一维p-Laplacian方程的共振性态
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摘要
p-Laplace方程是一类比较重要的微分方程模型,它来自于非牛顿流体问题及非线性弹性问题等.本文讨论了一维p-Laplace方程在共振点附近无穷多个次调和解的存在性,并给出了方程大初值解无界的条件以及周期解与无界解共存的条件.
一维p-Laplace方程作为平面等时Hamilton系统的扰动来看,其等时部分不具有对状态变量的齐性,因此我们采用作用-角变换的方法,把“非齐性”平均,研究在此变换下新的Hamilton系统.当方程的扰动项有界时,我们证明了作用-角变换下的Poincar(?)映射的扭转性;当方程的扰动项无界时,此时的Poincar(?)映射不足以清楚地反映扭转,因此我们考虑作用-角变换下的后继映射的扭转性.最后利用Poincar(?)-Birkhoff扭转定理得到方程的无穷多周期解的存在性.
当非等时项很小时,我们证明了在一定条件下,方程的所有大初值的解均无界.此外,我们还利用拓扑度理论证明了方程2π周期解的存在性.最后,本文给出了方程周期解和无界解共存的条件.
p-Laplacian equation is an important model of differential equation from non- Newtonian fluid theory and nonlinear elasticity.In this paper,we will investigate the resonant phenomena of 1-dimensional p-Laplacian equation.Based on a generalized version of the Poincar(?)-Birkhoff twist theorem by J.Franks and W.Ding,we prove the existence of infinite subharmonic solutions for the equation.Moreover,we establish the coexistence of periodic solution and unbounded solution for the equation.
From the view-point of a perturbation of the isochronous Hamiltonian system in the plane,the isochronous part for p-Laplacian equation is non-homogeneous in the variables. So we introduce some new action-angle variables to overcome this difficult.In the case of the perturbed term being bounded,we prove that Poincar(?) map of the equation has a twist property on some annulus.In the case of the perturbed term being unbounded,we don't know whether there is a twist property of Poincar(?) map or not.So we consider the successor map in the new action-angle coordinates and prove the twist property of the successor map. Then we obtain the existence of infinitely many periodic solutions for p-Laplacian equation by using Poincar(?)-Birkhoff twist theorem in both cases.
When the perturbed term is small enough,we prove that all of solutions with large amplitude are unbounded under some conditions.Moreover,we also prove the existence of 2πperiodic solution for p-Laplacian equation by using Topological degree.At last,we give the condition for the coexistence of periodic solution and unbounded solution for the equation.
一维p-Laplace方程作为平面等时Hamilton系统的扰动来看,其等时部分不具有对状态变量的齐性,因此我们采用作用-角变换的方法,把“非齐性”平均,研究在此变换下新的Hamilton系统.当方程的扰动项有界时,我们证明了作用-角变换下的Poincar(?)映射的扭转性;当方程的扰动项无界时,此时的Poincar(?)映射不足以清楚地反映扭转,因此我们考虑作用-角变换下的后继映射的扭转性.最后利用Poincar(?)-Birkhoff扭转定理得到方程的无穷多周期解的存在性.
当非等时项很小时,我们证明了在一定条件下,方程的所有大初值的解均无界.此外,我们还利用拓扑度理论证明了方程2π周期解的存在性.最后,本文给出了方程周期解和无界解共存的条件.
p-Laplacian equation is an important model of differential equation from non- Newtonian fluid theory and nonlinear elasticity.In this paper,we will investigate the resonant phenomena of 1-dimensional p-Laplacian equation.Based on a generalized version of the Poincar(?)-Birkhoff twist theorem by J.Franks and W.Ding,we prove the existence of infinite subharmonic solutions for the equation.Moreover,we establish the coexistence of periodic solution and unbounded solution for the equation.
From the view-point of a perturbation of the isochronous Hamiltonian system in the plane,the isochronous part for p-Laplacian equation is non-homogeneous in the variables. So we introduce some new action-angle variables to overcome this difficult.In the case of the perturbed term being bounded,we prove that Poincar(?) map of the equation has a twist property on some annulus.In the case of the perturbed term being unbounded,we don't know whether there is a twist property of Poincar(?) map or not.So we consider the successor map in the new action-angle coordinates and prove the twist property of the successor map. Then we obtain the existence of infinitely many periodic solutions for p-Laplacian equation by using Poincar(?)-Birkhoff twist theorem in both cases.
When the perturbed term is small enough,we prove that all of solutions with large amplitude are unbounded under some conditions.Moreover,we also prove the existence of 2πperiodic solution for p-Laplacian equation by using Topological degree.At last,we give the condition for the coexistence of periodic solution and unbounded solution for the equation.
引文
[1]丁同仁,常微分方程定性方法的应用,高等教育出版社,北京,2004.
[2]Alonso J.M.and Ortega R.,Roots of unity and unbounded motions of an asymmetric oscillator,J.Differential Equations 143(1998),201-220.
[3]Liu B.,Boundedness in asymmetric oscillations,J.Math.Anal.Appl 231(1999),355-373.
[4]Lazer A.C.and Leach D.E.,Bounded perturbations of forced harmonic oscillators at resonance,Ann.Mat.Pura Appl.82(1969),49-68.
[5]Alonso J.M.and Ortega R.,Unbounded solutions of semilinear equations at resonance,Nonlinearity 9(1996),1099-1111.
[6]Dancer E.,Boundary-value problems.for weakly nonlinear ordinary differential equations,Bull.Austral Math.Soc.15(1976),321-328.
[7]Fabry C.and Fonda A.,Nonlinear resonance in asymmetric oscillators,J.Differential Equations 147(1998),58-78.
[8]Fabry C.and Mawhin J.,Oscillations of a forced asymmetric oscillator at resonance,Nonlinearity 13(2000),493-505.
[9]Qain D.,Infinity of subharmonics for asymmetric Duffing equations with the Lazer-Leach -Dancer condition,J.Differential Equations 171(2001),233-250.
[10]Qain D.,Resonance phenomena for asymmetric weakly nonlinear oscillator,Science in China 45-A(2002),214-222.[11]Fonda A.,Positively homogeneous Hamiltonian systems in the plane,J.Differential Equations 200(2004),162-184.
[12]Fabry C.and Man(?)sevich R.,Equations with a p-Laplacian and an asymmetric nonlinear term,Discrete Contin.Dynam.Systems 7(2001),545-557.
[13]Liu B.,Boundedness ofsolutions for equations with p-Laplacian and an asymmetric nonlinear term,J.Differential Equations 207(2004),73-92.
[14]Mawhin J.,Resonance and nonlinearity:a survey,Ukrainian Mathematical Journal 59(2007),197-214.
[15]Zhang M.,Nonuniform nonresonance of semilinear differential equations,J.Differential Equations 166(2000),33-50.
[16]Amster P.and N(?)poli De P.,Landesman-Lazer type conditions for a system of p-Laplacian like operators,J.Math.Anal.Appl.326(2007),1236-1243.
[17]Ming X.,Wu S.and Liu J.,Periodic solutions for the 1-dimensional p-Laplacian equation,J.Math.Anal.Appl.325(2007),879-888.
[18]Franks J.,Generalizations of the Poincar(?)-Birkhoff theorem,Ann.of Math.128(1988),139-151.
[19]Ding W.,A generalization of the Poincar(?)-Birkhoff theorem,Proc.Amer.Math.Soc.88(1983),341-346.
[20]Qian D.and Torres P.J.,Periodic motions of linear impact oscillators via the successor map,SIAM &Math.Anal.36(2005),1707-1725.
[21]Ding T.and Zanolin F.,Periodic solutions of Duffing's equations with superquadratic potential,J.Differential Equations 79(1992),328-378.
[22]Jacobowitz H.,Periodic solutions of x~〃+f(x,t)=0 via the Poincar(?)-Birkhoff theorem,J.Differential Equations 20(1976),37-52.
[23]Ortega R.,Asymmetric oscillators and twist mappings,J.London Math.Soc.53(1996),325-342.
[2]Alonso J.M.and Ortega R.,Roots of unity and unbounded motions of an asymmetric oscillator,J.Differential Equations 143(1998),201-220.
[3]Liu B.,Boundedness in asymmetric oscillations,J.Math.Anal.Appl 231(1999),355-373.
[4]Lazer A.C.and Leach D.E.,Bounded perturbations of forced harmonic oscillators at resonance,Ann.Mat.Pura Appl.82(1969),49-68.
[5]Alonso J.M.and Ortega R.,Unbounded solutions of semilinear equations at resonance,Nonlinearity 9(1996),1099-1111.
[6]Dancer E.,Boundary-value problems.for weakly nonlinear ordinary differential equations,Bull.Austral Math.Soc.15(1976),321-328.
[7]Fabry C.and Fonda A.,Nonlinear resonance in asymmetric oscillators,J.Differential Equations 147(1998),58-78.
[8]Fabry C.and Mawhin J.,Oscillations of a forced asymmetric oscillator at resonance,Nonlinearity 13(2000),493-505.
[9]Qain D.,Infinity of subharmonics for asymmetric Duffing equations with the Lazer-Leach -Dancer condition,J.Differential Equations 171(2001),233-250.
[10]Qain D.,Resonance phenomena for asymmetric weakly nonlinear oscillator,Science in China 45-A(2002),214-222.[11]Fonda A.,Positively homogeneous Hamiltonian systems in the plane,J.Differential Equations 200(2004),162-184.
[12]Fabry C.and Man(?)sevich R.,Equations with a p-Laplacian and an asymmetric nonlinear term,Discrete Contin.Dynam.Systems 7(2001),545-557.
[13]Liu B.,Boundedness ofsolutions for equations with p-Laplacian and an asymmetric nonlinear term,J.Differential Equations 207(2004),73-92.
[14]Mawhin J.,Resonance and nonlinearity:a survey,Ukrainian Mathematical Journal 59(2007),197-214.
[15]Zhang M.,Nonuniform nonresonance of semilinear differential equations,J.Differential Equations 166(2000),33-50.
[16]Amster P.and N(?)poli De P.,Landesman-Lazer type conditions for a system of p-Laplacian like operators,J.Math.Anal.Appl.326(2007),1236-1243.
[17]Ming X.,Wu S.and Liu J.,Periodic solutions for the 1-dimensional p-Laplacian equation,J.Math.Anal.Appl.325(2007),879-888.
[18]Franks J.,Generalizations of the Poincar(?)-Birkhoff theorem,Ann.of Math.128(1988),139-151.
[19]Ding W.,A generalization of the Poincar(?)-Birkhoff theorem,Proc.Amer.Math.Soc.88(1983),341-346.
[20]Qian D.and Torres P.J.,Periodic motions of linear impact oscillators via the successor map,SIAM &Math.Anal.36(2005),1707-1725.
[21]Ding T.and Zanolin F.,Periodic solutions of Duffing's equations with superquadratic potential,J.Differential Equations 79(1992),328-378.
[22]Jacobowitz H.,Periodic solutions of x~〃+f(x,t)=0 via the Poincar(?)-Birkhoff theorem,J.Differential Equations 20(1976),37-52.
[23]Ortega R.,Asymmetric oscillators and twist mappings,J.London Math.Soc.53(1996),325-342.