Asymptotic behavior of radial solutions for a semilinear elliptic problem on an annulus through Morse index

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1 J. Dfferetl Equtos Asymptotc behvor of rdl solutos for semler ellptc problem o ulus through Morse dex P. Esposto,G.Mc,, Sjb Str b,p.n.srkth c Dprtmeto d Mtemtc, Uverstà degl Stud om Tre, Lrgo S. Leordo Murldo, om, Itly b Deprtmet of Mthemtcs, Id Isttute of Scece, Bglore-560 0, Id c TIF Cetre, PB 34, IISc Cmpus, Bglore-560 0, Id eceved October 006 Avlble ole My 007 Abstrct We study the symptotc behvor of rdl solutos for sgulrly perturbed semler ellptc Drchlet problem o ulus. We show tht Morse dex formtos o such solutos provde complete descrpto of the blow-up behvor. As by-product, we exhbt some suffcet codtos to gurtee tht rdl groud stte solutos blow-up d cocetrte t the er/outer boudry of the ulus. 007 Elsever Ic. All rghts reserved. MSC: 35J0; 35J5; 35J60 Keywords: Blow-up lyss; Morse dex; Mout Pss theorem. Itroducto I ths pper, we study the symptotc behvor s λ + of rdl solutos to the problem: { u + λv xu = u p Ω, u>0 Ω, u = 0 o Ω, Frst d secod uthors re supported by MUST, project Vrtol methods d oler dfferetl equtos. * Correspodg uthor. E-ml ddresses: esposto@mt.urom3.t P. Esposto, mc@mt.urom3.t G. Mc, sjb@mth.sc.eret. S. Str, srkth@mth.tfrbg.res. P.N. Srkth /$ see frot mtter 007 Elsever Ic. All rghts reserved. do:0.06/j.jde

2 P. Esposto et l. / J. Dfferetl Equtos where p>, Ω := {x N : < x < } s ulus d V : Ω s rdl smooth potetl bouded wy from zero: f V>0. Ω The strtg pot of our lyss s the followg, esy to prove, fct: sce H0,rd Ω, the spce of H0 Ω-rdl fuctos, s compctly embedded to Lp+ Ω for y p>, rdl solutos u λ of blow-up L Ω,.e.mx Ω u λ + s λ + smlr blow-up occurs geerl dom Ω s well, f N = d <p<+ or N 3 d <p N+ N. It s the qute terestg, lso vew of exstece, to detfy the lmtg equto, to uderstd the ture of the blow-up set d to descrbe the symptotc profle of u λ : throughout the pper, λ + d the mx Ω u + u correspodg soluto of. Actully, we oly kow of pper by Dcer [4] where some symptotc lyss of s crred over. It s lmted to the cse V d p subcrtcl; by mes of ODE techques, Dcer shows tht, for λ lrge, the oly postve rdl soluto s the rdl groud stte, d t tkes ts uque mxmum o sphere whose rdus goes to. I some ppers [,] by Ambrosett, Mlchod d N the kowledge of the lmtg equto s used to obt exstece. Amog other thgs, for potetls V stsfyg they foud [] solutos u λ blowg up s λ + o spheres of sutble rdus. Frst, they troduce uxlry potetl see lso [3] Mr := r V θ r, θ = p + p 3 here d wht follows we freely wrte x s x d Vxs V x. The, usg costructve methods bsed o oler Lypuov Schmdt reducto, they buld solutos u λ whch blowup t the er boudry f M >0 s well s solutos whch blow-up t spheres whose rdus s strct locl mxmum or mmum of M. More geerl, the Ambrosett, Mlchod d N work mkes cler the crucl role of the crtcl set : M = { [, ]: Ṁ 0, Ṁ 0 }. 4 At lest geerclly, y pot M should be good cddte for beg blow-up rdus,.e. for the exstece of λ,u solutos such tht λ +, mx u r + r δ s +, δ>0. Oe of our m results s tht blow-up rdus hs to belog to M. Actully, the symptotc lyss we develop ths pper reles o Morse dex ssumpto. Gve solutos λ,u wth λ + we wll ssume u hve uformly bouded Morse dex,.e. k N such tht, f W s ler subspce of H0,rd Ω d, for some N, Ω v + λ Vxv pu p v < 0, v W \{0}, the dm W k. As cosequece of Theorem 3., of Corollry 3. d Theorem 4. we hve the followg: 5

3 P. Esposto et l. / J. Dfferetl Equtos Theorem.. Let λ +,u be solutos to stsfyg 5. The, up to subsequece, there re k k d pots,, =,...,k, wth the followg propertes: re the uque pots of mxmum of u, u +, coverge to pots M, ot ecessrly dstct; furthermore, u 0 uformly wy from {,..., k }. We recll tht rdl groud stte soluto lwys stsfes 5: t hs exctly Morse dex oe H0,rd Ω see [5]. Thus, s by-product of Theorem., we obt, geerlzg [4], explct sequece of solutos blowg up o sphere compre wth []: Theorem.. Let u λ be rdl groud stte soluto of.forλ lrge, u λ hs uque pot of mxmum λ d u λ λ +. Furthermore, f λj, the Ṁr > 0 r, ] = whle Ṁr < 0 r [, =, Ṁ<0 < Ṁ Ṁ = 0. Thus, y cse, M. Flly, u 0 uformly wy from. The pper s orgzed s follows. I Secto we troduce blow-up pproch to detfy the lmt profle problem. I Secto 3 we obt the crucl globl estmte 9 whch wll llow us Secto 4 to loclze the blow-up set. I Appedx A, we brefly dscuss the lmtg problem d preset Pohozev-type detty.. Locl profle I ths secto we gve complete detfcto of the lmt profle problem d ts spectrl propertes. Let U be the uque soluto see Appedx A of the problem Ü + p + U = U p, 6 0 <Ur U0 =. Proposto.. Let λ,u be solutos of wth u stsfyg 5. Let, be such tht u +. Let = u p p d U r = u r + for r I, where I =,. Assume tht The, for subsequece, we hve tht + : u = mx { r } u. 7, +, 8 λ V 9 p + d U U Cloc s +, where U s the soluto of 6. Moreover

4 4 P. Esposto et l. / J. Dfferetl Equtos = U > 0, ψ C0 [, + ] : ψ x + λ V pu p ψ x dx < 0 lrge. 0 Ω Proof. Frst, we rewrte polr coordtes: ü N u = u p λ Vru,, r u > 0,, u = u = 0. Sce s pot of locl mxmum, we hve 0 ü = u p λ V u, d hece, deoted ωv := [mx Ω V ][m Ω V ], t results λ V u p = λ V 0, λ Vr ωv. Pssg evetully to subsequece, we c ssume λ V μ, L 0, L s +, for some μ [0, ], L 0,L [0, + ]. Flly, otce tht U stsfes the equto: Ü N U = U p λ r + V r + U, r I, U 0 =, U 0 = 0, U r > 0, r I, U = 0, r I. 3 I the sequel, we wll deote by A the Lebesgue mesure of set A. st Step: For y closed bouded tervl I wth 0 I, there exsts C = C I >0: U C, I I C N. 4 Set J = I I. Sce I s bouded, 7 mples U r U 0 = for I d r J. Hece, by, 3: U r = U r U 0 r 0 Ü tr dt N [ + ωv ] mx U s + r s J mx r J U r + N [ + ωv ] I, d the: mx r J U r N [ + ωv] I for I. I tur, ths mples

5 P. Esposto et l. / J. Dfferetl Equtos U r U s r s 0 Ü s + tr s dt N [ + ωv ] mx U t + r s t J N [ + ωv ] r s r, s J, I,.e. 4 holds wth C = mx{n [ + ωv][ I +]+, U C, I I : < I }. d Step: L 0 = L =+ d U U Cloc s +. Assume tht L 0 < +. The, by 4, U s uformly bouded C, [,], for y >0. Sce L 0 < + mples L =+, we c ssume, up to subsequece d dgol process, tht U U Cloc [ L 0, + d the L 0 > 0 where: Ü + μu = U p L 0, +, 0 Ur U0 = L 0, +, U L 0 = 0 vew of 7, 3. Sce U s eve see Appedx A, UL 0 = 0 d the UL 0 = 0 becuse U 0. Hece U 0, cotrdcto. Thus L 0 =+. Smlrly, L =+. 3rd Step: μ = p+ d 0 holds. As show Appedx A, U postve mples ts eergy s opostve: 0 HU, U:= U μu + p + U p+ U 0 μ U 0 + p + U p+ 0 = p + μ. Hece μ p+.now,μ> p+ mples see Appedx A U s postve, possbly costt, perodc soluto d there s coutble fmly of fuctos φ j C0 wth mutully dsjot supports such tht, for some δ>0, t results φ j + μφ j pup φj dr δ<0. Let φ j, r = φ j r, so tht supp φ j, = + supp φ j re dsjot for dfferet j s d coted { j x + j },forsome j > 0. Moreover, f := lm + log some subsequece, by Steps we get: φj, + λ Vr pu p φ j, Ω = r N φ j, + λ Vr pu p φ j,

6 6 P. Esposto et l. / J. Dfferetl Equtos = Supp φ j r + N [ φ j + N λ V r + pu p ] φ j φ j + μ pup φj δ<0 j. Ths cotrdcts 5 d hece μ = p+. As for 0, just otce tht, by 6 we hve U + U p + pup = p U p+ < 0 see A. Appedx A d hece, by desty, there exst = U d ψ C0 [,] such tht ψ + ψ p + pup < 0. As bove, we see tht ψ r = ψ r stsfes the requremets 0. Ths eds the proof of Proposto.. 3. Globl behvor Oce the lmt profle problem 6 hs bee detfed d the locl behvor roud blowup sequece hs bee descrbed, our ext tsk s to provde globl estmtes: we wll show tht the sequece u decys expoetlly wy from blow-up pots d we wll prove tht the umber of blow-up sequeces cot exceed k, the upper boud for the Morse dex of the u s. We hve the followg globl result: Theorem 3.. Let λ,u be solutos of stsfyg 5. Up to subsequece, there exst,...,k, k k k gve 5, wth = u p 0 such tht λ V s + =,...,k, 5 p + C =,...,k, 6 + j 0 s +, j =,...,k, j, 7 j u = mx u, 8 { r } u r C p k = for some γ,c >0 d + s +. e γ r r,, N, 9

7 P. Esposto et l. / J. Dfferetl Equtos Proof. The proof s dvded to two steps. st Step: There exst k k sequeces,...,k stsfyg 5 8 such tht: [ lm lm sup + + p ] mx u r {d r } = 0, 0 where d r = m{ r : =,...,k} s the dstce fucto from {,...,k }. Frst of ll, let be pot of globl mxmum of u : u = mx r, u r. Sce 8 clerly holds for, Proposto. pples, d 9 provdes exctly 5. If 0 lredy holds for, the we tke k = d the clm s proved. If ot pssg to subsequece δ>0, + : p mx u r δ>0. { r } Now, pplcto of Proposto. gves, evetully for subsequece, p u r + = U r Ur uformly o bouded sets U soluto of 6. By the decy of U see A., there s δ > 0 such tht Ur δ for r δ. Hece, usg, we see tht j gve j j : j j d p j mx u j r δ. { δ r j j j } j Ths, jotly wth gves j p Hece, for y j: mx u j r = p { r j δ j } j mx u j r δ>δ { r j j } j p j mx u j r j. 3 { δ r j j j } j j { r j j } j : uj j = mx u j r δ p j. 4 { r j δ } j By 4 we get j := u j j p j δ p, d sce j j we see tht 6 s fulflled, s well s 7 becuse j j j j. Ths equlty d 3 mply 8: I fct u j j = mx { r [ j j δ ]δ p } j u j r.

8 8 P. Esposto et l. / J. Dfferetl Equtos r j [j δ ]δ p j r j j j [ j δ ]δ p j j j [ j δ ] j = δ j. Up to the subsequece j, thus 6 8 hold true for {, }, d, f {, } lso stsfy 0, we re fshed. Otherwse, we terte the bove rgumet: gve s sequeces,...,s, let us deote d r = m{ r : =,...,s}. If 5 8 re stsfed, but 0 s ot, we hve δ>0, + : p mx u r δ {d r } d, by ssumptos 6 8 d Proposto.: [ ] θ C, : θ, p u r + = p p U r p θ Uθ r 5 uformly o bouded sets. By A., θ Uθ r < δ for r δ. Now thgs go s bove, replcg r wth d r. Flly, the rgumet eds fter t most k terto, becuse Proposto. pples to y sequece, =,...,k, provdg, for lrge, rdl fuctos ψ C 0 Ω such tht 0 holds wth supp ψ { x + },forsome>0. By 7 we get tht ψ,...,ψk hve dsjot compct supports for y lrge d the k k. d Step: Let,...,k be s the frst step. The there re γ,c >0 such tht: u r C p k = e γ r r,, N. By 0, for >0 lrge d, t results recll tht ωv := [mx Ω V ][m Ω V ] p mx u r {d r } p + ωv d hece u p r p+ωv {d r }. O the other hd, by 5 we get p, λ Vr [ ωv ] λ V p + ωv. Hece, the followg holds true: there re >0 d such tht, f, the [ λ Vr u p r ] p + ωv > 0 fd r. 6

9 P. Esposto et l. / J. Dfferetl Equtos Now, cosder the ler opertor: L φ = φ + λ Vr u p r φ, φ C Ω. Notce tht L u = 0. Sce u > 0Ω, L stsfes the mmum prcple y dom Ω see [6]. Let γ>0 d φ r = e γ r. By 6, for lrge t results L φ = φ [ γ + N r γ r r + λ Vr u p r ] > 0 f d r, γ 8p+ωV d, γ. I ddto, by 5 we hve e γ φ r p u r r= ± = p u ± θ p U±θ > 0. The Φ := e γ p k = φ stsfes L Φ u >0 { d r > } d Φ u > 0 o { d r = } { } r =, otce tht, by 6 7 {d r > } re dsjot tervls for, d the, by mmum prcple u Φ {d r > },f s lrge d. Tht s Sce u r e γ p k = e γ r f d r d. 7 u r mx u = p e γ p Ω k = e γ r f d r d, 7 holds for y r, d. Thus, for some C e γ 9 holds true for y d the proof s ow complete. As by-product, the umber of pots of locl mxmum s cotrolled by 5: Corollry 3.. Let λ, u be solutos of stsfyg 5. Up to subsequece, u hs, for lrge, exctly k pots of locl mxmum,...,k, k k, where,...,k re gve by Theorem 3.. Proof. By 6 u p λ Vru < 0 r {d r }, for lrge d fxed d. Hece, by ll the pots of locl mxmum of u sty, for lrge, the rego d r. We re led to show tht,...,k re, for lrge, the oly pots of locl mxmum of u d r.

10 0 P. Esposto et l. / J. Dfferetl Equtos By cotrdcto, let s be pots of locl mxmum of u, wth 0 < s,for some k. Sce 0 s the oly crtcl pot of the lmt fucto U,bytheCloc covergece of U to U we get s := s 0s +. By 3 d 5 we get: Ü s = U p s λ V s U s p + > 0. The, s s strct locl mxmum d hece there s locl mmum t some t strctly betwee s d. However, s for s, t should be t := t 0s + d Ü t <0 for lrge, cotrdcto. 4. Locto of the blow-up set I cocetrto pheome, the role of the modfed potetl Mr gve 3 hs bee poted out ppers of Ambrosett, Mlchod d N [,], whe delg wth the sme equto ether N or bll/ulus N wth homogeeous Drchlet boudry codto. To show by symptotc pproch the role of Mr, we wll combe the results the prevous secto wth Pohozev-type detty see Appedx A. Let us strt wth some symptotc estmtes for u, solutos of. By Corollry 3. u hs, up to subsequece, exctly k pots of locl mxmum,...,k, wth, sy, [, ], =,...,k.letj ={j =,...,k: j }. We hve the followg: Lemm 4.. Let gr be some smooth fucto o [, ]. Let q>.fx {,...,k} d deote I δ := [ δ, + δ], where δ>0 s so smll tht I δ {,..., k }={ }. The I δ gru q = g j J j p q p U q + o 8 where o 0 s +. I prtculr, there holds: u p+ = k = p+3 p U p+ + o. 9 Proof. Let d r := m{ r : =,...,k}. Gve>0, 8, 6 d 7 mply tht, for, {d r } re mutully dsjot tervls d { d r }, d I δ { d r } { = r j }. j J By 9 we kow tht u q C q p k j= e j r qγ. Thus

11 I δ gru q = P. Esposto et l. / J. Dfferetl Equtos I δ {d r } gru q + I δ {d r } gru = q q gru + O p j J { r j } = j q p+ p j J k q g r j + j j q U e j= Iδ {d r } qγ r j + O q p { r j } k j= j { r j } e qγ r j. Up to subsequece, by 6 we c ssume tht /j θ j [ C, ] for y j =,...,k. Sce U j U Cloc for y j =,...,k, we fd, log some subsequece q p+ lm p + Iδ u q = g q p+ p θj j J { r θ j } U q + O k j= { r θ j } e qγ r θ j. Sedg to fty, we get, log the sme subsequece, q p+ lm p + Iδ u q = g q p+ p θj j J U q. Sce we foud the sme vlue log y coverget subsequece, d recllg the defto of θ j, the proof of 8 s complete. Flly, sce by 9 u 0s + uformly fr wy from {,..., k }, 8, wth q = p + d g, mples 9. The symptotc expsos Lemm 4., combed wth the Pohozev detty A.3, leds to the detfcto of, =,...,k: Theorem 4.. For y =,...,k M, where M s gve 4. Proof. Gve =,...,k, frst cosder the cse,.letiδ be s Lemm 4.. By 5, 9 λ u 0 uformly wy from the s d ellptc regulrty estmtes mply the sme for u. Thus we see, pluggg = δ, b = + δ A.3, tht: N 3 +δ u p+ + λ p + δ +δ δ r V N V u + N 3 +δ δ N r u 0

12 P. Esposto et l. / J. Dfferetl Equtos s +. By 5 d 8 we get s + here o 0s + +δ δ u p+ = j J j p+3 p U p+ + o, +δ δ u r = λ O j J j p+3 p, λ +δ δ r V N V u = V p + V N U + o. Hece, lso mkg use of the relto U p+ = p+3 4 U see A., we get j J j p+3 p 0 = N 3 p + [ 4 = N p + 3 p + [ = N + p + 3 p = p p + 3p + U p+ N p + U + V p + V p + p p + + V ] V p p + 3p + U V θ N Ṁ. U U V ] p + V U Cosder ow the cse =. Let Iδ be s bove. As before, λ u + u 0s + t + δ. Tkg =, b = + δ A.3, we see tht: N 3 +δ u p+ + λ p + +δ N 3 +δ u p+ + λ p + + N 3 +δ r V N V +δ N r u u 0 u + N 3 r V N V u +δ N r u

13 P. Esposto et l. / J. Dfferetl Equtos s +. Argug s bove, we get tht 0 p N + p + 3 p + 3p + p V V U. Hece, Ṁ 0, d = M holds. Cse = c be delt smlrly, gettg ow = M. Hece, the theorem s completely estblshed. Appedx A A.. Phse ple lyss of the lmtg equto Let U be C -soluto of the equto Ü + μu = U p U, d Ur, Ur the correspodg prmetrzed orbt the phse ple. Let Hu,v:= v + Gu, Gu := μ u + p + u p+ be the eergy fucto; t s coserved qutty: h HUr, Ur s the eergy of the orbt U, U. Sce level sets {Hu,v= h} re compct, U s globlly defed. For smplcty, we wll cosder the cse μ>0 cse μ = 0 c be delt smlr d smpler wy. Drect specto o the level sets of H gves: {Hu,v= h>0} s closed orbt eclosg the ustble equlbrum 0, 0; {u, v: u>0, Hu,v= 0} s homoclc orbt, symptotc to 0, 0; {u, v: u>0, Hu,v<0} s closed orbt eclosg the stble equlbrum μ p, 0. As cosequece, U postve mples: HU, U 0. From ow o we wll ssume U0 =, U0 = 0 otce tht U s eve, becuse t stsfes the sme Cuchy problem s Ũr:= U r. I ths cse, HUr, Ur p+ μ 0ff μ p+,sou postve mples μ p+. Cse μ> p+ : U hs fte Morse dex. From bove: U s postve perodc soluto. I cse U μ p = U0 =, the lerzed equto t U s v + p v = 0. Let, b be such tht the frst egevlue of the Drchlet problem s smller th p. Let ϕ be the correspodg postve egefucto. After settg ϕ 0 outsde, b, we see tht ϕ p ϕ < 0. Let U μ p.letu r = m Ur. By the bove dscusso, 0 <U r < μp d hece G U r < 0. If T s perod of U, I k := [ r + kt, r + k + T ], ϕ k := [U U r]χ Ik, the ϕ k + μϕ k pup ϕk = [ U p μu pu p μ U U r ] ϕ k dr. I k I k

14 4 P. Esposto et l. / J. Dfferetl Equtos But U p r μur pu p r μur U r= G Ur G Ur[Ur U r] G U r becuse G s covex o 0, +. Thus we hve ϕ k + μϕ k pup ϕk G U r T 0 [ U U r ] < 0. By desty, we c replce the ϕ k wth C0 -fuctos wth mutully dsjot supports. Cse μ = p+ : expoetl decy. Zero eergy mples U, U s homoclc to the zero equlbrum. Also, U s eve d U r > 0 > Ur r >0. We clm tht C >0: Ur Ce p+ r, r p + U = + p + U p+. A. Ths follows from the coservto of eergy: U p+ U U p+. Sce U <0o0, + d Ur 0sr +, we get tht: Ur Ur = l Ur = U p + p r p + r s r +. Hece, there exst C>0 d >0 lrge so tht Ur Ce p+ for r. I smlr wy, we c get expoetl decy t. The coservto of eergy gves expoetl decy for U s well, d by tegrto o yelds: U = p+ U U p+. Multplyg 6 by U d tegrtg o, we obt tht U = U + U p+. A. p + Tkg the dfferece of these lst two reltos, A. follows. A.. A Pohozev-type detty Lemm A.. Let u be rdl soluto of. Let <b. The u = b u b + r up+ p + λ rvu + N 3 uu + N 3 N u b r + N 3 b u p+ + λ p + b r V N V u + N 3 b N r u. A.3

15 P. Esposto et l. / J. Dfferetl Equtos Proof. Multply, wrtte polr coordtes, by r u d tegrte o [,b]: b u p λv u r u = A tegrto by prts gves b Hece, we obt: b u p λv u r u = r u p+ ü N u r u = r b r u N 3 b u. p + λ Vu b p + b u p+ + λ u = b u p+ u b + r p + λ b Vu + N 3 b u p + b u p+ + λ Multplyg by u d tegrtg o [,b], we get: b d so u p+ λv u = b b b ü N u u = uu r b V + r Vu. V + r Vu. A.4 b b + u = uu + N u b b N + r r u + u N u b b r b N r u u p+ λv u. A.5 Isertg A.5 A.4, we flly get A.3. efereces [] A. Ambrosett, A. Mlchod, W.-M. N, Sgulrly perturbed ellptc equtos wth symmetry: Exstece of solutos cocetrtg o spheres. I, Comm. Mth. Phys [] A. Ambrosett, A. Mlchod, W.-M. N, Sgulrly perturbed ellptc equtos wth symmetry: Exstece of solutos cocetrtg o spheres. II, Id Uv. Mth. J [3] M. Bdle, T. D Aprle, Cocetrto roud sphere for sgulrly perturbed Schrödger equto, Nol. Al. Ser. A [4] E.N. Dcer, Some sgulrly perturbed problems o ul d couterexmple to problem of Gds, N d Nreberg, Bull. Lodo Mth. Soc [5] N. Ghoussoub, Dulty d Perturbto Methods Crtcl Pot Theory, Cmbrdge Trcts Mth., vol. 07, Cmbrdge Uv. Press, Cmbrdge, 993. [6] D. Glbrg, N.S. Trudger, Ellptc Prtl Dfferetl Equtos of Secod Order, secod ed., Sprger-Verlg, 983.