帶消失位勢(shì)Choquard方程解的存在性

蔡明建

(中南民族大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)學(xué)院,武漢430074)

帶消失位勢(shì)Choquard方程解的存在性

蔡明建

(中南民族大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)學(xué)院,武漢430074)

研究了一類(lèi)非局部Schr?dinger方程解的存在性.運(yùn)用山路引理和Ekland變分法，利用泛函幾何結(jié)構(gòu)和極小化序列得到了方程解的存在性.首次將半線性橢圓方程的相關(guān)結(jié)果推廣到Choquard型消失位勢(shì)Schr?dinger方程.

Choquard方程；山路引理；Ekland變分法；消失位勢(shì)

1 主要問(wèn)題及結(jié)果

本文考慮一類(lèi)帶消失位勢(shì)Choquard方程：

(1)

Choquard型方程解的存在性的研究，近年來(lái)一直吸引著研究者的興趣.文[3]中Moroz和Schaftingen考慮了基態(tài)解的存在性，文[4]討論該類(lèi)方程解的存在集中性，其他相關(guān)解的存在性問(wèn)題見(jiàn)文[5-7].本文的主要結(jié)果是定理1.

定理1 假設(shè)條件(i)～(ⅳ)，(H1)～(H4)成立,那么方程(1)在D1,2(RN)中至少存在兩個(gè)非平凡解.

2 預(yù)備知識(shí)

與方程組(1)所對(duì)應(yīng)的能量泛函可以寫(xiě)為：

其中

由Hardy-Littlewood不等式[8]知：

(2)

當(dāng)u∈E滿足：

設(shè)E為一實(shí)的希爾伯特空間，泛函I∈C1(E,R).我們說(shuō){un}為I的P.S.序列:如果當(dāng)n→∞時(shí),有I(un)→c,I′(un)→0.泛函I在指標(biāo)c∈R處滿足P.S.條件是指,上述{un}存在一個(gè)收斂的子序列.若I(u)=c,I′(u)=0時(shí)，稱(chēng)u為I在E上的臨界點(diǎn)，c為I的臨界值.

引理1I∈C1(E,R)且I滿足：

(1) 存在β,ρ>0使得I(u)≥β對(duì)任意‖u‖=ρ時(shí)都成立；

(2) 存在e∈E,‖e‖>ρ且使得I(e)<0.

由此可知,存在足夠小的ρ>0使得‖u‖=ρ時(shí)：

由此可知當(dāng)t→+∞時(shí)，I(tu)<0成立，故證存在e∈E,‖e‖>ρ且使得I(e)<0.

3 定理1的證明

在這一部分將證明主要結(jié)果定理1.

引理2 如果(i)～(ⅳ)，(H1)～(H4)成立，則有I滿足P.S.條件.

證明1)任取{un}?E滿足I(un)→c并且I′(un)→0,那么{un}有界.

事實(shí)上：

由條件(H4)可知存在c1，使得：

因此：

c(1+‖u‖)≥k‖u‖2-c1,可知{un}有界.

‖un-u‖2=I′(un)-I′(u),un-u-K(x)(f(un)-f(u))(un-u)dx-Q(un-u,un-u)dx.

由文[1,2],有：

(3)

(4)

n→∞時(shí)，

(5)

由文[9]有：

(6)

這里當(dāng)n→∞時(shí)，

I′(u)-I′(un),un-u→0.

(7)

由(5)～(7)式可知n→∞時(shí)，‖un-u‖→0.

綜上可知，方程(1)在D1,2(RN)中至少存在兩個(gè)非平凡解u1,u2.

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ExistenceofSolutionsofChoquardTypeEquationswithVanishingPotentials

CaiMingjian

(College of Mathematics and Statistics,South-Central University for Nationalities, Wuhan 430074,China)

In this paper，a class of nonlocal Schr?dinger equations were studied. The Mountain Pass Lemma and Ekland variational principle were used to prove the existence of solutions. Some new results about Choquard type Schr?dinger equations with vanishing potentials were obtained by related results in the semilinear elliptical equations.

Choquard type equations; Mountain Pass Lemma；Ekland variational principle；vanishing potentials

2017-09-12

蔡明建(1981-)，男，講師，博士，研究方向：偏微分方程，E-mail: cmj9904@mail.scuec.edu.cn

中央高校基本科研業(yè)務(wù)費(fèi)專(zhuān)項(xiàng)資金資助項(xiàng)目(CZQ12014)

O175.25

A

1672-4321(2017)04-0149-03