渐近线性p-Kirchhoff型方程解的多重性

王田娥,李 健,牛太阳,李俊杰

(吉林农业大学 信息技术学院,长春 130118)



渐近线性p-Kirchhoff型方程解的多重性

王田娥,李 健,牛太阳,李俊杰

(吉林农业大学 信息技术学院,长春 130118)

考虑有界区域上p-Kirchhoff型方程在Dirichlet边界条件下解的存在性,应用山路定理得到了当非线性项满足渐近线性增长条件时p-Kirchhoff型方程两个非平凡解的存在性.

多重性;山路定理;p-Kirchhoff型方程

0 引 言

Kirchhoff[1]在研究弹性弦的自由振动时,提出了如下模型:

一般称该模型为Kirchhoff型方程,它在非牛顿力学、弹性理论和生物数学等诸多领域应用广泛.文献[2-5]研究了Kirchhoff方程所对应的稳态方程:

(1)

本文考虑如下p-Kirchhoff型方程:

(2)

其中:Ω是N(N≥3)中有界光滑区域;Δpu=div(u),1

(3)

目前,对p-Laplacian方程边值问题的研究已有许多结果[6-9].

对于问题(2),当非线性项f满足各种增长条件时,应用变分法对其进行研究已得到了丰富的结果.文献[10-12]研究了问题(2)在非线性项f满足超线性次临界增长时解的存在性与多重性;文献[13-14]在临界增长情形研究了问题(2)解的存在性;文献[15]在非线性项满足渐近线性增长时得到了问题(2)解的多重性.本文进一步研究在非线性项f满足渐近线性增长时问题(2)解的多重性.

1 主要结果

对于特征值问题

已知其存在一列特征值0<λ1<λ2≤λ3≤…≤λn→+∞,其中第一特征值λ1是简单的、孤立的特征值,具有相应的特征函数φ1>0.假设:

(H1)存在常数m2>m1>0,使得m1≤M(s)≥m2,∀s∈+;

(H2)存在s1>0,使得M(s)=m2,∀s>s1;

(H5)存在μ1,μ2∈(λ1,+∞),使得

关于x∈Ω一致成立.

定理1如果(H1)～(H5)满足,则问题(2)至少有一个正解和一个负解.

2 定理1的证明

定义

定义1如果

显然若u为J的临界点,则u是问题(2)的弱解.

定理2[16]设X为实Banach空间,Φ∈C1(X,)并且满足(PS)条件.此外,存在ρ,α,β∈(0,+∞)和u0∈X,使得

定义

首先证明泛函J+具有山路几何,即:

引理1在定理1的假设下,有:

因此,应用Sobolev嵌入与Poincare不等式,并结合假设(H1)和q>p,可得常数δ1,C1>0,使得

从而存在充分小的ρ>0使得结论1)成立.

又由(H1)可得

因此,存在充分大的t1,使得u1=t1φ1满足结论2).

下面证明J+满足(PS)条件.

引理2在定理1的假设下,泛函J+满足(PS)条件.

证明:对任意的c>0,假设{un}n∈⊂满足:

(4)

(5)

由式(5)可得

取v=un,由假设(H3)～(H5)知,存在α>0,使得

因此,z0是问题

(6)

ζφ1(x)≤z0(x), ∀x∈Ω.

(7)

取φ=δφ1,β∈(λ1,λ1+ε),则有

(8)

应用上下解方法,再结合式(7),(8),可得问题

由假设(H3)～(H5)与Sobolev嵌入定理可得

(9)

记

则

此外,由|((J+)′(un),un-u0)|→0,并结合式(9)可得Qn→0.再结合弱收敛un⇀u0与不等式

下面证明定理1.应用定理2,J+存在临界点u+满足J+(u+)≥η>0;再应用最大值原理可知u+>0.因此u+为问题(2)的正解.类似地,问题(2)存在负解u-<0.

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(责任编辑：赵立芹)

MultiplicityofSolutionsofAsymptoticallyLinear
p-KirchhoffTypeEquations

WANG Tian’e,LI Jian,NIU Taiyang,LI Junjie

(CollegeofInformationTechnology,JilinAgriculturalUniversity,Changchun130118,China)

This paper deals with the existence of solutions forp-Kirchhoff type equations in bounded domains under Dirichlet boundary condition.When the nonlinearity is asymptotically linear at infinity,there exist two nontrivial solutions of thep-Kirchhoff type equation which can be proved with the aid of the mountain pass theorem.

multiplicity;mountain pass theorem;p-Kirchhoff type equation

10.13413/j.cnki.jdxblxb.2015.03.05

2014-10-13.

王田娥(1977—),女,汉族,硕士,讲师,从事微分方程和最优化的研究,E-mail:443988941@qq.com.通信作者:李 健(1981—),男,汉族,博士,讲师,从事空间推理和微分方程的研究,E-mail:liemperor@163.com.

吉林省青年科研基金(批准号:20130522110JH)、吉林省重点科技攻关项目(批准号:20140204045NY)和吉林省教育厅“十二五”科学技术研究项目(批准号:[2014]第468号).

O175.25

：A

：1671-5489(2015)03-0372-05