一類帶有多重Hardy項(xiàng)和多重強(qiáng)耦合Hardy-Sobolev臨界項(xiàng)的橢圓方程組的正解

康東升，李 靜，徐良順

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

一類帶有多重Hardy項(xiàng)和多重強(qiáng)耦合Hardy-Sobolev臨界項(xiàng)的橢圓方程組的正解

康東升，李 靜，徐良順

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

研究了一類帶有多重Hardy項(xiàng)和多重強(qiáng)耦合Hardy-Sobolev臨界項(xiàng)的橢圓方程組，運(yùn)用集中緊性原理和山路定理，控制Hardy項(xiàng)系數(shù)和強(qiáng)耦合臨界項(xiàng)指數(shù)，證明了在一定條件下方程組正解的存在性，首次把帶有多重Hardy項(xiàng)的臨界橢圓方程的相關(guān)方法應(yīng)用到帶有多重Hardy項(xiàng)和強(qiáng)耦合Hardy-Sobolev臨界項(xiàng)的橢圓方程組.

多重Hardy項(xiàng)；強(qiáng)耦合Hardy-Sobolev臨界項(xiàng)；集中緊性原理；山路定理

AbstractIn this paper, an elliptic system is investigated, which involves multiple Hardy-type terms and strongly-coupled critical Hardy-Sobolev terms. By the concentration compactness principle and the Mountain Pass Theorem and by controlling the coefficients of Hardy-type terms and the exponents of strongly-coupled critical terms, the existence of positive solutions to the system is verified under certain conditions, and for the first time, the related methods for the critical elliptic equation involving multiple Hardy-type terms are extended to the system of critical elliptic equations involving multiple Hardy-type terms and strongly-coupled Hardy-Sobolev critical terms.

Keywordsmultiple Hardy-type terms; strongly-coupled critical Hardy-Sobolev term; concentration compactness principle; Mountain Pass theorem

1 相關(guān)知識(shí)

本文研究如下帶有多重Hardy項(xiàng)和多重強(qiáng)耦合Hardy-Sobolev臨界項(xiàng)的橢圓方程組：

(1)

其中Ω?N(N≥3)是有界光滑區(qū)域，方程組中的參數(shù)滿足如下假設(shè)：

方程組(1)與下面的Hardy-Sobolev不等式密切相關(guān)[1]：

(2)

關(guān)于對(duì)應(yīng)的最佳常數(shù)和極值函數(shù)，參見文獻(xiàn)[2]和[3]．當(dāng)t=2時(shí)(2)式就變成了著名的Hardy不等式[1]：

(3)

則J∈C1(H×H,).如果(u,v)≠(0,0)滿足

則稱(u,v)∈H×H是方程組(1)的一個(gè)解.此時(shí)由橢圓方程的正則性理論可知:

(4)

(5)

其中D1,2(N)是N)關(guān)于范數(shù)的完備化空間. 由文獻(xiàn)[3]可知Sμ,t與a無關(guān),并且Sμ,t有如下形式的達(dá)到函數(shù)：

不失一般性，假設(shè)：

(M3) 存在l,1≤l≤k,使得0≤λl=μl<μ*，并且

本文的主要結(jié)果可以歸納為下面的定理1.

定理1 假設(shè)條件(M1)，(M2)和(M3)成立，則方程組(1)在H×H中至少存在一個(gè)正解.

2 正解的存在性

首先需要建立幾個(gè)定義和引理.

設(shè)泛函I∈C1(H×H,).如果對(duì)于任意滿足下列條件的{(un,vn)}?H×H：

I(un,vn)→c,I′(un,vn)→0 在(H×H)*中，{(un,vn)}在H×H中都存在一個(gè)強(qiáng)收斂的子列，則稱I滿足(P.S.)c條件.

證明當(dāng)n→∞時(shí)，假設(shè){(un,vn)}?H×H滿足：

J(un,vn)→c

其中δx是點(diǎn)x處的Dirac質(zhì)量.由Hardy-Sobolev不等式，得出：

(6)

因此有：

(7)

于是由(6)、(7)式可得：

另一方面，推出：

證明證明過程與文獻(xiàn)[6]的定理5類似，這里略去.

引理3[7]當(dāng)ε→0時(shí)，有：

引理4 設(shè)：

則在定理1的假設(shè)條件下有：

證明考慮下面兩個(gè)函數(shù)：

當(dāng)ε→0時(shí)，由引理3可知，存在不依賴于ε的常數(shù)C1，C2>0，使得：

C1<τε

為了簡單，假設(shè)：

tmin=min{tj:1≤j≤k,j≠l}∈(0,4).

直接通過計(jì)算得：

μl<μ*?2b(μl)>tmin.

(8)

另外，

由(8)式和引理3可得：

另一方面，

由引理2和3可以得出：

(9)

再由(9)式、引理3可得：

Γ={γ∈C([0,1],H)|γ(0)=0,J(γ(1))<0}.

對(duì)任意u∈H{0}，由不等式(2)和(3)可得：

J(un,vn)→c,J′(un,vn)→0.

由引理4可得出：

由引理1知在{(un,vn)}中存在子序列(仍記為{(un,vn)})，使得在H×H中(un,vn)→(u,v). 因此J存在一個(gè)臨界點(diǎn)(u,v)滿足方程組(1). 設(shè)u+=max{u,0},v+=max{v,0}，分別用u+,v+來替換方程組(1)中等式右邊的u,v，重復(fù)上述過程可以得到方程組(1)的一個(gè)非負(fù)解(u,v)．由最大值原理[9]可以推出在Ω{0}中u,v>0.定理1證畢．

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PositiveSolutionstoanEllipticSystemInvolvingMultipleHardy-TypeTermsandMultipleStrongly-CoupledCriticalHardy-SobolevTerms

KangDongsheng,LiJing,XuLiangshun

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

O175.25

A

1672-4321(2017)03-0137-05

2017-04-11

康東升(1967-)，男，教授，博士，研究方向：偏微分方程，E-mail:dongshengkang@scuec.edu.cn

國家自然科學(xué)基金資助項(xiàng)目(11601530);中南民族大學(xué)研究生科研創(chuàng)新項(xiàng)目(2017sycxjj083)