在性  

A Neumann Problem for Quasilinear Elliptic Equation with Logarithmic Nonlinearity in a Ball

作　　者：崔周进[1] 徐兵[1] 

机构地区：[1]解放军理工大学数理系,南京211101

出　　处：《科技导报》2009年第2期43-46,共4页Science & Technology Review

基　　金：江苏省教育厅自然科学基金项目(08KJB110005)

摘　　要：研究了球内带非线性对数项的拟线性椭圆方程-div(|▽u|p-2▽u)=logu+h(x)uq带纽曼边值问题解的存在性,推广了De Queiroz的相关结论,De Queiroz研究的是p=2时解的存在性。利用双摄动理论,首先对参数0<ε<1考虑一组逼近问题-div(|▽u|p-2▽u)=logεu2+uε+uε+εε+h(x)uq解的存在性。由于不能直接利用Poincaré不等式去求解上述逼近问题,所以对于每个0<r<R,定义另外一个区域ArR:=BR\Br,考虑在ArR上逼近问题解的存在性,当r→0+时可以得到逼近问题解的存在性。最后令ε→0,求出逼近问题解的极限,得到所研究问题存在一个径向的正解u∈C1(BR\{0})∩C(BR)。The nonlinear quasilinear elliptic problem -div（｜△u｜^p-2△u）=logu＋h（x）u^q in a ball BRlelong to Rn with Neumann boundary conditions on BR are studied. For each 0 〈ε〈1, a family of ＇ approximate problems -div（｜△u｜^p-2△u）=log（u^2＋εu＋ε/u＋ε）＋h（x）u^q is considered. Then, it is natural to look for a family of solutions of this problem and then to pass the limit as ε→0 to obtain a solution to the first problem. Because we cannot use Poincare inequality to solve the second problem directly by variational methods, for each 0 〈r〈R, ArR：=BR／B^-r is defined. Consider the third family of the problem, a solution to the second problem is obtained when r→0^＋. By means of a double perturbation argument, a positive radial solution u ∈C^1（B^-R／{0}）∩C（B^-R） is obtained.

关 键 词：奇异椭圆方程 纽曼边值问题 正则性 双摄动理论 

分 类 号：O175.25[理学—数学]