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【摘要】<正>The authors show the regularity of weak solutions for some typical quasi-linear elliptic systems governed by two p-Laplacian operators. The weak solutions of the following problem with lack of compactness are proved to be regular when α(x) and α,β,p, q satisfy some conditions: where Ω(?) RN (N≥3) is a smooth bounded domain.

【刊名】Chinese Annals of Mathematics


互1 .Introduetion In this paper,we eonsider the regularity ofthe weak solutionsofthefollowingquasi一linear elliPtie systems 、,,尹 ,l . ,l 了..、 一△;tL+a(x)}。!“一‘Iv{口+’。=!。},’一“。, 一△。。+a(x)}。l“+”vi口一’v=}v}q’一“。, 。(x)=v(x)=O, x〔几, x任几, x〔口几, where几〔几N(N全3)15 a smooth bounded domain,么;15 the介L叩laeian operator, namely△;。主div(!,。!”一2勺。).In addition a+r>0,尽+1>0 anda(x)c LOO(几). following inequality 15 valid: a+1 We assllme thatl三尹<N,1三q<N, the Positive eonstantsa,刀,P,q andN,the 一下丁一+ 刀 尽+1 一一,丁一圣1, q’ 1 ZU HU,Y.X.&Ll.J wlzel℃户*arlolq*are the eritieal Sobolev exponents,i·e.,尸* 二梁里a;1(la*=户二仁 丈、一P通;、一7 Analo (‘xistell(、〔、 90,15 res,llts have been obtained for single elliptie eqliation(se(: As长),th(、 arzd regularity reslilts for some quasi一linear elliptie equations or system、(),IR入 wer、·仆r to【2,3{and tlie referenees therein.The eondition(1.2)ean guarantee that th〔、〔”‘ ft;,1(、tionale()rresponding with the problem(1.1)defined as follows exists ‘恤川一宁力场!,·解牛力肠,?血一岁户止一‘击 一宁加q‘dx·加·)}·}。一:,一、二. By variati(川al methods,the eritieal Point theory and Lions’eolleexitratioiz(、()l,zpa(生,le、、 pri;leiPle.we(:an prove the existeneeofnontrivialweaksolutionstotheProblern(1.1)51叮lilal, t()tlle arguments of}41.Thus,it 15 neeessary to show the regularity()ftllewoaksoluti(),,、t() I)1、()l)le;11(1 .1).In this paper,we obtain the regularity result by applying Morse,5 iteroti、·(、 s(henl〔、t()tllis elliPtie systems. 价、hall denote by叫,尹(‘2)the sobolev spaee obtained as the〔losure ofC(f(‘2)int于1。 nornl日。11“=几}甲。l”dx.For simplieity,we denote大,·dx by丁·dx and deno‘e diff(.re,“ I)〔)siti、·e corzstants by C whieh may ehange frotn line to line.The natural spaee setting f()r ou,、prol)le,,1 15 the spaeeE三叫,尹(几)x叫、q(几).We prove first any weak 501,,,注(),: (、‘,))‘E belongs toL刃(几)xL戈(‘2),then(。,:)belongs toC‘,“(几)、“‘(几),l,>(), 刀>0. Now,W映state our main results in this PaPer. Theorem 1.1.Let(。,。)〔E占e the oeak solot乞onoj尸roble爪(1.1),ands叩poso tha云 (1 .2)holds aod亡人。扣艺之。切乞。9 eood葱tzoo ho之ds 月+l Q*一(尽+1) P*一(a+l) < 。+1 (1‘3) 了一P* <一 Th。,‘(,,,,;)任去戈(‘2)xL加(几) Theoreml,2. xL况(公2).。:o:e口。。: Under the cond‘t乞ons of Theo二。1·1,it holde that(},。},!v二!)任L关(‘2) 亡here。劣落s亡亡切。尹05落tz,,e eons云a。忿s拼>0,。>0、。eh悉ha艺(,,、,)〔 C’,肛(贝)xC’,“(几). 、,、 卜了、 LllLI 日a Example 1.1.TakeN=4,P=叮=3,and leta+1=尽+1=4.Oize ean see that c()nditiolls(].2)and(1.3) Example 1.2.When hold. N=3,夕=g=2,ifa+1=尽+1=3,then(l‘2)and(13) satisfied.Heli(、e tho elassieal solutions()f the problenl(1.1)are obtained. REGULARITY RESUUrS FOR SOME QUASI一LINEAR ELLIPTIC SYSTEMS 121 写2 .ProofS of Main Results Proof of Theorem 1.1.First we supPoseu七0,v全0 ands where、15 to be determined.In aw即similar to〔11,let、:=刀·。 and =“吕任L”(几), whereL‘R+, 曰1, 叭卜L >一u 。(x), L, ifu三L, ifu>L. J!气t 一一 、、、尹了 X /..气 L 廿 Take刀任‘了(几),叼三1 ifx任B几(x。),Vxo任豆fixed:叼三0 ifx彭B尺+r(x。),ando<r<R such that!v。!三告·Then v、:=。·。玄1·v。+。·。f‘(,。+(s一l),。:), f,v、…“·:2一‘/},。}··…贾“一‘’“· +‘一/。…梦“一”(}v·}·+(一‘,·‘v二,·,dx· (2 .1) sinee(。,:)。刀15 a weak solution of the prob一em(一1),for any沪。叫,夕(fl),it follows that /!v·!一v·勺,“·+f·(·)、·“。一‘,:“一、“一f二’一‘、“一 Let、=。p。·。彭‘一‘’,then,。叫,夕(。).By(2.2)we get (2 .2) p(s一1)一1 .uL 舰 P 叼 L︸i 塑由 五一奴 一 p U 甲 一 S P + 一 沁L U p 刀 P U V 产/J N艺i=1 +,,v·}一舞舞。一影一‘’ “·+f·(·)。一}:“一全“一l’、 一/二’。…彭“一‘’“二 (2 .3) Then for any‘>0,we deduee that f.,…。二贾一‘’“·+,(一‘,f‘甲二,·:二彭’一‘’dx :一f睿}二}夕一舞舞。一彭一‘)d· 一f·‘·)。一‘!:“·‘·贾‘一‘’“·+了二‘。…贾一l’“· :j,!甲·!一’。一2二彭“一‘’(·,,·,202+C‘,。,2二2,“· +f‘·(·,‘。·…“·‘,·‘·彭‘一‘,dx+j二’。…彭“一‘’“· 一f.,·、·。二贾‘一‘’“·+砚f.,·!一’。一2}v。一’二澎“一‘’“· +j,·(·,,。一,·,“·‘·全‘一‘’“·+/二’。…全“一‘’“二 122 HU,K X.&Ll,次 Cho()se/二告,then j .芍·!·。二f/(“一‘’dx+、(一‘)f}vl名·}·。二:‘“一‘’“· f}甲·,一。一2.甲。.2一全‘一‘’d· C <一 +/!·(·)}。一}·,”+1·彭“一‘’“·+/二‘。…彭”一‘’“二 Notice that for any£>0,the following inequality holds f、vU}一}甲:}2·俨一2一彭“一‘’“·:·f},·!·。二影“一‘’“·+二/}v。、一梦“一1’“· Wel飞lay ehoose‘sueh that 几l、J、. J斗tJ八0 … q自9‘,召 夕l!r‘ j{甲·尸。二:‘“一‘’dx+‘、(一‘,/.,二}·。二彭“一‘)dx :Cj.v7,,·…贾“一‘’dx+/}·(·)}。·…‘:}“一:(“一‘)dx+f二’。一 心“一‘’dx. By(2 .1)and(24),we obtain 关!甲?·}·d一关,,(。。·玄‘,‘尹dx :〔卜一‘五‘}?。,··…全“一‘’+。二(·,,·、“·‘…1·势“一‘)+。…‘二影“一”,“· Le‘s一干,‘hell、一2,号。L,(。)By(2·5)and Sobo‘ev‘nequa,‘ty,we have !关(。?必二分)”‘“·]丹 :口(誓功}伽‘·俨砰一 :‘加卿角阿 +la(二)ln尸。“+‘回月十‘呀一尹+。,俨’·砰一尹)d: N、〕ti(‘e七hat 五。)}。尹‘一贾一dx一无 (沪俨心一尹 R+r 一’一、:!厂(。U锐分)二以·{券!Z。二、)史尹 一J、去J‘J召倪+。 Simil找rly,fi)r‘、+1全P,we have _尤。一{:奸1·叮一dx一/n_(、·心一)·一}·}口一血 三{五(”二宁,·’“·]弄}儿,+, (。。+‘一夕}:}“+‘)恙 ,户由奋尹 u劣」 Clearly 尸甲(。+1一p) r 丈一 回 P巾 F巾(召+一) p‘一p “·:(尤:!·’、·)害(五:} p甲(口+1) p*一仪+i dx) p晰一(。+1少 尸*一尸 (2 .7) 厂无 REGULARITY RESUI月,5 FOR SOME QUASI一LINEAR ELLIPTIC SYSTEMS 123 Here we have used(1 .3)and(。,。)〔E in the last ine明ality.As for the easeo<a+1<p, the same result holds after the arguments similar to(2.11).By the eontinuity of integral, for any£o>0 we may ehooseR,尹sueh that 关“”’“x<‘”’ JB尺+, p*(0+1一p) tLP*一P 产*f口+z) p*一p dx<£0. 回 /儿 sinee。。叫,,(几),it follows tha‘。。Lp‘(几)·Take:0 such‘ha‘ _P*J“1 .已n一二, U4 一 P*一p C then 五。一}”·‘二f一“·+五俨二’二f一“·兰告 过卫.,子 p、夕才甲lr Ur夕忆为峨l 力护」 Substituting it into(2.6),we get }五(。…夕)·‘“·]升三C五,v。‘·… <C. LettingL丹oo,by Fatou’5 lemma,we have 产*2、J (/_u兮dx)”’三C. 、J刀R

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