Heisenberg群上擬線性橢圓方程解的多重性

賈 高, 郭露倩, 張龍杰

(上海理工大學(xué)理學(xué)院,上海 200093)

Heisenberg群上擬線性橢圓方程解的多重性

賈 高, 郭露倩, 張龍杰

(上海理工大學(xué)理學(xué)院,上海 200093)

在Heisenberg群上研究一類擬線性橢圓方程邊值問題解的多重性.在全空間中,假設(shè)方程的主導(dǎo)系數(shù)及導(dǎo)數(shù)有界,而方程的非線性項具有超線性增長.由于在該假設(shè)下,方程所對應(yīng)的泛函是連續(xù)的,但沒有可微性,因此必須使用不光滑臨界點理論.首先,介紹不光滑臨界點理論中的弱斜率、臨界點、(PS)c條件等概念和相關(guān)的基本引理;其次,研究泛函的臨界點的性質(zhì),利用非線性泛函理論、Fatou引理、Lebesgue控制收斂定理和Brezis-Browder定理證明(PS)c序列的強收斂性質(zhì);最后,借助推廣的山路引理得到該邊值問題具有無窮多個解,且這些解是彼此分離的.

擬線性橢圓方程;不可微泛函;不光滑臨界點理論;Heisenberg群

近年來,在歐氏空間中,關(guān)于擬線性橢圓方程解的存在性和多重性問題已有許多學(xué)者進行研究[1-5],在Heisenberg群上關(guān)于半線性或擬線性次橢圓方程的研究也主要集中于對單個解的存在性[6-9].本文在Heisenberg群上研究一類擬線性橢圓方程解的多重性.

1 問題的提出

所謂Heisenberg群 HHN是指在瓗N×瓗N×瓗上定義如下的群運算式中,?表示通常瓗N中的內(nèi)積,則這個運算賦予 HHN李群結(jié)構(gòu).

定義1 Heisenberg群上Laplace算子被定義

本文研究下面問題解的存在性和多重性.

利用變分法來研究式(1),尋求其弱解等價于找如下泛函I:E→瓗的臨界點

定義2 稱u是泛函I的臨界點,如果u∈E,使得〈I′(u),h〉=0,?h∈E∩L∞( HHN).

下面給出如下基本條件:

a.A(·,·): HHN×瓗→瓗滿足對于每個s∈瓗,A(η,s)關(guān)于η是可測的,對幾乎所有的η∈ HHN, A(η,s)關(guān)于s是屬于C1的;

b.存在0＜α＜β＜+∞,使得α≤A(η,s)≤β, |As(η,s)|≤β,a.e.η∈ HHN和?s∈瓗;

本文的主要結(jié)論是:

定理1 若A(η,s)和f(η,s)分別滿足條件a—d和條件e—f,則存在序列{un}?E∩L∞( HHN)是式(1)的弱解,且有I(un)→+∞.

2 預(yù)備知識和基本引理

為了得到定理1,建立下列引理.首先介紹引理1(參考文獻[10]中的定理1.4),它是C1泛函經(jīng)典結(jié)論的推廣(參考文獻[13]中的定理9.12).

引理1 設(shè)X是無限維Banach空間,f:X→瓗為連續(xù)的偶泛函,且對任意c∈瓗,I滿足(PS)c條件.更進一步假設(shè):

a.對于任意有限維子空間W?E,?瓗＞0,使得?u∈W:u=瓗?f(u)≤f(0);

為了證明泛函I滿足(PS)c條件,還需要引入(C-P-S)c條件的定義.

引理2 設(shè)c為一實數(shù),若I滿足(C-P-S)c條件,則I滿足(PS)c條件.

證明方法是基本的,可以參考文獻[11],這里

〈I′(u),h〉=0,?h∈E∩L∞( HHN)(7)即u是I的一個臨界點.

證明方法可參考文獻[10]中的引理2.3,此處省略.

下面的引理是Brezis-Browder定理[9]的向量形式,其證明可參考文獻[8].

3 定理1的證明

容易驗證泛函I是連續(xù)且為偶泛函.進一步由引理2,對于?c∈瓗,泛函I滿足(PS)c條件.

首先,證明泛函I滿足引理1的假設(shè)a.

設(shè)W是空間E的有限維子空間,且u∈W使得I(u)≥0.于是由假設(shè)b得到

由條件e和條件f,存在m(η)∈L∞( HHN)滿足m (η)＞0,a.e.η∈ HHN和一個正常數(shù)C,使得

因為θ＞2,故集合{u∈W,I(u)≥0}在E上有界,說明引理1的假設(shè)a成立.

其次,證明泛函I滿足引理1的假設(shè)b.

由條件b和條件f,對u∈E,有

[1] Aouaoui S.Multiplicity of solutions for quasilinear elliptic equations in瓗N[J].Journal of Mathematical Analysis and Applications,2010,370(2):639-648.

[2] Gazzola F.Positive solutions of critical quasilinear elliptic problems in general domains[J].Abstract and Applied Analysis,1998,3(1/2):65-84.

[3] Salvatore A.Some multiplicity results for a superlinear elliptic problem in瓗N[J].Topological Methods in Nonlinear Analysis,2003,21:29-39.

[4] 趙美玲,賈高.加權(quán)Sobolev空間中奇異擬線性橢圓方程共振問題[J].上海理工大學(xué)學(xué)報,2012,34(6):598-603.

[5] 劉向平,章國慶.帶非線性邊界的p-Laplacian問題的多重解[J].上海理工大學(xué)學(xué)報,2013,35(5):449-451.

[6] Garofalo N,Lanconelli E.Existence and nonexistence results for semilinear equations on the Heisenberg group[J]. Indiana University Mathematical Journal,1992,41(1): 71-98.

[7] Niu PC.Nonexistence for semilinear equations and systems in the Heisenberg group[J].Journal of Mathematical Analysis and Applications,1999,240(1):47-59.

[8] Squassina M.Existence of multiple solutions for quasilinear diagonal elliptic systems[J].Electronic Journal of Differential Equations,1999(14):1-12.

[9] Brezis H,Browder F E.Sur une propriata des espaces de Sobolev[J].Comptes Rendus Mathématique Académie des Sciences Paris A-B,1978,287(1): 113-115.

[10] Canino A.Multiplicity of solutions for quasilinear elliptic equations[J].Topological Methods in Nonlinear Analysis,1995,6(2):357-370.

[11] Douik H.Variational methods in nonsmooth analysis and quasilinear equations[D].Germany,North Rhine Westphalia:Aachen University,2003.

[12] Ladyaenskaya O A,Uralceva N N.Linear and quasilinear elliptic equations[M].New York:Academic Press,1968.

[13] Rabinowttz P H.Minimax methods in critical point theory with applications to differential equations[M]. Providence:American Mathematical Society,1986.

(編輯:董 偉)

Multiplicity of Solutions of Quasilinear Elliptic Equations on Heisenberg Group

JIAGao, GUOLuqian, ZHANGLongjie
(College of Science,University of Shanghai for Science and Technology,Shanghai 200093,China)

The multiplicity of solutions for a class of quasilinear elliptic boundary value problems on the Heisenberg group was concerned.In the whole space,the main coefficients and their derivatives were assumed to be bounded,and the nonlinear term satisfies superlinear growth conditions.Under the above assumptions,the functional is continuous but not differentiable in the whole space.So,the nonsmooth critical point theory should be applied.The concepts about weak slope,critical point,(PS)cconditions and some fundamental lemmas in the nonsmooth critical point theory were introduced.The properties of the critical point of the functional were analysed.The strong convergence of the(PS)csequences was proved by using nonlinear functional theory,Fatou’s lemma,Lebesgue’s dominated convergence theorem and Brezis-Browder theorem.Moreover,by virtue of the generalized Mountain Pass lemma,the existence of infinite weak solutions of the boundary value problem was confirmed and these solutions are separable from one another.

quasilinear elliptic equation;nondifferentiable functional;nonsmooth critical point theory;Heisenberg group

O 175.25

A

1007-6735(2015)03-0210-05

10.13255/j.cnki.jusst.2015.03.002

2014-03-13

國家自然科學(xué)基金資助項目(11171220);滬江基金資助項目(B14005)

賈 高(1960-),男,教授.研究方向:非線性分析及應(yīng)用.E-mail:gaojia89@163.com