Bubbling Phenomena of Certain Palais-Smale Sequences of m-harmonic Type Systems. Libin Mou and Changyou Wang

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1 Bubblg Pheoe of Cert Pls-Sle Sequeces of -Hroc Type Systes Lb Mou d Chgyou Wg Abstrct I ths pper, we study the bubblg pheoe of wek soluto sequeces of clss of degeerte qusler ellptc systes of -hroc type We prove tht, uder pproprte codtos, the eergy s preserved durg the bubblg process The results pply to -hroc ps fro fold Ω to hoogeeous spce, d to -hroc ps wth costt volues, d lso to cert Pls-Sle sequeces Itroducto d M Results The Pls-Sle P-S codto for fuctol E s turl ssupto for the exstece of crtcl pot of E; ths codto sys tht y Pls-Sle sequece {u } e, sup E u < d DE u 0, s hs strogly coverget subsequece I y terestg cses where Pls-Sle codto fls, people dscovered the bubblg pheoe of cert Pls-Sle sequeces Geerlly spekg, the flure of strog covergece s due to the loss of eergy, d bubblg pheoeo refers tht the lost eergy ws recovered or cptured by few bubbles solutos of the blow-up equto developed durg the lt process or bubblg process For refereces o bubblg pheoe d relted probles, see [SU] [Jj] [S] [Pt] [Qj] o hroc ps o surfces; [BN] [S2] o seler ellptc equtos, d [Wh] [BC] [S3] o H-systes, whch descrbe surfces of costt e curvtures Suppose Ω, g s Re fold The spce W, Ω, R k cossts of ll fuctos u : Ω R k wth fte eergy: E u = Du dω, Ω where dω deotes the volue eleet of Ω, ofte beg otted Du = Du,, Du k s the dfferetl of u We ssue tht, k 2 tegers, d Ω = We re terested bubblg pheoe of zg sequeces d Pls-Sle sequeces of the eergy E subject to cert costrts The Euler-Lgrge equto c ofte be wrtte s dv Du 2 Du = f u, Du, where u W, Ω, R k We ssue the followg hypotheses Mthetcs Subject Clssfcto:35J60, 58E20, 46E35 Key Words: Wek covergece, Pls-Sle sequece, -hroc p, volue costrt, hoogeeous spce, Hrdy spce, BMO spce

2 H-I f : R k R k R k s sooth fucto tht c be wrtte s: 2 f u, Du = q α u, Du Db α u, α= where α, : R k R k R k2 d b α : R k R k2 re sooth vector-vlued fuctos such tht u, Du C Du, b u C for soe costt C Here C y deped o u Copoetwse, 2 becoes f u, Du = k q j= α= l= l αj u, Du x l b αj u, =,, k Ths ssupto wll be used, together wth others, to show tht fu, Du s the locl Hrdy spce, better th L I few terestg cses, ths ssupto turlly holds; see [Wh] [Hf] [El] [MY] [TW], d the exples below Equto s uderstood the wek sese: For ll φ C 0 Ω, R k, there holds 3 Du 2 DuDφ = f u, Du φ Ω We ssue tht s coforlly vrt the followg sese H-II Suppose Φ : Ω, h Ω, g s coforl dffeoorphs, the 3 holds wth u replced by u Φ, φ by φ Φ d Ω by Ω Note tht, sce the eergy Ω Du s coforlly vrt, the Euler-Lgrge equto of Ω Du subject to costrt s coforlly vrt the sese of H-II, s log s the costrt s closed uder coforl trsfortos The -hroc p equtos d the H-systes, defed below, re two such exples Ths property ples tht the equto keeps the se for uder coforl trsfortos, especlly uder dltos d trsltos We cosder sequece {u } = W, Ω, R k stsfyg perturbed equtos of : Ω 4 dv Du 2 Du = f u, Du + h, h 0 W,, for =, d u u W,, but u u W, The study of the covergece behvor of {u } leds to the below-up equto: 5 dv Du 2 Du = f u, Du, for u W, R, R k Ths equto s obted s lt of the equtos stsfed by properly rescled {u } see 45 Here R plys the role of tget spces of Ω A otrvl soluto u of 2

3 5 s clled bubble By the coforl vrce of the equto 5 d the coforl equvlece of R d S uder the stereogrphc projecto, bubble u s detfed wth ũ W, S, R k, whch stsfes 5 o S Suppose u s regulr sy C ; ths s the cse f dv u, Du = 0, by Th 26 d Prop 3 [MY], the ũ s regulr o S except t the orth pole of the projecto Suppose f u, Du Du = 0, the by Theore 5 [MY], ũ c be exteded cross the orth s regulr soluto o the whole S The vlue of ũ t the orth pole wll be deoted sply by u Note tht both codtos dv u, Du = 0 d f u, Du Du = 0 re turlly stsfed by the two exples below A uque property of the bubbles s ufor lower boud for ther eergy Tht s, there s µ > 0 such tht for y otrvl bubble u, R Du dx µ It s well-kow tht for wekly covergece sequece {u } W, Ω, R k, u u strogly f d oly f the eergy coverges: E u E u, see [El] for exple Our result descrbes the covergece behvors of cert wekly coverget sequeces, d ccouts for ll the eergy loss wth fte uber of bubbles Theore Suppose tht H-I d H-II re stsfed If {u } W, Ω, R k s sequece tht stsfes 4 d for soe p >, h, dv u, Du 0 W,p, s, the there exst soluto u C Ω, R k of, fte uber l of bubbles ω C S, R k, l sequeces of pots { } Ω, l sequeces of postve ubers { λ }, l, d subsequece of {u }, stll deoted by {u }, such tht l E u = E u + l = E ω, { } λ 2 For j, x s, 3 u l = λ j, λj λ ω λ, j λ +λj ω W 0 s, We ow pply ths result to two exples: -hroc ps to hoogeeous spce d - hroc ps wth costt volue Applcto to -Hroc Mps Let Ω d N q be two closed Re folds Assue N s hoogeeous, d soetrclly ebedded to soe R k, k 2 Deote by W, Ω, N the set of ll v W, Ω, R k wth v x N for e x Ω Slrly, for gve u W, Ω, N, W, Ω, T u N deotes the spce of ll v W, Ω, R k wth v x T ux N for e x Ω A -hroc p s crtcl pot of E u W, Ω, N, whch stsfes E u dv Du 2 Du + Du 2 A u Du, Du = 0, 3

4 [ ], W, Ω, T u N = W, 0 Ω, T u N where = R k A Pls-Sle sequece {u } W, Ω, N of E the stsfes, d A s secod fudetl for of N d l sup E u < where h 0 W, Ω, R k dv Du 2 Du = Du 2 A u Du, Du + h, Uder slghtly stroger codto o h, Theore ples the followg Theore 2 Suppose tht Ω s closed fold, N s hoogeeous fold d {u } W, Ω, N s Pls-Sle sequece such tht for soe p >, sup h W,p < The there exst -hroc p u C Ω, N d fte uber l of -hroc ps ω C S, N, l sequeces of pots { } Ω, l sequeces { λ } of postve ubers, l, d subsequece of {u }, stll deoted s {u }, such tht l E u = E u + l = E ω, { } λ 2 For j, x, λj λ j λ, j s, λ +λj 3 u l W = ω λ ω 0 s, The codto tht h s bouded W,p for soe p > c ot be droped, s show by the exple [Pt] I poeer work, Scks-Uhlebeck frst developed [SU] the blow-up ethod to study perturbed eergy fuctol As pplcto, they obted the exstece of l ersos to Re fold Struwe [S] obted slr result for clss of solutos of hroc p het flows o surfces Jost [Jj] descrbed the bubblg process of -x schee for ps fro surfce to closed fold; see lso the pper of Prker [Pt] Bethuel [Bf] showed tht the wek lts of Pls-Sle sequeces of eergy of ps o surfce re lso hroc ps, but he dd ot descrbe the bubblg process Qg [Qj] descrbed the bubblg behvor of cert Pls-Sle sequeces of ps fro surfce to stdrd spheres Our result c be cosdered s geerlzto of these results to hgher deso cses I forthcog pper [Wc], the secod uthor proves the se result s Theore 2 for ps fro surfce to geerl copct fold N As corollry, we obt the strog covergece of cert Pls-Sle sequeces Corollry 3 Suppose tht Ω s closed fold, N s hoogeeous fold d {u } W, Ω, N s Pls-Sle sequece stsfyg the codtos Theore 2 The there exsts 4

5 subsequece of {u } strogly coverget to -hroc p u C Ω, N, provded y oe of the followg codtos holds: If l sup Ω Du dx < Ω Du + µ, where µ > 0 s the lrgest lower boud of -eergy of o-costt -hroc ps fro S to N 2 N supports strctly covex fucto f, ely, there exsts uber c 0 > 0 such tht where h s the etrc o N Hess f c 0 h, Applcto to -Hroc Mps wth Costt Volues Let Ω R be sooth do A -hroc p u W, Ω, R + wth costt volue s crtcl pot of Ω Du subject to costt volue eclosed by the coe geerted by u Ω wth vertex 0 R + The volue c be expressed s V u = u u u, + R where u u s the cross product of dervtves u = u x A -hroc p wth costt volue stsfes the equto 6 dv Du 2 Du = Hu u, for soe costt H See [MY] for detls It hs bee otced tht the rght hd sde f u, Du = Hu u c be wrtte the for 2 Specfclly, we hve u,, f = H + û,, u + x,, x = l= where =,, + u +2 = u, d It s esy to check tht l u+, x l u,, l = H +l û, u + ˆ,, u + x,, ˆx l, x dv = l= l x l = 0 for y u W, R, R + ; see [Db] I the cse Ω = R, W, S, R + c be detfed wth W, R, R + through the stereogrphc projecto π : R S Suppose u stsfes 6 o R, 5

6 the d u π lso stsfes 6 o S \ {orth pole} By the regulrty results [MY], both u d u π hve to be regulr C,α, 0 < α < o R d o S respectvely, d u = u πorth pole essts We stte result of pplcto of Theore to the cse Ω = S : Theore 4 Suppose {u } W, S, R + s bouded sequece of solutos of 7 dv Du 2 Du = Hu u + h, wth h 0 W,p, s, for soe p > The there exst soluto u C S, R k of 6, fte uber l of bubbles ω C S, R k, l sequeces of pots { } S, l sequeces of postve ubers { λ }, l, d subsequece of {u }, stll deoted by {u }, such tht l E u = E u + l = E ω { } λ 2 For j, x, λj λ j λ, j s λ +λj 3 u l W = ω λ ω 0 s, Rerk I the cse Ω, bubblg pheoe lso occur d c be descrbed the slr er s Theore I such cse, oe hs to clude bubbles o the hlf spce or equvletly, o the ut bll B, through coforl trsforto, whch re solutos of 5 o R + R or o B, resp d costt o R + {0} o B, resp For soe equtos, there re o otrvl bubbles o the hlf spce or the ut bll d the coclusos of Theore hold for Ω wth or wthout boudry Ths s the cse for 6 whe = 2, where Wete [Wh2] proved tht the oly soluto of u = 2u u 2 W,2 0 D, R 3 s 0, where D = { x R 2 : x < } Wete s ths result ws used the pper of Brezs d Coro [BC], whch estblshed the bubblg pheoe of soluto sequeces {u } W,2 0 D, R 3 stsfyg u = 2u u 2 + f wth f 0 W,2 For hroc ps o dsc wth costt boudry vlues, Lere proved tht they ust be costt [Ll] It would be terestg to kow whether the results of Lere d Wete hold hgher desol cses For exple, suppose u W, B, R + stsfes 6 or d u = u = 0 o B, where B = {x R : x < } Is u 0? Ths s closely relted to the uque cotuto proble for -Lplc equtos The de of proof for ll these Theores s bsed o two steps: frst, we prove the so-clled ɛ- copctess le whch s cosequece of the ɛ-cotuty esttes; secod, we study the proble bout bubbles over bubbles d show tht there s o eergy cocetrtg eck regos to be specfed below Sectos 2, 3, 4 The pper s orgzed s follows I 2, we prove the ɛ-cotuty estte As corollry, we prove the regulrty of the solutos beg cosdered I 3, we obt ɛ-copctess le d soe coprso les tht re ecessry for the proof of Theore I 4, we prove the results, by lyzg the cocetrto desty of the eergy, usg vrous esttes we obted Sectos 2 d 3 Sce we oly cosder the cses wthout boudry, ll the eeded esttes re of locl verso For ths reso d for splcty, Sectos 2 d 3, we ssue Ω s sooth do R wth 6

7 stdrd etrc As for ottos, costts re geerclly deoted by C or C ; they y chge fro le to le We wll deote by o qutty or sequece tht goes to 0 s the vrble dex goes to fty B r x deotes the bll R cetered t x d of rdus r; B r = B r 0 2 ɛ-cotuty Esttes The hypothess I o f s used to show tht f u, Du s Hrdy spce Hloc For the defto d propertes of Hloc Ω, R d BMOR, R, plese see [Ss] or [CLMS] d the refereces there Here we hve Proposto 2 Suppose u W, Ω, R k such tht h dv u, Du W,p for soe p > The u, Du Db u H loc Ω, R k Moreover, for y copct subset K Ω, there s costt C K such tht 2 u, Du Db u H K C K [ Du,Ω + h W,p ] Proof Ths ws essetlly proved [CLMS] To trce the estte, we sketch the proof Tke φ C0 R, R such tht φ = d spt φ B 0, For gve K Ω, x K d R r < d x, Ω, we defe φ r y = r φ y x r The by tegrto of prts, = R [ u, Du Db u] φ x = [ u, Du Db u] y φ r y dy R b u b u x,r h φr + b u b u x,r u, Du Dφ r = I + II R Here b u x,r = B r b u We estte I d II seprtely x II Cr + CM B rx Du b u b u x,r Du q q x M Du s s x, where q d s re chose so tht < q <, s = xl fucto defed by { M f x = sup B r x q +q B rx d < s < Here M f s the locl f : B r x Ω } By the soorphs theore of : W,p 0 W,p, we fd H W,p 0 Ω, R k such tht H = h wth H W,p C h W,p It follows 7

8 I = b u b u x,r Hφ r R = DHDb u φ r DH R R b u b u x,r Dφ r Therefore, I Cr DH Du + Cr + DH b u b u x,r = III + IV B r x B r x For IV, choose < q < p, < s < such tht s = q +q The [ ] IV C M q DH q M s Du s As for III, tke < α < d β = α such tht < β < p The III CM Du α α M DH β β By the Hrdy-Lttlewood xu theore see Ste [Se], Hölder equlty, d the defto of H K, we hve u, Du Db u H K C Du + C DH + C Ω C Du,Ω + C K K C Du,Ω + C K h W,p K K M DH β /β /p DH p + C K K M DH β p β /p It ws proved Toro-Wg [TW] tht y -hroc p to Re hoogeeous spce hs Hölder cotuous grdet The se result ws show [MY] for the solutos u of tht stsfy H- d dv u, Du = 0 Now we prove eergy decy le, whch wll be used to show C 0 cotuty of solutos u of 4 Le 22 Suppose u W, B, R k s soluto of 22 dv Du 2 Du = f u, Du + h, wth h = dv u, Du, where f stsfes H- d h, h W,p for soe p > The there exst postve ubers ɛ 0, θ 0 <, α 0 < d C 0, depedet of u, such tht for every x B d r < 4, f Du ɛ B 2 0, the [ 23 Du θ 0 Du + C 0 h W + ] h r α 0,p W,p B r x B 2r x 8

9 Proof For gve x B 2 d r < 4, tke η C 0 B 2r x, [0, ] such tht η = B r x Deote A x, r = B 2r x \ B r x d u x,2r = Ax,r u Multplyg 22 by η u u x,2r d tegrtg, we get tht 24 Du 2 Du D η u u x,2r = B f u, Du η u u x,2r + B hη u u x,2r = I + II B Deote Eu, 2r = B 2r x Du By Pocre equlty, Hölder equlty d the defto of BMO, we hve η u u x,2r W CEu, 2r;, η u u x,2r BMO C Eu, 2r Cɛ 0 ; It follows η u u x,2r Cr η u u x,2r R B 2r x / Cr B 2r x D η u u x,2r Cr Eu, 2r Cɛ 0 25 II C h W, B 2r x η u u x,2r W, C h W, B 2rx Eu, 2r Cɛ 0 h W, B 2r x I order to estte I, we defe f u, Du = Eu, 2r f u, Du d pck up p, p The Le 2 ples tht f u, Du Hloc B d for K = spt η we hve [ f u, Du H K C Eu, 2r Du + ] h 26 W,p B 2r x B 2rx C + C Eu, 2r h W,p B 2r x The we hve I = Eu, 2r f u, Du η u u x,2r B = Eu, 2r η f u, Du µ η u u x,2r R + µeu, 2r η u u x,2r R 9

10 Here µ = η f u, Du / η Use Proposto [S] [TW] to coclude tht η f u, Du µ H R wth 27 η f u, Du µ H R C + f u, Du H K C [ + Eu, 2r h µ Cr f u, Du H K By the dulty of H d BMO, I c be estted s follows W,p B 2r x I CEu, 2r η f u, Du µ H η u u x,2r BMO + µ Eu, 2r η u u x,2r CɛEu, 2r + Cɛ h W,p B 2r x For the left hd sde of 24, we hve η Du + η Dη Du 2 Du u u x,2r B B Du C Du B r Ax,r Here we used Hölder d Pocre equltes Use Hölder equlty for the h d h ters 25 d 27 d cobe We obt Du C { α 0 = p B r x Ax,r Du + Cɛ 0 B 2r x Du + Cɛ 0 h W,p B rα0, }, p p Add C B Du rx to the bove equlty We hve Du θ 0 Du + Cɛ 0 h, h W,p B rα0, B r x B 2r x where θ 0 = C+Cɛ 0 C+ <, f we choose ɛ 0 < 2C Hece 23 holds Theore 23 There exst postve costts ɛ 0, δ 0 < d C such tht f u W, B, R k stsfes the codtos Le 22, d B Du ɛ 0, the u C δ0 B/2, R k d u C δ 0B/2 C Proof For every x B 2 d r < 4, f we defe F x, r = B rx Du the Le 22 ples tht F x, r θ 0 F x, 2r + Cr α 0, where C depeds o p, p, ɛ 0, h, h W,p B By Le 823 Glbrg-Trudger [GT], there re ubers β 0 0, d R 4, such tht for ll r R, r β0 r α0 F x, r C F x, R + r R 0 ]

11 By the Morrey s decy le see Morrey [Mc], u C δ0 B/2 for δ0 u C δ 0B/2 C = {α 0, β 0 } d Corollry 24 Suppose u W, Ω, R k s soluto of 22 dv Du 2 Du = f u, Du + h, wth h = dv u, Du, where f stsfes H- d h, h W,p for soe p > The there exsts postve uber δ 0, such tht for every u C,δ 0 Ω, R k Proof For every pot Ω, sy 0 Ω, we tke sll uber r > 0 such tht B r Du < ɛ 0 s Theore 23 Cosder u r W, B, R k defed by u r x = urx The u r stsfes 22 wth f beg replced by fu r, Du r d h by r h wth B Du r < ɛ 0 Therefore, Theore 23 ples tht u r C δ 0 B /2 for soe δ 0 So u C δ 0 B r/2 C,δ 0 regulrty c be obted by cosderg the equto stsfed by Du, s expled [MY] We ed ths secto wth suffcet codto for strog covergece of Pls-Sle sequeces Proposto 25 Let {u } W, Ω, R k be sequece stsfyg 4 If u u L loc Ω, R k, the u u W, loc Ω, R k I prtculr, u s soluto of Proof It suffces to prove tht {u } s Cuchy sequece W, I order to do tht, let ξ be y loc cut-off fucto Ω Multplyg both equtos of u d u l by ξ 2 u u l d subtrctg oe fro the other, we get Du 2 Du Du l 2 Du l D ξ 2 u u l 28 = Ω It s esy to see tht the rght hd sde stsfes + f u, Du f u l, Du l ξ 2 u u l Ω Ω h h l ξ 2 u u l RHS h h l W, ξ 2 u u l W, + Du + Du l ξ 2 u u l L 0, s, l Therefore, by the covexty of tegrd d the strog covergece of u L see [HLM], for exple, the left hd sde of 28 B r Du Du l + o s l, It follows Du Du l 0, s l, B r 3 ɛ-copctess Les Ths secto cossts of soe preprtory les for the proof of the Theore

12 Le 3 Suppose the hypotheses H-I d H-II hold The there s ɛ 0 > 0 such tht f {u } W, B, R k stsfes 3 dv Du 2 Du = f u, Du + h, wth both h dv u, Du, h W, 0, sup h, h W,p < for soe p > d B Du ɛ 0, the u cots subsequece tht coverges to u W, loc B, R k u C B, R k d stsfes Proof By pssg to subsequece, we y ssue tht u u wekly W, d strogly L By Theore 23, for y 0 < r <, u C δ 0 Br C for soe δ 0 d C depedet of Therefore, by Arzel-Ascoll s copctess theore, we c extrct subsequece of {u } deoted by {u } such tht u u C 0 B r, R k Now pply Proposto 25 to get the cocluso We wll copre the eergy of the solutos to 4 wth the eergy of -hroc fuctos Frst, we clculte the eergy of the hroc fucto o ulus wth costt boudry vlues We hve For 0 < R < R 2, let A R, R 2 = {x R : R x R 2 } d A R, R 2 = B R B R2 Le 32 Suppose tht u W, A R, R 2, R k stsfes 32 dv Dv 2 Dv = 0, A R, R 2, u BR =, u BR2 = b Here d b re costts R k The 33 AR,R 2 Dv = b log R 2 R Proof Fro the covexty of AR,R 2 Dv, we kow tht the soluto of 32 s uque Sce the vlues of v o boudry re costt, we cosder the rdl soluto v x = v x The the equto of v becoes 34 v r + r v r = 0, v R =, v R 2 = b It follows v x = C + b log R 2 R log x for soe C d AR,R 2 Dv = R2 R v r r dr b = log R log R 2 2 R R 2

13 Le 33 Suppose f stsfes hypotheses H-I d H-II d u W, Ω, R k stsfes dv Du 2 Du = f u, Du + h wth h dv u, Du, h W,p for soe p > Ω Du ɛ 0, d for Ω Ω, defe There exst ɛ 0 > 0 d C such tht f dv Dv 2 Dv = 0, Ω ; v Ω = u Ω The 35 Du,Ω C Dv,Ω + C h W,p, Proof Multplyg both equtos of u d v by u v d subtrctg the, we get tht Du 2 Du Dv 2 Dv 36 D u v = h u v + f u, Du u v = I + II Ω Ω Ω Wth the detfcto of h wth H for H W,p 0, by Hölder equlty d Sobolev ebeddg theore, we hve I C DH D u v Ω C DH,Ω D u v,ω C h W, Ω D u v,ω Let ξ C 0 Ω, R be such tht ξ o Ω The g by Proposto [S] [TW], we hve II = f u, Du ξ u v R C f u, Du µ H R u v BMO R + µ ξ u v C Du,Ω + h D u v W,p,Ω Here µ = ξ ξf u, Du, d we used Sobolev equlty d the fct tht µ s less th the Hrdy or of f u, Du Therefore 36 ples D u v,ω C Du,Ω + h, h D u v W,p,Ω e, 37 D u v,ω C Du,Ω + h, h W,p 3

14 O the other hd, Du,Ω 2 D u v,ω + Dv,Ω Usg 37, we get Du,Ω C Du,Ω + C Dv,Ω + C h, h W,p Ω Choose ɛ 0 sll eough, the 35 follows Flly, we gve the proof tht the eergy of bubbles hs lower boud Proposto 34 Suppose tht H-I d H-II re stsfed { } µ = f Du : u : S R k s o-costt soluto of 5 > 0 S Proof Suppose µ = 0, the by ts defto, there exsts sequece of o-costt solutos {u } : S R k such tht Du S 0 By the regulrty results [MY][TW], {u } re C,α wth u C,α C wth 0 < α <, C depedet of Sce u re ot costt, there exst {p } S such tht Du p 0 Usg the coforl vrce of -eergy, we c ssue, by coposg u wth sutble coforl trsforto of S, tht Du p = for soe fxed p S By pssg to subsequece, we c ssue u u C W, Ths ples tht u =costt d Du p =, cotrdcto 4 Proofs of M Theores Proof of Theore Wthout loss of geerlty, we ssue tht there exsts u W, Ω, R k such tht u u wekly W, d strogly L For clrty, we dvde the proof to four steps Step Defe the cocetrto set Σ Ω by } Σ = r>0 {x Ω : l f Du > ɛ 0, B r x where ɛ 0 s the se uber s Theore 23 It follows fro stdrd rguet tht Σ s fte subset of Ω; see [SU] or [TW] Furtherore, for y x 0 / Σ, there exsts r 0 > 0 such tht l f Du ɛ 0 B r0 x 0 Thus the ɛ-copctess Le 3 ples tht subsequece of {u }, stll deoted by {u }, stsfes u u W, loc B r 0 x 0 C 0 loc B r0 x 0 It follows u u W, loc C 0 loc Ω \ Σ 4

15 I prtculr, u stsfes the equto o Ω \ Σ d so o Ω solted sgulrtes re reovble By the regulrty results [TW][MY], u s C,α o Ω Step 2 As Brezs-Coro [BC], we defe, for 0 < δ < 2 {d x, x : 2 l}, 4 Q t = sup Du x B δ x x+tb It s esy to see tht there exst sequeces { } Bδ x x d λ 0 such tht 42 Q λ = Defe the resclg fuctos by +λ B Du = ɛ 0 2 v : λ Ω \ { } R k by v x = u + λ x, the by coforl vrce of the eergy, 43 Dv = λ Ω\{ } Ω Du 44 B x Dv ɛ 0 2, for ll x R d wth equlty whe x = 0 Moreover, 45 dv Dv 2 Dv = f v, Dv + λ h Fro 43 to 45, Le 3 pples to v We hve for soe ω W, R, R k, 46 v ω Cloc 0 W, loc R, R k I prtculr, B Dω = ɛ 0 2 So ω s otrvl bubble Repetg ths process to x 2,, x l, we get bubble solutos ω : S R k d sequeces { } x, d λ 0, s, such tht 47 u + λ ω Cloc 0 W, loc R, R k Step 3: Defe w = u l = λ exp We cl tht ] [ω λ ω, where deotes orth pole S d λ = 48 Dw = Du l Ω Ω = R Dω + o 5

16 I fct, for 0 < δ < 2 {d x, x j : j} we hve 49 Dw = B δ x Dw + B R λ Dw B δ x \B R λ = I + II / Note tht ω d ω j hve dsjot supports for j It follows by chge of vrbles, 40 Dω j = o B R λ Hece 4 I = Dw B R λ = D u ω B R λ λ + o = D u + λ ω B R O the other hd, by locl strog covergece of u + λ to ω W,, we hve the followg detty see pge Evs [El] provded tht R s chose to be suffcetly lrge 42 D w + λ ω = Du + λ Dω + o B R B R B R 49 = Du B R λ Dω + o R It follows fro tht 40 I = Du B R λ Dω + o R O the other hd, for j 44 B δ x \B R λ j Dω j λ j B δ x Dω j = o R \B j δλ j Dω j λ j We choose R suffcetly lrge so tht 45 R \B R Dω = o 6

17 44 d 45 ply tht 46 II Du + o B δ x \B R λ Cobg 40 wth 46, we get, for l, tht B δ x Dw = B δ x Du Dω + o R O Ω \ l = B δ x, we hve u u W, Whch ples tht Ω\ l = B δx Dw = Ω\ l = B δx So, the cobto of 47 d 48 ples the cl 48 Step 4 Du + o Cse : If l Ω Dw = Ω Du, the we lredy coclude tht w u W, Ω, R k Otherwse, we hve Cse 2: l Ω Dw > Ω Du Let Σ Ω be the set of cocetrto of w I fct, fro 48 we kow tht Σ Σ We ssue tht Σ = {x,, x l } for soe l l Pck x Σ, the we hve 49 l sup t 0 l / Defe Q for w, the there exst { 420 Q λ = +λ B t x Dw = ɛ > 0 } Bδ x d { } λ R such tht { ɛ Dw = B 2, ɛ 0 2 It s esy to see tht λ λ d λ 0 otherwse, there exsts λ 0 > 0 such tht Dw ɛ x +λ 0 B 2, whch cotrdcts wth 49; oreover, x otherwse, there exsts cocetrte pot outsde Σ We defe w : λ Ω \ R k by w x = w + λ x, s Step 2 The 42 sup λ Ω\ Dw = sup Ω } Dw < 422 B x { ɛ Dw 2, ɛ 0 2 }, x R ; wth equlty for x = 0 We y ssue tht w ω wekly W, loc R, R k We ow prove the followg 7

18 Cl : For l, 423 x { λ λ, λ λ, } λ + λ 2: w ω W, loc R, R k d ω s o-trvl bubble Proof of the Cl: It suffces to prove 423 for = Suppose t dd ot hold, there would exst R < such tht R λ λ +λ R d R The λ +λ Dω Dω B +R2 +2Rλ B = D u + λ ω B R 2 +2R 0, whch cotrdcts to 420 So 423 holds To prove the Cl 2, we cosder the two cses of 423 Cse : There exsts M > 0 such tht 424 ether M λ λ M, λ + λ, or λ λ, λ + λ I ths cse, + λ B R + λ B R = for lrge R > 0 Therefore, Dω + λ Dω o R \B R d B x B R λ { Du ɛ + λ 2, ɛ 0 2 } + o, x R, wth equlty for x = 0 So we pply le 3 to coclude tht u + λ ω W, loc R, R k for soe ω : R R k, whch s bubble Cse b: There exsts M < such tht λ λ = = x The, for ll α > 0 d l 425 R \B α Dω + λ λ, M For the splcty, we ssue tht λ R \B αλ /λ λ Dω = o, whch ples { Du ɛ + λ B x\b α 2, ɛ 0 2 } + o, x R, wth equlty t x = 0 Ag, by Le 3, we get u + λ ω W, loc R \ B α d ω s o-costt soluto, whch exteds to R by lettg α Moreover, for y R > 0, 426 B R D w + λ ω = 8 B α D w + λ ω + o

19 = O the other hd, for β > B αλ D w ω + o λ B βλ Dw = o, 428 B βλ Dω λ = Dω = o B βλ /λ Therefore, f we deote A, βλ, αλ = Bαλ \ Bβλ, the 429 D w + λ ω = D B R A,βλ,αλ We choose α so sll d β so lrge tht 430 Dω = o, D ω ω 0 = o B α B α w ω + o λ 43 B β D u + λ ω = o The 432 B R D w + λ ω = If we defe v o A, βλ, αλ { ɛ 2, ɛ 0 2 A,βλ,αλ Du + o, } + o dv Dv Dv = 0, v = u, o A, βλ, αλ The, fro le 33, we hve 434 A,βλ,αλ Du C A,βλ,αλ Dv + C h W,p I prtculr, 435 D w + λ ω C B R A,βλ,αλ Dv + o 9

20 Now, we defe f d g o A, βλ, αλ by dv Df Df = 0, f = u ω 0, o Bαλ f = u ω, o B βλ dv Dg Dg = 0, g = ω 0, o Bαλ g = ω, o B βλ The, fro d Proposto 32, we hve, wth A = A, βλ, αλ, 438 Df = o, Dg = ω ω 0 A A log αλ βλ By the lty of v, we kow tht Dv A A Df + Dg A It follows fro 438 d λ /λ, for fxed α d β, 439 A,βλ,αλ Dv = o Now 439 d 435 ply B R D w + λ ω = o Ths copletes proof of the cl We c repet ths rguet for w er other pots Σ to get bubbles ω j : R R k d { } j Ω, λ j R for l + j l + l such tht 2 Theore holds d 48 holds wth l replced by l + l, w j + λ j ω j W, loc [ R ], R k Sce ech ω : S R k hs t lest eergy µ, ths bubblg process tertes fter tes, where C = l Ω Du Notce tht the rguet keeps ll eergy durg the bubblg process The coclusos of theore follow C µ Proof of Corollry 3 Suppose the cocluso were ot true The, fro theore, there exsts t lest oe o-costt -hroc p ω : R N such tht l Ω Du Ω Du + µ, whch cotrdcts to the ssupto 2: I ths cse, we wll show tht y -hroc p ω fro S to N s costt I fct, f f s the gve covex fucto o N, the we hve the followg ch rule see, Jost [Jj]: 430 dv Dω 2 D f ω c 0 Dω 20

21 The we tegrte ths equlty over S to coclude tht E ω = 0 Proof of Theore 2 We strt by recllg tht o Re hoogeeous fold for detls, we refer to Hele [Hf] or Toro-Wg [TW], tht there exst q felds Y α α q d q sooth kllg vector felds X α α q o N such tht for y y N d v T y N, we hve q v = X α, v Y α I prtculr q Du 2 Du = Du 2 Du, X α u Y α u / Note the property of X α α q tht for α q 43 dv Du 2 Du, X α u = h, X α u α= α= Therefore, we c wrte the -hroc equto s dv Du 2 Du = Sce h 0 W,p for soe p > + q h, X α u Y α u α= q Du 2 Du, X α u Y α u α= = h + q α u, Du b α u α= d X α, Y α re sooth u, t follows tht h 0 W,p The coforl vrce of the -hroc p equto follows fro tht of for soe p > Ω Du Therefore, the codtos of Theore re stsfed, d so the coclusos of Theore 2 hold Proof of Theore 4 I ths cse, we lredy observed tht f = Hu u c be wrtte s f = l= l u+ x l wth dv = 0 Moreover, the blow-up equto s coforlly vrt I prtculr, codtos of Theore re stsfed d hece the coclusos follow 2

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