具Neumann边值条件p(t)-Laplacian微分方程多解的存在性
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摘要
利用Ricceri变分原理,讨论一维p(t)-Laplacian微分方程-(|u′|~(p(t)-2) u′)′+|u|~(p(t)-2) u=λf(t,u)的Neumann边值问题,证明了该问题在一定条件下至少存在3个弱解.
Using Ricceri variational principle,we discussed the Neumann boundary value problem of one dimensional p(t)-Laplacian differential equation-(|u′|~(p(t)-2) u′)′+ |u|~(p(t)-2) u=λf(t,u),and proved that the problem had at least three weak solutions under certain conditions.
Using Ricceri variational principle,we discussed the Neumann boundary value problem of one dimensional p(t)-Laplacian differential equation-(|u′|~(p(t)-2) u′)′+ |u|~(p(t)-2) u=λf(t,u),and proved that the problem had at least three weak solutions under certain conditions.
引文
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[7]FAN Xianling,ZHANG Qihu.Existence of Solutions for p(x)-Laplacian Dirichlet Problem[J].Nonlinear Anal:Theor,Meth Appl,2003,52(8):1843-1852.
[8]HARJULEHTO P,HSTP,KOSKENOJA M.The Dirichlet Energy Integral on Intervals in Variable Exponent Sobolev Spaces[J].Z Anal Anwendungen,2003,22(4):911-923.
[9]ZHANG Qihu,LIU Xiaopin,QIU Zhimei.Existence of Solutions for Weighted p(t)-Laplacian System Multi-point Boundary Value Probelms[J].Nonlinear Anal:Theor,Meth Appl,2009,71(9):3715-3727.
[10]MIHAILESCU M.Exsitence and Multiplicity of Solutions for a Neumann Problem Involving the p(x)-Laplace Operator[J].Nonlinear Anal:Theor,Meth Appl,2007,67(5):1419-1425.
[11]RICCERI B.On a Three Critical Points Theorem[J].Arch Math,2000,75(3):220-226.
[12]KIM I H,KIM Y H,PARK K.Existence of Three Solutions for Equations of p(x)-Laplace Type Operators with Nonlinear Neumann Boundary Conditions[J/OL].Bound Value Probl,2016-10-21.https://doi.org/10.1186/s13661-016-0688-2.
[13]SHI Xiayang,DING Xuanhao.Exsitence and Multiplicity of Solutions for a General p(x)-Laplacian Neumann Problem[J].Nonlinear Anal:Theor,Meth Appl,2009,70(10):3715-3720.
[14]JI Dehong.Positive Solutions for Four-Point Boundary Value Problem Involving the p(t)-Laplacian[J].Qual Theory Dyn Syst,2016,15(1):39-48.
[15]范先令,赵敦.广义Orlicz-Sobolev空间Wk,p(x)(Ω)[J].甘肃教育学院学报(自然科学版),1998,12(1):1-6.(FAN Xianling,ZHAO Dun.On the Generalized Orlicz-Sobolev Space Wk,p(x)(Ω)[J].Journal of Gansu Education College(Natural Sciences),1998,12(1):1-6.)
[16]MIHAILESCU M,RADULESCU V.A Multiplicity Result for a Nonlinear Degenerate Problem Arising in the Theory of Electrorheological Fluids[J].Proc Math Phys Eng Sci,2006,462:2625-2641.
[17]ZEIDLER E.Nonlinear Functional Analysis and Its ApplicationsⅡ/B[M].Berlin:Springer-Verlag,1990.
[2]WANG Youyu,GE Weigao.Multiple Positive Solutions for Multipoint Boundary Value Problems with One-Dimensional p-Laplacian[J].J Math Anal Appl,2007,327(2):1381-1395.
[3]LIAN Hairong,GE Weigao.Positive Solutions for a Four-Point Boundary Value Problem with the p-Laplacian[J].Nonlinear Anal:Theor,Meth Appl,2008,68(11):3493-3503.
[4]WANG Youyu,ZHAO Meng,HU Yinping.Triple Positive Solutions for a Multi-point Boundary Value Problem with a One-Dimensional p-Laplacian[J].Comput Math Appl,2010,60(6):1792-1802.
[5]李圆晓,魏英杰,高文杰.拟线性二阶方程三点边值问题对称正解的存在性[J].吉林大学学报(理学版),2010,48(1):1-8.(LI Yuanxiao,WEI Yingjie,GAO Wenjie.Existence of Symmetric Positive Solutions to the Three-Point Boundary Value Problem of Quasilinear Second Order Equation[J].Journal of Jilin University(Science Edition),2010,48(1):1-8.)
[6]FENG Xingfang,FENG Hanying,TAN Huixuan.Existence and Iteration of Positive Solutions for Third-Order Sturm-Liouville Boundary Value Problems with p-Laplacian[J].Appl Math Comput,2015,266(1):634-641.
[7]FAN Xianling,ZHANG Qihu.Existence of Solutions for p(x)-Laplacian Dirichlet Problem[J].Nonlinear Anal:Theor,Meth Appl,2003,52(8):1843-1852.
[8]HARJULEHTO P,HSTP,KOSKENOJA M.The Dirichlet Energy Integral on Intervals in Variable Exponent Sobolev Spaces[J].Z Anal Anwendungen,2003,22(4):911-923.
[9]ZHANG Qihu,LIU Xiaopin,QIU Zhimei.Existence of Solutions for Weighted p(t)-Laplacian System Multi-point Boundary Value Probelms[J].Nonlinear Anal:Theor,Meth Appl,2009,71(9):3715-3727.
[10]MIHAILESCU M.Exsitence and Multiplicity of Solutions for a Neumann Problem Involving the p(x)-Laplace Operator[J].Nonlinear Anal:Theor,Meth Appl,2007,67(5):1419-1425.
[11]RICCERI B.On a Three Critical Points Theorem[J].Arch Math,2000,75(3):220-226.
[12]KIM I H,KIM Y H,PARK K.Existence of Three Solutions for Equations of p(x)-Laplace Type Operators with Nonlinear Neumann Boundary Conditions[J/OL].Bound Value Probl,2016-10-21.https://doi.org/10.1186/s13661-016-0688-2.
[13]SHI Xiayang,DING Xuanhao.Exsitence and Multiplicity of Solutions for a General p(x)-Laplacian Neumann Problem[J].Nonlinear Anal:Theor,Meth Appl,2009,70(10):3715-3720.
[14]JI Dehong.Positive Solutions for Four-Point Boundary Value Problem Involving the p(t)-Laplacian[J].Qual Theory Dyn Syst,2016,15(1):39-48.
[15]范先令,赵敦.广义Orlicz-Sobolev空间Wk,p(x)(Ω)[J].甘肃教育学院学报(自然科学版),1998,12(1):1-6.(FAN Xianling,ZHAO Dun.On the Generalized Orlicz-Sobolev Space Wk,p(x)(Ω)[J].Journal of Gansu Education College(Natural Sciences),1998,12(1):1-6.)
[16]MIHAILESCU M,RADULESCU V.A Multiplicity Result for a Nonlinear Degenerate Problem Arising in the Theory of Electrorheological Fluids[J].Proc Math Phys Eng Sci,2006,462:2625-2641.
[17]ZEIDLER E.Nonlinear Functional Analysis and Its ApplicationsⅡ/B[M].Berlin:Springer-Verlag,1990.