王田娥,李 健,牛太陽,李俊杰

(吉林農(nóng)業(yè)大學(xué) 信息技術(shù)學(xué)院,長春 130118)

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漸近線性p-Kirchhoff型方程解的多重性

王田娥,李 健,牛太陽,李俊杰

(吉林農(nóng)業(yè)大學(xué) 信息技術(shù)學(xué)院,長春 130118)

考慮有界區(qū)域上p-Kirchhoff型方程在Dirichlet邊界條件下解的存在性,應(yīng)用山路定理得到了當(dāng)非線性項滿足漸近線性增長條件時p-Kirchhoff型方程兩個非平凡解的存在性.

多重性;山路定理;p-Kirchhoff型方程

0 引 言

Kirchhoff[1]在研究彈性弦的自由振動時,提出了如下模型:

一般稱該模型為Kirchhoff型方程,它在非牛頓力學(xué)、彈性理論和生物數(shù)學(xué)等諸多領(lǐng)域應(yīng)用廣泛.文獻(xiàn)[2-5]研究了Kirchhoff方程所對應(yīng)的穩(wěn)態(tài)方程:

(1)

本文考慮如下p-Kirchhoff型方程:

(2)

其中:Ω是N(N≥3)中有界光滑區(qū)域;Δpu=div(u),1

(3)

目前,對p-Laplacian方程邊值問題的研究已有許多結(jié)果[6-9].

對于問題(2),當(dāng)非線性項f滿足各種增長條件時,應(yīng)用變分法對其進(jìn)行研究已得到了豐富的結(jié)果.文獻(xiàn)[10-12]研究了問題(2)在非線性項f滿足超線性次臨界增長時解的存在性與多重性;文獻(xiàn)[13-14]在臨界增長情形研究了問題(2)解的存在性;文獻(xiàn)[15]在非線性項滿足漸近線性增長時得到了問題(2)解的多重性.本文進(jìn)一步研究在非線性項f滿足漸近線性增長時問題(2)解的多重性.

1 主要結(jié)果

對于特征值問題

已知其存在一列特征值0<λ1<λ2≤λ3≤…≤λn→+∞,其中第一特征值λ1是簡單的、孤立的特征值,具有相應(yīng)的特征函數(shù)φ1>0.假設(shè):

(H1)存在常數(shù)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

關(guān)于x∈Ω一致成立.

定理1如果(H1)～(H5)滿足,則問題(2)至少有一個正解和一個負(fù)解.

2 定理1的證明

定義

定義1如果

顯然若u為J的臨界點,則u是問題(2)的弱解.

定理2[16]設(shè)X為實Banach空間,Φ∈C1(X,)并且滿足(PS)條件.此外,存在ρ,α,β∈(0,+∞)和u0∈X,使得

定義

首先證明泛函J+具有山路幾何,即:

引理1在定理1的假設(shè)下,有:

因此,應(yīng)用Sobolev嵌入與Poincare不等式,并結(jié)合假設(shè)(H1)和q>p,可得常數(shù)δ1,C1>0,使得

從而存在充分小的ρ>0使得結(jié)論1)成立.

又由(H1)可得

因此,存在充分大的t1,使得u1=t1φ1滿足結(jié)論2).

下面證明J+滿足(PS)條件.

引理2在定理1的假設(shè)下,泛函J+滿足(PS)條件.

證明:對任意的c>0,假設(shè){un}n∈?滿足:

(4)

(5)

由式(5)可得

取v=un,由假設(shè)(H3)～(H5)知,存在α>0,使得

因此,z0是問題

(6)

ζφ1(x)≤z0(x), ?x∈Ω.

(7)

取φ=δφ1,β∈(λ1,λ1+ε),則有

(8)

應(yīng)用上下解方法,再結(jié)合式(7),(8),可得問題

由假設(shè)(H3)～(H5)與Sobolev嵌入定理可得

(9)

記

則

此外,由|((J+)′(un),un-u0)|→0,并結(jié)合式(9)可得Qn→0.再結(jié)合弱收斂un?u0與不等式

下面證明定理1.應(yīng)用定理2,J+存在臨界點u+滿足J+(u+)≥η>0;再應(yīng)用最大值原理可知u+>0.因此u+為問題(2)的正解.類似地,問題(2)存在負(fù)解u-<0.

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(責(zé)任編輯：趙立芹)

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,漢族,碩士,講師,從事微分方程和最優(yōu)化的研究,E-mail:443988941@qq.com.通信作者:李 健(1981—),男,漢族,博士,講師,從事空間推理和微分方程的研究,E-mail:liemperor@163.com.

吉林省青年科研基金(批準(zhǔn)號:20130522110JH)、吉林省重點科技攻關(guān)項目(批準(zhǔn)號:20140204045NY)和吉林省教育廳“十二五”科學(xué)技術(shù)研究項目(批準(zhǔn)號:[2014]第468號).

O175.25

：A

：1671-5489(2015)03-0372-05