Entire Solution of a Singular Semilinear Elliptic Problem

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1 JOURNAL OF MATEMATICAL ANALYSIS AND APPLICATIONS 200, ARTICLE NO 0218 Etie Sluti f a Sigula Semiliea Elliptic Pblem Ala V Lai ad Aihua W Shae Depatmet f Mathematics ad Statistics, Ai Fce Istitute f TechlgyENC, 2950 P Steet, Wight-Pattes AFB, Ohi Submitted by Jh Laey Received Decembe 19, 1994 The auths pve that the sigula semiliea elliptic equati u pž x u 0, 0, p x 0, x R, 3, has a uique psitive C 2 ŽR lc sluti that decays t ze ea pvided tž t dt, whee Ž t max pž x 0 x t Futheme, they shw that this cditi p is ealy ptimal 1996 Academic Pess, Ic INTRODUCTION We study the sigula semiliea elliptic equati u pž x u 0, x R, Ž 1 whee u satisfies hmgeeus Diichlet buday cditis The imptace f this pblem i scietific applicatis has bee widely ecgized Žsee 7 I paticula, i the case 1, the pblem aises i the study f buday laye equatis f the class f -Newtia fluids amed pseudplastic ude the classical cditis f a steady flw ve a semi-ifiite flat plate Žsee 1 Csideed i the ctext f patial diffeetial equatis Ž 1, the abve equati has bee the subject f much study The equati has a uique classical sluti i a buded dmai, whee pž x is a sufficietly egula fucti which is psitive 5 Thee exist etie psitive slutis with Ž 0, 1 f pž x sufficietly egula 3, 4 This is geealized t all 0 via the uppe ad lwe sluti methd 6 the methds 2 * Reseach suppted i pat by NSF Gat DMS X96 $1800 Cpyight 1996 by Academic Pess, Ic All ights f epducti i ay fm eseved 498

2 SINGULAR SEMILINEAR ELLIPTIC PROBLEM 499 I this pape we shw the existece ad uiqueess f a psitive etie sluti t Ž 1 vaishig at ifiity ude elaxed decay ad psitivity cditis the fucti pž x, taig the ealy esults by 3, 4, 6 t ame a few, as special cases I the last secti, we pit ut that u decay cditi pž x is ealy ptimal MAIN RESULTS Ž Ž We cside Eq 1 i which the egative C R fucti p satisfies cmbiatis f the fllwig hyptheses: Ž 1 Wheeve pž x 0, 0 pž x 0BŽ x,, whee BŽ x, is the ball f adius ceteed at x ; 1a p x 0 xr 2 tž t dt, whee Ž t max pž x 0 xt 3 lim Ž f sme Ž 0, 1 ; 1Ž2 3a t Ž 0 t dt Ž 4 Thee exists a psitive cstat C such that CŽx pž x x R TEOREM 1 yptheses Ž 1a ad Ž 2 ae sufficiet f Eq Ž 1 t 2 hae a uique psitie glbal sluti u x C ŽR 2 lc aishig at ifiity Futheme, if als satisfies Ž 3, the už x decays at the ate f at least x Ž2Ž1 ea ifiity, whee 0 1 TEOREM 2 Ude the same cditis as gie i Theem 1 except that Ž 1a is eplaced by Ž 1, Eq Ž 1 has a uique psitie sluti haig the same decay ppeties ea ifiity as the sluti gie i Theem 1 I de t pve the existece f a sluti t Ž 1, we eed t emply a cespdig esult by Laze ad McKea 5 f buded dmais We state it hee as a lemma LEMMA 1 Let R, 1, be a buded dmai with smth Ž 2 buday f class C,01 If p C Ž, pž x 0 f all 2 x ad 0, the thee exists a uique fucti u C C such that už x 0 f all x ad u is a sluti f Ž 1 aishig We w pve Theem 1 Pf Fist we pve the uiqueess Suppse u ad ae slutis f Ž 1 bth vaishig at ifiity We eed ly shw that u f the a simila agumet ca be made t pduce u fcig u Suppse u 0 at sme pit Sice u 0 as x, we w that

3 500 LAIR AND SAKER max Ž u exists ad is psitive At that pit, we have u, 0Ž R u pž xžu 0, a ctadicti ece u T pve existece, let u be the uique sluti t the equati u pž x u 0 f x, Ž 2 which vaishes the sphee x This is justified by Lemma 1 Futheme, we defie u 0 f x Usig the maximum piciple agumet as de abve f the uiqueess, it is easy t shw that u u R We w pve thee exists a psitive fucti C 2 ŽR 1 f which u R T d s, we see a psitive adially symmetic sluti w f w Ž 1 Ž Ž 3 Ž 1Ž1 x R, f which lim w 0 The w will have the equied ppeties Ideed, usig Ž 3, it is clea that satisfies which yields 1 Ž 1 Ž 1 2 Ž 1 Ž, Ž I a mae simila t u uiqueess pf abve, it is agai a staight- fwad agumet t shw that u f x ad, hece, f all R T btai w, we itegate Ž 3 Sice w is adial, Ž 3 becmes Ž Itegatig 4 yields 1 w w Ž 1 Ž 4 wž K Ž Ž d d, whee the cstat K must be chse t esue that w 0 ad lim w 0 We ca chse K d d, 5

4 SINGULAR SEMILINEAR ELLIPTIC PROBLEM 501 pvided the idicated itegal is fiite We w pve that it is, i fact, fiite Itegati by pats pduces 1 1 Ž d d 1 d 2 1 Ž 2 Ž d d d Ž 2 Ž d Ž d Ž 6 Ž Nw, usig L pital s ule, we evaluate the limit f the ight side f 6 as We have 2 1 lim Ž d Ž d lim 0 1 Ž d 2 0 Ž d 2 lim Ž d Ž d Ž 2 Ž Thus 3 has a psitive C R sluti that appaches ze as x Nw we have a buded iceasig sequece u u u 1 1 f all x R, whee u is the sluti f Ž 2 that vaishes x, ad 2 C vaishig at ifiity Thus thee exists a fucti, say už x, such that u u x pitwise i R Clealy u 2 We claim that u C Ž R ad thus is a classical sluti f Ž 1 lc The pf is me less stadad Žsee, f example, 5 F cmpleteess, we pvide the utlie hee Let x R, 0, be abitay Let be a C fucti that is equal t 1BŽ x, 2 ad ze utside BŽ x, We have Ž u 2up, 1, whee p u u F fixed ad x, we ca chse N s that B x, B 0, N The u p x u N ece u N p x u p x u N Tgethe with u, we have that the L -m f p is buded idepedet f, N Thus f N, we ca wite u u B u q, 7

5 502 LAIR AND SAKER whee B 2u, q u pu 2u Clealy the L - ms f B ad q ae buded idepedet f Itegatig Ž 7 ve BŽ x, yields 2 Ž BŽ x, BŽ x, u dx B u q dx c B Ž u dxc 1 2 BŽ x, ž / BŽ x, c u dx c, whee c 1, c 1, c2 ae sme cstats idepedet f, N Thus we get 2 2 u c 2c It the fllws that the L 2 ŽBŽ x, 1 2 -m f Ž u is buded idepedet f, N ece the L 2 ŽBŽ x, 2 - m f u is buded idepedet f, N Lettig 1 be a C fucti that is equal t 1 BŽ x, 4 ad ze utside BŽ x, 2 ad, pceedig i the same lie f agumet Žsee 5, we cclude that 2 uc It the fllws immediately that už x is a sluti f Ž 1 lc T cmplete the pf we cside w, whee 0 is t be detemied Lettig ad successively applyig L pital s ule yields 2 lim w lim c Ž, whee c 1Ž 2, 2 ece the assumpti that lim Ž is sufficiet f w t decay at the ate f 2 ea ifiity, whee 0 1 Thus the sluti už x has a decay ate ea ifiity f at least that f the uppe Ž2 Ž1 bud x, which is This cmpletes the pf Rema Ou decay cditi Ž 3 pž x is i geeal less estictive tha the cmmly used cditi Ž 3a Žsee 3, 4, 6, f example I paticula, suppse the limit i Ž 3 is psitive The thee is a psitive cstat c f which c f lage, which implies that Ž 3a fails t hld f ay 0 O the the had, if Ž 3a hlds, u uique sluti is pecisely that f the abve-metied esults ad, hece, has the same decay ate It may als be ted that u theem des t equie hypthesis 4, which is athe cmmly used cditi We w pceed with Theem 2 Pf We cside u p Ž x u 0 Ž 8

6 SINGULAR SEMILINEAR ELLIPTIC PROBLEM 503 i R, whee p Ž x pž x Žx, 1, 2,, is ay smth fucti f which Ž t 0, f t 0 ad satisfies Ž 2 We have p 0xR By Theem 1, Ž 8 has a uique psitive sluti u 0 as x Clealy ž / Ž x u1 pž x u10 The maximum piciple the implies that u u deceasig sequece 1 u u u u Thus we have a Let už x be the pitwise limit fucti f the sequece u 4 1 We have 2 uc ad u x 0 xr 2 We w pve that u C ŽR lc ad is csequetly a sluti f Eq Ž 1 T shw the smthess f the limitig fucti už x, we attempt t fllw the same agumet as i Theem 1 Nte that w u sluti sequece u 4 1 is mte deceasig ad thus des t have a uifm psitive lwe bud lie the un i Theem 1 weve, the same agumet will g thugh, pvided that a psitive lwe bud is assumed f ay ball i R T achieve this, we eed t cside ly tw cases cceig the eighbhd f a abitaily chse pit x : Ž i pž x 0; Ž ii pž x 0 I what fllws we shw that i eithe case we ca fid a ball ceteed at x such that the sequece u 4 is uifmly buded Suppse pž x 0 By Ž 1, thee exists a ball BŽ x, such that pž x 0BŽ x, Let x be ay pit BŽ x, ad BŽ x, be a ball ceteed at x f adius that des t ctai ay f the zes f pž x By Lemma 1, Ž 1 has a uique psitive sluti, which we call u, i BŽ x, vaishig the buday We have u pž x xu 0 That is, u is a lwe sluti f Ž 8 Applicati f the maximum piciple yields that u u f all Wite mi BŽx, 2 u 0 Sice is buded, it ca be cveed by fiitely may such balls Let be the miimum f all such The we have u f all alg the buday We claim that this is i fact tue f all x If t, say Ž u 0 at sme pit i f sme The Ž u must attai a psitive maximum at a pit i, whee Ž u 0 O the the had, u 0 p x x u0, a ctadicti Thus we have a uifm lwe bud f u 4 1 i Suppse pž x 0 Sice pž x is ctiuus, we ca fid a ball f sme adius, say 0, ceteed at x such that pž x 0 xbž x, By Lemma 1, Ž 1 has a uique psitive sluti, say u, i BŽ x, that vaishes BŽ x, Clealy u is a lwe sluti f Ž 8 The maximum

7 504 LAIR AND SAKER piciple agai implies that u u f all We ca assume that u u i BŽ x, 2 f sme Sice x was abitaily chse, we have shw that f ay x R, thee exists a ball f sme psitive adius ceteed at this pit whee the sequece u 4 1 has a uifm psitive lwe bud By the aalysis give abve, we have that už x 2 C ŽR ad už x is a psitive sluti f Ž 1 lc T pve uiqueess, we assume that u ad ae tw psitive slutis f Ž 1 that vaish at ifiity As i Theem 1, we eed ly shw that u xr Let z Ž 1 1, whee 0 ad x Suppse thee is a pit at which u z 0 Sice u z 0asx,we w that max Ž u z R exists ad is psitive At that pit we have Ž u z, 0 u z p x u Ž Ž 1 Ž 10, a ctadicti ece u z That is, u Ž1 1 f all 0, which yields u 2 IS NEARLY OPTIMAL Theem 1 shws that Ž 2 is sufficiet f the existece f the uique sluti f Eq Ž 1 The fllwig theem shws that cditi Ž 2 is ealy ecessay TEOREM 3 R ad satisfies Ž The Eq 1 Pf Suppse p is a psitie adial fucti that is ctiuus 0 tpž t dt has psitie adial sluti that decays t ze ea ifiity Suppse Ž 1 has such a sluti, u Ž The 1 u Ž u Ž pž u Ž Ž Itegatig this equati as we did 4 pduces 1 1 už už 0 pž u Ž d d Ž 9 Sice u is psitive, we get 1 1 pž u Ž d duž 0 f all 0 Ž 10

8 SINGULAR SEMILINEAR ELLIPTIC PROBLEM 505 Thus the left side f this equati has a fiite limit as weve, Eq Ž 9 yields u Ž u0 Ž f all 0 Usig this i iequality Ž 10 pduces 1 1 u Ž 0 pž d duž 0 f all 0 weve, we ca use itegati by pats ad L pital s ule Žas we did i pvig that the itegal i Ž 6 is fiite t ewite this as ctadictig the hypthesis lim 1 tpž t dt u Ž 0, 0 REFERENCES 1 A J Callegai ad A Nachma, A liea sigula buday value pblem i the they f pseudplastic fluids, SIAM J Appl Math 30 Ž 1980, R Dalmass, Slutis d equatis elliptiques semi-lieaies siguliees, A Mat Pua Appl 153 Ž 1988, A Edels, Etie slutis f sigula elliptic equatis, J Math Aal Appl 139 Ž 1989, T Kusa ad C A Swas, Etie psitive slutis f sigula semiliea elliptic equatis, Japa J Math 11 Ž 1985, A C Laze ad P J McKea, O a sigula liea elliptic buday value pblem, Pc Ame Math Sc 111 Ž 1991, A W Shae, O sigula semiliea elliptic equatis, J Math Aal Appl 173 Ž 1993, J S Wg, O the geealized Emde-Fwle equati, SIAM Re 17 Ž 1975,