一类含渐近线性的奇异椭圆边值问题正解的存在性  

Existence of Positive Solution to a Class of Problems of Singular Elliptic Boundary Value with Asymptotical Linearity

作　　者：宋叔尼[1] 刘霞[1] 

机构地区：[1]东北大学理学院,辽宁沈阳110004

出　　处：《东北大学学报（自然科学版）》2007年第10期1514-1516,1520,共4页Journal of Northeastern University(Natural Science)

基　　金：国家自然科学基金资助项目(50534020)

摘　　要：利用临界点理论,研究了一类含有渐近线性项和奇异项的半线性椭圆方程的边值问题.首先,利用椭圆算子特征值的性质,结合函数f(u)的渐近线性,证明了椭圆边值所对应的泛函J在凸闭集Γε={u∈C10(-Ω)|u≥εφ1}上满足PS条件.其次,利用Banach空间中的常微分方程理论,证明了对任意的a∈R+,J在Γε上具有收缩性,并利用Schauder型条件,证明了Γε是泛函J的一个下降流不变集.最后,对于u∈Γε,证明了J(u)是下方有界的.从而得到了奇异椭圆方程的边值问题至少存在一个正解的结论.According to the critical point theory, a class of problems of elliptic boundary value with an asymptotically linear term and singular term is studied. It is proved that the functional J corresponding to the elliptic boundary value satisfies PS condition on the convex closed set Гε=｜u∈C0^1（Ω^-）｜u≥εφ1｜ by the property of elliptic operator eigenvalue in combination with the asymptotical linearity of the function f（ u ）. Then it is also proved that J is retractable to a α∈R^＋, by the ordinary differential equation theory in Banach space. Furthermore, Гε is proved an invariant set of decent flow of J by Schauder condition, and J （ u ） is proved lower bounded for u ∈Гε A conclusion is therefore reasoned out that there is a positive solution at least to the problems of singular elliptic boundary value.

关 键 词：奇异椭圆方程 边值问题 渐近线性 临界点理论 正解 

分 类 号：O177.92[理学—数学]