东北大学学报（自然科学版）

东北大学学报(自然科学版) ›› 2007, Vol. 28 ›› Issue (10): 1514-1516+1520.DOI: -

• 论著 • 上一篇    下一篇

一类含渐近线性的奇异椭圆边值问题正解的存在性

宋叔尼;刘霞;   

  1. 东北大学理学院;东北大学理学院 辽宁沈阳110004;辽宁沈阳110004
  • 收稿日期:2013-06-24 修回日期:2013-06-24 出版日期:2007-10-15 发布日期:2013-06-26
  • 通讯作者: Song, S.-N.
  • 作者简介:-
  • 基金资助:
    国家自然科学基金资助项目(50534020)

Existence of positive solution to a class of problems of singular elliptic boundary value with asymptotical linearity

Song, Shu-Ni (1); Liu, Xia (1)   

  1. (1) School of Sciences, Northeastern University, Shenyang 110004, China
  • Received:2013-06-24 Revised:2013-06-24 Online:2007-10-15 Published:2013-06-26
  • Contact: Song, S.-N.
  • About author:-
  • Supported by:
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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.

关键词: 奇异椭圆方程, 边值问题, 渐近线性, 临界点理论, 正解

Abstract: 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 ΓΕ = {u ∈ C01 (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 ∈ R+ on ΓΕ by the ordinary differential equation theory in Banach space. Furthermore, ΓΕ is proved an invariant set of decent flow of J by Schauder condition, and J(u) is proved lower bounded for u ∈ΓΕ. A conclusion is therefore reasoned out that there is a positive solution at least to the problems of singular elliptic boundary value.

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