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一类带p-Laplacian的Sturm-Liouville型2m点边值问题三个正解的存在性
引用本文:赵俊芳,葛渭高.一类带p-Laplacian的Sturm-Liouville型2m点边值问题三个正解的存在性[J].数学物理学报(A辑),2011,31(1):92-102.
作者姓名:赵俊芳  葛渭高
作者单位:1.中国地质大学(北京)信息工程学院 北京 100083|2.北京理工大学理学院数学系 北京 |100081
基金项目:国家自然科学基金(10926051)和教育部博士专项基金(20050007011)资助
摘    要:该文考虑了下面的具一维$p$\,-Laplacian算子的多点边值问题 $ \left\{ \begin{array}{rl} &;\disp (\phi_{p}(x'(t)))'+h(t)f(t,x(t),x'(t))=0,\hspace{3mm}01,~\alpha_{i}>0,~\beta_{i}>0,~0<\sum\limits_{i=1}^{m-1}\alpha_{i}\xi_{i}\leq1,~ 0<\sum\limits_{i=1}^{m-1}\beta_{i}(1-\eta_{i})\leq1,~0=\xi_{0} <\xi_{1}<\xi_{2}<\cdots<\xi_{m-1}<\eta_{1}<\eta_{2}<\cdots<\eta_{m-1}<\eta_{m}=1,~i=1,2,\cdots,m-1.$ 通过运用锥上的不动点定理, 该文得到了至少三个正解的存在性. 有趣的是文中的边界条件是一个新型的Sturm-Liouville型边界条件, 这类边值问题到目前为止还很少被研究.

关 键 词:多点  正解  边值问题  不动点定理  p-Laplacian
收稿时间:2008-04-16
修稿时间:2009-07-14

Triple Positive Solutions of 2m-point Boundary Value Problem of Sturm-Liouville Type with one Dimensional p-Laplacian
Zhao Junfang,Ge Weigao.Triple Positive Solutions of 2m-point Boundary Value Problem of Sturm-Liouville Type with one Dimensional p-Laplacian[J].Acta Mathematica Scientia,2011,31(1):92-102.
Authors:Zhao Junfang  Ge Weigao
Affiliation:1.School of Information Engineering, China University of Geosciences, Beijing 100083|2.Department of Mathematics, Beijing Institute of Technology, Beijing 100081
Abstract:In this paper, we are concerned with the following multi-point boundary value problem with one-dimensional p-Laplacian $$   \left\{  \begin{array}{rl} &\disp (\phi_{p}(x'(t)))'+h(t)f(t,x(t),x'(t))=0,\hspace{3mm}0<t<1,\&\disp x'(0)-\sum_{i=1}^{m-1}\alpha_{i}x(\xi_{i})=0,\ \ x'(1)+\sum_{i=1}^{m-1}\beta_{i}x(\eta_{i})=0,  \end{array}  \right. $$ where $\phi_{p}(s)=|s|^{p-2}s,~p>1,~\alpha_{i}>0,~\beta_{i}>0,~0<\sum\limits_{i=1}^{m-1}\alpha_{i}\xi_{i}\leq1,~ 0<\sum\limits_{i=1}^{m-1}\beta_{i}(1-\eta_{i})\leq1,~0=\xi_{0} <\xi_{1}<\xi_{2}<\cdots<\xi_{m-1}<\eta_{1}<\eta_{2}<\cdots<\eta_{m-1}<\eta_{m}=1,~i=1,2,\cdots,m-1.$ By using a fixed point theorem in a cone, we obtain the existence of three positive solutions at least. The interesting point is that the boundary condition is a new kind of Sturm-Liouville type boundary condition, which has rarely been treated up to now.
Keywords:Multi-pointzz  Positive solutionszz  Boundary value problemzz  Fixed point theoremzz     p-Laplacianzz
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