p(t)-Laplace方程解的存在性*

陈 颂,闫晓芳

(永城职业学院 基础部,河南 永城 476600)

0 引言

p-Laplace方程是一类椭圆微分方程.含有p-Laplace算子的微分方程解的存在性问题受到了广泛的关注,相关研究结果可参见文献[1-3]等.p(t)-Laplace方程首先是从非线性弹性理论中引发出来的,p(t)-Laplace算子比p-Laplace算子有更复杂的非线性性质,相关结果可参见文献[4-6]等.

本文讨论了如下p(t)-Laplace方程解的存在性问题:

(1)

其中p∈C(,)是T-周期函数,对于t∈有p(t)>1,f∈C(×N,N),且f(t,u)关于t是T-周期的.

1 定理

定理如果存在r>0和线性向量场V,使得对任意的u∈N,〈Vu,u〉≥0,当且仅当u=0时,〈Vu,u〉=0,且对所有的t∈[0,T],u∈N,当〈Vu,u〉=r时,都有〈f(t,u),Vu〉>0,〈f(t,u),V*u〉>0,则以上方程至少有一个解u满足|u(t)|≤r,其中t∈[0,T].

令g∈C([0,T]×N,N).设X是Banach空间,定义映射A:X→X*:

定义映射G:X→X*:

引理[7]令p为定理中所定义的,A,g,G均为上文定义.若存在一个正常数M,使得

|g(t,u)|≤M,

(2)

∀t∈[0,T],∀u∈N,

则映射A+G:X→X*是满的.特别地,存在u∈X使得A(u)+G(u)=0,且u是以下问题的解:

(3)

定义映射Q=Qr:N→N:

令g(t,u)=f(t,Q(u))-Q(u).则g∈C(×N,N).

则由引理,映射A+G:X→X*是满的.特别地,存在u∈X使得A(u)+G(u)=0,且u是以下问题的解:

(4)

现在证明u满足|u(t)|≤r,∀t∈[0,T].首先证明以下论断.

论断存在t∈[0,T],使得|u(t)|≤r.

证明反证法.设对所有的t∈[0,T]都有|u(t)|>r.考虑函数

φ(t)=〈|u′(t)|p(t)-2u′(t),Vu(t)〉,

t∈[0,T],

故

φ′(t)=|u′(t)|p(t)+

〈f(t,Qu(t)),Vu(t)〉+

〈u(t)-Qu(t)〉,t∈[0,T].

因为|Qu(t)|=r,有

〈f(t,Qu(t)),Vu(t)〉≥0,

又因

故

〈u(t)-Qu(t),Vu(t)〉=

(|u(t)|-r)|u(t)|>0.

故

φ′(t)>0, ∀t∈[0,T].

(5)

这表明φ′(t)在[0,T]上严格增,这和φ(0)=φ(T)矛盾.故论断成立.

2 定理的证明

假设u不满足|u(t)|≤r,∀t∈[0,T],则存在t∈[0,T],使得|u(t)|>r.假设u在上定义,且为T-周期的,则u∈C1(,N).设

由假设,可以找到σ<τ使得

h(σ)=maxh(t),

|u(τ)|=r,t∈,

且当t∈[σ,τ]时,|u(t)|>r.故|u(σ)|>r且h′(σ)=0.

令

ψ(t)=〈u(t),Vu(t)〉,

ψ1(t)=〈u′(t),Vu(t)〉,

ψ2(t)=〈u(t),Vu′(t)〉.

类似于式(5)的证明,有

将V换成V*,同理可得

〈u′(t),V*u(t)〉′>0, ∀t∈[σ,τ],

则

〈V′u(t),u(t)〉′>0, ∀t∈[σ,τ],

故

故

ψ″(t)=〈u(t),Vu(t)〉″=

(〈u′(t),Vu(t)〉′+

〈u(t),Vu′(t)〉′=

因此,ψ′(t)在[σ,τ]上严格增.由于

因此,对t∈[σ,τ],有h′(t)

故u满足

|u(t)|≤r,∀t∈[0,T].

故可推出Qu(t)=u(t).因此u是问题(4)的解,故u是问题(1)的解.定理得证.

参考文献:

[1] MAWHIN J. Some boundary value problems for Hartman-type perturbations of the ordinary vectorp-Laplace[J]. Nonlinear Analysis: Real World Applications, 2000, 40(1/8): 497-503.

[2] KNOBLOCH H W. On the existence of periodic solutions for second order vector differential equations[J]. Journal of Differential Equations, 1971, 9(1): 67-85.

[3] ZEIDLER E. Nonlinear Functional Analysis and Its Applications, Part 2 B: Nonlinear Monotone Operators[M]. New York: Springer-Verlag, 1990.

[4] FAN Xianling, WU Hongqing, WANG Feizhi, et al. Hartman-type results forp(t)-Laplace systems[J]. Nonlinear Analysis: Real World Applications, 2003, 52: 585-594.

[5] FAN Xianling, ZHAO Yuanzhang, ZHAO Dun, et al. Compact inbedding theorems with symmetry of Strauss-Lions type for the spaceW1p(x)(Ω)[J]. Journal of Mathematical Analysis and Applications, 2001, 255: 333-348.

[6] FAN Xianling, ZHAO Dun. On the spacesLp(x)(Ω) andWm,p(x)(Ω)[J]. Journal of Mathematical Analysis and Applications, 2001, 263: 424-446.

[7] FAN Xianling, FAN Xing. A knobloch-type result forp(t)-Laplace systems[J]. Journal of Mathematical Analysis and Applications, 2003, 282: 453-464.

[8] MUSIELAK J. Orlicz Spaces and Modular Spaces: Lecture Notes in Mathematics, Vol. 1034[R]. Berlin: Springer-Verlag, 1983.

[9] ADAMS R A. Sobolev Spaces[M]. New York: Academic Press, 1975.

[10] ODEN J T. Qualitative Methods on Nonlinear Mechanics[M]. New Jersey Englewood Cliffs: Prentice-Hall, 1986.