具有臨界非線性項(xiàng)的p-雙調(diào)和方程無(wú)窮多小解的存在性

苗鳳華,宋玥薔，周晨星

(1.長(zhǎng)春師范大學(xué) 數(shù)學(xué)學(xué)院,長(zhǎng)春 130032;2.長(zhǎng)春師范大學(xué) 科研處,長(zhǎng)春 130032)

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具有臨界非線性項(xiàng)的p-雙調(diào)和方程無(wú)窮多小解的存在性

苗鳳華1,宋玥薔2，周晨星1

(1.長(zhǎng)春師范大學(xué) 數(shù)學(xué)學(xué)院,長(zhǎng)春 130032;2.長(zhǎng)春師范大學(xué) 科研處,長(zhǎng)春 130032)

利用一個(gè)新的對(duì)稱山路引理研究一類具有臨界非線性項(xiàng)的p-雙調(diào)和方程,得到了該問(wèn)題無(wú)窮多個(gè)非平凡解的存在性,并證明了這些解序列趨近于零.

p-雙調(diào)和方程;對(duì)稱山路引理;無(wú)窮多解

0 引 言

考慮如下p-雙調(diào)和方程:

(1)

其中:Ω?N是一個(gè)有界光滑區(qū)域稱為p-雙調(diào)和算子,當(dāng)p=2時(shí)為通常的雙調(diào)和算子;p*=Np/(N-2p)為如下Sobolev嵌入的臨界指標(biāo):

(2)

目前,關(guān)于臨界指數(shù)增長(zhǎng)問(wèn)題解的研究已取得了許多結(jié)果[1-7].但對(duì)含有p-雙調(diào)和算子問(wèn)題的研究報(bào)道較少[8-10],特別是對(duì)帶有臨界非線性項(xiàng)的p-雙調(diào)和算子無(wú)窮多個(gè)小解存在性的研究目前尚未見(jiàn)報(bào)道.本文利用一個(gè)新的對(duì)稱山路引理[11]研究問(wèn)題(1),獲得了問(wèn)題(1)無(wú)窮多個(gè)小解的存在性,并且這些解趨近于零.利用集中緊性原理[12]克服嵌入失去緊性條件所帶來(lái)的困難,利用文獻(xiàn)[13]中截?cái)喾椒朔疚闹蟹蔷€性項(xiàng)是強(qiáng)不定的導(dǎo)致對(duì)稱山路引理不能直接應(yīng)用的困難.

假設(shè)f(x,u)滿足下列條件:

(H1)f(x,u)∈C(Ω×,),對(duì)任意的u∈,f(x,-u)=-f(x,u);

成立,則稱u∈H是問(wèn)題(1)的一個(gè)弱解.

利用標(biāo)準(zhǔn)證明方法,可以證明泛函I∈C1(H,),并且泛函I對(duì)應(yīng)的臨界點(diǎn)正好對(duì)應(yīng)問(wèn)題(1)的解.

1 緊性條件

(3)

進(jìn)而存在某個(gè)依賴于ε的正常數(shù)c(ε)>0,使得下列不等式成立:

(4)

引理1假設(shè)條件(H1)～(H3)成立,則對(duì)任意的λ>0,泛函I滿足(PS)c條件,這里

證明:令{un}為函數(shù)空間H中的(PS)序列,則由式(4)可得

于是,取定ε=(p*-p)/(2pp*λ)可知

(5)

其中o(1)→0,且M是某個(gè)正常數(shù).另一方面,由式(3)可得

(6)

不等式(5),(6)表明,(PS)序列{un}在空間H中有界.從而可抽取子列,不妨仍記為{un},使得un?u弱收斂于H,un→u幾乎處處收斂于Ω,

(7)

(8)

另一方面,由H?lder不等式和序列{un}的有界性可知

(9)

類似地,下列極限成立:

(10)

因此,由估計(jì)式(8)～(10)可得

(11)

這里用到I′(u)=0,從而可知序列{un}在空間H中強(qiáng)收斂到u.

2 主要結(jié)果

設(shè)X是一個(gè)Banach空間,記

Σ∶={A?X{0}:A是閉的并且在X中關(guān)于原點(diǎn)對(duì)稱}.

若A∈Σ,定義虧格γ(A)為

γ(A)∶=inf{m∈:?φ∈C(A,Rm{0}),-φ(x)=φ(-x)}.

如果對(duì)任意的m∈,不存在如上定義的φ,則約定γ(A)=+∞.令Σk為X中所有閉對(duì)稱子集A的全體,使得0?A且γ(A)≥k.

引理2(對(duì)稱山路引理)[11]設(shè)E是一個(gè)無(wú)限維空間,I∈C1(E,),如果下列條件成立:

2)對(duì)每個(gè)k∈,存在Ak∈Σk,使得

則下列結(jié)論成立:

(i)存在序列{uk},使得I′(uk)=0,I(uk)<0,并且{uk}趨于零;

其中A,B,C是某些正的常數(shù).

則易知χ(t)∈[0,1],并且χ(t)是C∞.令φ(u)=χ(‖u‖),考慮I(u)的擾動(dòng):

(12)

因此,可得:

引理3設(shè)G(u)由式(12)定義,則下列結(jié)論成立:

1)G∈C1(H,),G是偶的并且有下界;

引理4假設(shè)條件(H3)成立,則對(duì)任意的k∈,存在δ=δ(k)>0,使得γ({u∈H:G(u)≤-δ(k)}{0})≥k成立.

證明:證明方法類似于文獻(xiàn)[4],故略.

定理1假設(shè)條件(H1)～(H3)成立,則存在λ*>0,使得對(duì)任意的λ∈(0,λ*),問(wèn)題(1)有一列非平凡解{un},且當(dāng)n→∞時(shí),un→0.

注1如果定理1中沒(méi)有對(duì)稱性條件(即f(x,-u)=-f(x,u)),則可利用本文方法得到至少一個(gè)非平凡解的存在性.

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(責(zé)任編輯：趙立芹)

ExistenceofInfinitelyManySmallSolutionsforp-BiharmonicEquationwithCriticalNonlinearity

MIAO Fenghua1,SONG Yueqiang2，ZHOU Chenxing1

(1.CollegeofMathematics,ChangchunNormalUniversity,Changchun130032,China;2.DepartmentofScientificResearch,ChangchunNormalUniversity,Changchun130032,China)

A class ofp-biharmonic equations with critical nonlinearity to obtain infinitely many solutions by means of a version of the symmetric mountain pass theorem.Finally,we showed that this sequence of solutions converge to zero.

p-biharmonic equation;symmetric mountain pass theorem;infinitely many solutions

10.13413/j.cnki.jdxblxb.2015.03.04

2014-08-04.

苗鳳華(1968—),女,漢族,碩士,副教授,從事微分方程的研究,E-mail:mathfhmiao@163.com.通信作者:宋玥薔(1980—),女,漢族,碩士,從事微分方程的研究,E-mail:songyueqiang@sohu.com.

國(guó)家自然科學(xué)基金(批準(zhǔn)號(hào):11301038)、吉林省科技廳青年基金(批準(zhǔn)號(hào):20130522100JH)、吉林省教育廳“十二五”科學(xué)技術(shù)研究項(xiàng)目(批準(zhǔn)號(hào):吉教科合字[2013]第252號(hào))和吉林大學(xué)符號(hào)計(jì)算與知識(shí)工程教育部重點(diǎn)實(shí)驗(yàn)室開(kāi)放課題基金(批準(zhǔn)號(hào):93K172013K03).

O175.2

：A

：1671-5489(2015)03-0367-05