一类p-Laplacian方程和方程组边值问题正解的存在性
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摘要
本文主要研究了一类二阶微分方程组边值问题和一类奇异p-Laplacian方程及n维p-Laplacian方程组边值问题正解的存在性.本文共分为四章：
第一章,简述了问题产生的历史背景和现阶段的主要成果,并概述本文的主要工作.
第二章,主要运用Krasnosel'skii不动点定理,研究了二阶微分方程组边值问题在某些条件下一个正解的存在性.
第三章,主要运用Leggett-Williams定理研究了一类形如的奇异p-Laplacian方程边值问题,给出了在一定条件下具有三个解的充分条件.
第四章,利用不动点指数理论,建立了n维p-Laplacian方程组三点奇异边值问题一解、两解的存在定理.
This paper mainly researches on the existence of positive solutions of boundary value problems for a kind of second order differential equations and singular p-Laplacian equation and n-dimension p-Laplacian equations. This article is divided into four chapters:
The first chapter outlines the problem of the historical background and the main results of the present stage, outlines the main work of this paper.
ChapterⅡ, by using Krasnosel'skii fixed point theorem, it mainly researches on the existence of one positive solution of boundary value problems for second order differential equations.
ChapterⅢ, by using Leggett-Williams theorem, it mainly researches on the existence of three positive solutions for boundary value problem of one kind of p-Laplacian equation and also give the sufficient conditions of the existence of three positive solutions.
Chapter IV, by using fixed point index theorem, it mainly establishes the existence theorem of one positive solution and two positive solutions of three-point singular boundary value problems for n-dimensional p-Laplacian.
第一章,简述了问题产生的历史背景和现阶段的主要成果,并概述本文的主要工作.
第二章,主要运用Krasnosel'skii不动点定理,研究了二阶微分方程组边值问题在某些条件下一个正解的存在性.
第三章,主要运用Leggett-Williams定理研究了一类形如的奇异p-Laplacian方程边值问题,给出了在一定条件下具有三个解的充分条件.
第四章,利用不动点指数理论,建立了n维p-Laplacian方程组三点奇异边值问题一解、两解的存在定理.
This paper mainly researches on the existence of positive solutions of boundary value problems for a kind of second order differential equations and singular p-Laplacian equation and n-dimension p-Laplacian equations. This article is divided into four chapters:
The first chapter outlines the problem of the historical background and the main results of the present stage, outlines the main work of this paper.
ChapterⅡ, by using Krasnosel'skii fixed point theorem, it mainly researches on the existence of one positive solution of boundary value problems for second order differential equations.
ChapterⅢ, by using Leggett-Williams theorem, it mainly researches on the existence of three positive solutions for boundary value problem of one kind of p-Laplacian equation and also give the sufficient conditions of the existence of three positive solutions.
Chapter IV, by using fixed point index theorem, it mainly establishes the existence theorem of one positive solution and two positive solutions of three-point singular boundary value problems for n-dimensional p-Laplacian.
引文
[1]FINK A M. Positive solutions of second order systems of boundary value problems [J]. J.Math Anal Appl,1993,180:93-108
[2]姚庆六,吕海琛.一维奇异p-Laplacian方程的正解[J].数学学报,1998,41(6)：1255-1264
[3]丁卫平,刘玉记.一类p-Laplacian边值问题的多个正解[J].数学的实践与认识2004,34(5)：146-152
[4]徐夫义,苏华,王健.p-Laplaciari非线性奇异边值问题正解的存在性[J].山东大学学报(理学版),2006,41(5)：100-104
[5]Chan-Gyun Kim, James R. Ward. Nonresonance for a one-dimensional p-Laplacian with strong singularity[J].Applied Mathematics Letters,2011,24:1400-1404
[6]王峰,贾宝瑞等.具p-Laplacian的三阶边值问题正解的存在性[J].重庆工商大学学报(自然科学版),2011,28(2)：125-129
[7]Bing LIU. Positive Solutions of Three-Point Boundary Value Promblems for the One-Dimensional p-Laplacian with Infinitely Many Singularities[J].Applied Mathematics Letters,2004,17:655-661
[8]刘锡平,贾梅,葛渭高.p-Laplacian算子方程三点边值问题单调正解的存在性[J].吉林大学学报(理学版),2007,45(1)：49-55
[9]马德香,葛渭高.具p-Laplacian算子的三点边值问题正解的存在性[J].数学研究与评论,2007,27(2)：425-431
[10]田元生,刘春根.一维p-Laplacian算子方程三点奇异边值问题单调正解的多重性[J].应用数学学报,2008,31(4)：635-641
[11]田元生,刘春根.三阶p-Laplacian方程三点奇异边值问题三个正解的存在性[J].应用数学学报,2008,31(6)：1118-1127
[12]杨志林,孙经先.非线性二阶常微分方程组边值问题的正解[J].数学学报,2004,47(1)：111-118
[13]江波,夏大峰等.一类常微分方程组边值问题的求解方法[J].南京气象学院学报,2006,29(4)：576-579
[14]刘亚亚,夏大峰,赵慧芳.一类奇异二阶三点方程组正解的存在性[J].南京气象学院学报,2008,31(4)：599-602
[15]王斌,汪卫华等.带有p-Laplacian算子的二阶微分方程组多个正解的存在性[J].河北科技大学学报,2011,32(1)：15-19
[16]郭大钧.非线性泛函分析[M].济南：山东科学技术出版社,2002.
[17]郭大钧,孙经先.非线性积分方程[M].济南：山东科学技术出版社,1987.
[18]郭大钧,孙经先,刘兆理.非线性常微分方程泛函方法[M].济南：山东科学技术出版社,1995.
[19]肖建中,李刚.抽象分析基础[M].清华大学出版社,北京,2009.
[20]马如云.奇异二阶边值问题的正解[J].数学学报,1998,42(1)：1225-1230
[21]李仁贵,刘立山.二阶奇异非线性微分方程边值问题的正解[J].应用数学和力学,2001,22(2)：435-439
[22]程建纲.二阶边值问题的正解[J].数学学报,2001,44(3)：429-436
[23]孙伟平,葛渭高.一类非线性边值问题正解的存在性[J].数学学报,2001,44(4)：577-580
[24]沈文国.一类奇异二阶常微分方程三点边值问题的多个正解[J].华中师范大学学报(自然科学版),2007,41(2)：176-178
[25]姚庆六.关于一类二阶两点边值问题的正解存在性[J].中山大学学报(自然科学版),2003,42(4)：18-20
[26]姚庆六,秀芬.二阶非线性常微分方程的三点边值问题的一个存在定理[J].兰州大学学报(自然科学版)2009,39(1)：17-19
[27]李红玉,孙经先.奇异二阶常微分方程组边值问题的正解[J].应用泛函分析学报,2010,12(1)：21-26
[28]Guo D. Lakshmikantham V. Nonlinear problems in abstract cones[M]. Orlando:Academic Press,1988
[29]田元生,刘春根.带p-Laplacian算子三点奇异边值问题对称正解的存在性[J].数学物理学报,2010,30A(3)：784-792
[30]田元生,刘春根.三阶p-Laplacian方程三点奇异边值问题正解的存在性[J].工程数学学报,2009,26(3)：519-527
[31]梁月亮,续晓欣.三阶p-Laplacian边值问题解的存在唯一性定理[J].中北大学学报(自然科学版),2001,31(5)：443-447
[32]Jun Cheng, Shiping Lu. Periodic Solutions for a Third Order p-Laplacian Equation[J]. Journal of Anhui Normal University(Natural Science),2009,32:311-315
[33]Douglas R Anderson. Existence of solutions for a first-order p-Laplacian BVP on time scales[J].Nonlinear Analysis,2008,69:4521-4525
[34]Hanying Feng, Weigao Ge, Ming Jiang. Multiple positive solutions for m-point boundary-value problems with a one-dimensional p-Laplacian[J]. Nonlinear Analysis,2008, 68:2269-2279
[35]Dehong Ji, Weigao Ge. Multiple positive solutions for some p-Laplacian boundary value problems[J]. Applied Mathematics and Computation,2007,187:1315-1325
[36]Yong-Hoon Lee, Inbo Sim. Existence results of sign-changing solutions for singular one-dimensional p-Laplacian problems[J]. Nonlinear Analysis,2008,68:1195-1209.
[37]Chen Yang, Jurang Yan. Positive solutions for third-order Sturm-Liouville boundary value problems with p-Laplacian[J]. Computers and Mathematics with Applications,2010, 59:2059-2066
[38]田元生,刘春根.三阶p-Laplacian非齐次边值问题多重正解的存在性[J],数学进展,2011,40(1)：71-78
[39]张晓燕,孙经先.一维奇异p-Laplacian方程多解的存在性[J].数学物理学 报,2006,26A(1)：143-149
[40]Chengbo Zhai. The existence of positive solutions to mixed boundary value problems of p-Laplacian equations[J]. Acta Analysis Functionalis Applicata,2003,5(2):170-173
[41]李福义,范勇.非线性参数椭圆系统正解的存在性与多解性[J].数学学报,1999,42(4)：591-596
[42]刘斌.具p-Laplacian算子型奇异方程组边值问题正解的存在性[J].数学学报,2005,48(1)：35-50
[43]D.R.Kunninger, Haiyan wang. Existence and multiplicity of positive solutions for elliptic systems[J].Nonlinear Analysis, Theory, Methods&Applications,1997,29:1051-1060
[44]Yongping Sun. Optimal existence criteria for symmetric positive solutions to a three-point boundary value problem[J]. Nonlinear Analysis,2007,66:1051-1063
[2]姚庆六,吕海琛.一维奇异p-Laplacian方程的正解[J].数学学报,1998,41(6)：1255-1264
[3]丁卫平,刘玉记.一类p-Laplacian边值问题的多个正解[J].数学的实践与认识2004,34(5)：146-152
[4]徐夫义,苏华,王健.p-Laplaciari非线性奇异边值问题正解的存在性[J].山东大学学报(理学版),2006,41(5)：100-104
[5]Chan-Gyun Kim, James R. Ward. Nonresonance for a one-dimensional p-Laplacian with strong singularity[J].Applied Mathematics Letters,2011,24:1400-1404
[6]王峰,贾宝瑞等.具p-Laplacian的三阶边值问题正解的存在性[J].重庆工商大学学报(自然科学版),2011,28(2)：125-129
[7]Bing LIU. Positive Solutions of Three-Point Boundary Value Promblems for the One-Dimensional p-Laplacian with Infinitely Many Singularities[J].Applied Mathematics Letters,2004,17:655-661
[8]刘锡平,贾梅,葛渭高.p-Laplacian算子方程三点边值问题单调正解的存在性[J].吉林大学学报(理学版),2007,45(1)：49-55
[9]马德香,葛渭高.具p-Laplacian算子的三点边值问题正解的存在性[J].数学研究与评论,2007,27(2)：425-431
[10]田元生,刘春根.一维p-Laplacian算子方程三点奇异边值问题单调正解的多重性[J].应用数学学报,2008,31(4)：635-641
[11]田元生,刘春根.三阶p-Laplacian方程三点奇异边值问题三个正解的存在性[J].应用数学学报,2008,31(6)：1118-1127
[12]杨志林,孙经先.非线性二阶常微分方程组边值问题的正解[J].数学学报,2004,47(1)：111-118
[13]江波,夏大峰等.一类常微分方程组边值问题的求解方法[J].南京气象学院学报,2006,29(4)：576-579
[14]刘亚亚,夏大峰,赵慧芳.一类奇异二阶三点方程组正解的存在性[J].南京气象学院学报,2008,31(4)：599-602
[15]王斌,汪卫华等.带有p-Laplacian算子的二阶微分方程组多个正解的存在性[J].河北科技大学学报,2011,32(1)：15-19
[16]郭大钧.非线性泛函分析[M].济南：山东科学技术出版社,2002.
[17]郭大钧,孙经先.非线性积分方程[M].济南：山东科学技术出版社,1987.
[18]郭大钧,孙经先,刘兆理.非线性常微分方程泛函方法[M].济南：山东科学技术出版社,1995.
[19]肖建中,李刚.抽象分析基础[M].清华大学出版社,北京,2009.
[20]马如云.奇异二阶边值问题的正解[J].数学学报,1998,42(1)：1225-1230
[21]李仁贵,刘立山.二阶奇异非线性微分方程边值问题的正解[J].应用数学和力学,2001,22(2)：435-439
[22]程建纲.二阶边值问题的正解[J].数学学报,2001,44(3)：429-436
[23]孙伟平,葛渭高.一类非线性边值问题正解的存在性[J].数学学报,2001,44(4)：577-580
[24]沈文国.一类奇异二阶常微分方程三点边值问题的多个正解[J].华中师范大学学报(自然科学版),2007,41(2)：176-178
[25]姚庆六.关于一类二阶两点边值问题的正解存在性[J].中山大学学报(自然科学版),2003,42(4)：18-20
[26]姚庆六,秀芬.二阶非线性常微分方程的三点边值问题的一个存在定理[J].兰州大学学报(自然科学版)2009,39(1)：17-19
[27]李红玉,孙经先.奇异二阶常微分方程组边值问题的正解[J].应用泛函分析学报,2010,12(1)：21-26
[28]Guo D. Lakshmikantham V. Nonlinear problems in abstract cones[M]. Orlando:Academic Press,1988
[29]田元生,刘春根.带p-Laplacian算子三点奇异边值问题对称正解的存在性[J].数学物理学报,2010,30A(3)：784-792
[30]田元生,刘春根.三阶p-Laplacian方程三点奇异边值问题正解的存在性[J].工程数学学报,2009,26(3)：519-527
[31]梁月亮,续晓欣.三阶p-Laplacian边值问题解的存在唯一性定理[J].中北大学学报(自然科学版),2001,31(5)：443-447
[32]Jun Cheng, Shiping Lu. Periodic Solutions for a Third Order p-Laplacian Equation[J]. Journal of Anhui Normal University(Natural Science),2009,32:311-315
[33]Douglas R Anderson. Existence of solutions for a first-order p-Laplacian BVP on time scales[J].Nonlinear Analysis,2008,69:4521-4525
[34]Hanying Feng, Weigao Ge, Ming Jiang. Multiple positive solutions for m-point boundary-value problems with a one-dimensional p-Laplacian[J]. Nonlinear Analysis,2008, 68:2269-2279
[35]Dehong Ji, Weigao Ge. Multiple positive solutions for some p-Laplacian boundary value problems[J]. Applied Mathematics and Computation,2007,187:1315-1325
[36]Yong-Hoon Lee, Inbo Sim. Existence results of sign-changing solutions for singular one-dimensional p-Laplacian problems[J]. Nonlinear Analysis,2008,68:1195-1209.
[37]Chen Yang, Jurang Yan. Positive solutions for third-order Sturm-Liouville boundary value problems with p-Laplacian[J]. Computers and Mathematics with Applications,2010, 59:2059-2066
[38]田元生,刘春根.三阶p-Laplacian非齐次边值问题多重正解的存在性[J],数学进展,2011,40(1)：71-78
[39]张晓燕,孙经先.一维奇异p-Laplacian方程多解的存在性[J].数学物理学 报,2006,26A(1)：143-149
[40]Chengbo Zhai. The existence of positive solutions to mixed boundary value problems of p-Laplacian equations[J]. Acta Analysis Functionalis Applicata,2003,5(2):170-173
[41]李福义,范勇.非线性参数椭圆系统正解的存在性与多解性[J].数学学报,1999,42(4)：591-596
[42]刘斌.具p-Laplacian算子型奇异方程组边值问题正解的存在性[J].数学学报,2005,48(1)：35-50
[43]D.R.Kunninger, Haiyan wang. Existence and multiplicity of positive solutions for elliptic systems[J].Nonlinear Analysis, Theory, Methods&Applications,1997,29:1051-1060
[44]Yongping Sun. Optimal existence criteria for symmetric positive solutions to a three-point boundary value problem[J]. Nonlinear Analysis,2007,66:1051-1063