SINGLE-POINT CONDENSATION AND LEAST-ENERGY SOLUTIONS

Transcription

1 SINGLE-POINT CONDENSATION AND LEAST-ENERGY SOLUTIONS XIAOFENG REN AND JUNCHENG WEI Abstract. We prove a cojecture raised i our earlier paper which says that the least-eergy solutios to a two dimesioal semiliear problem exhibit sigle-poit codesatio pheomea as the oliear expoet gets large. Our method is based o a sharp form of a well-kow borderlie case of the Sobolev embeddig theory. With the help of this embeddig, we ca use Moser iteratio scheme to carefully estimate the upper boud of the solutios. We ca also determie the locatio of the codesatio poits. 1. Itroductio I this work, we shall cofirm the cojecture raised i our earlier paper [8]. We study { u + u p = 0 i (1.1) u = 0 o where p > 1 ad is a smooth bouded domai i R 2. We cosider the so-called least-eergy solutios of (1.1) obtaied by miimizig fuctioal (1.2) J p : {v W 1,2 0 () : v p+1 = 1} R where J p (v) = v 2. Stadard variatioal argumets show that the miimum of J p is achieved by a positive fuctio i {v W 1,2 0 () : v p+1 = 1} ad a positive scalar multiple of the fuctio solves (1.1). Let us deote such a leasteergy solutio by u p. I [8] we proved Theorem 1.1. There exist C 1, C 2, idepedet of p, such that for p large eough. 0 < C 1 < u p L () < C 2 < To uderstad the shape of u p for large p, let (1.3) v p = u p. up p 1991 Mathematics Subject Classificatio. Primary 35B40, 35A08, 35A15; Secodary 34A34. 1

2 2 XIAOFENG REN AND JUNCHENG WEI For a sequece v p of v p we defie the blow-up set S of v p to be the subset of such that x S if there exist a subsequece, still deoted by v p, ad a sequece x i with (1.4) v p (x ) ad x x. We use #S to deote the cardiality of S. We also defie a peak poit P for u p to be a poit i such that u p does t vaish i L orm i ay small eighborhood of P as p. It turs out later that the set of the peak poits of u p is same as the set of blow-up poit of v p. About the blow-up set of v p we proved the followig theorem i [8]. Theorem 1.2. (1) For ay smooth bouded domai ad a sequece v p of v p with p there exists a subsequece of v p such that the blow-up set S of that subsequece is cotaied i ad has the property 1 #S 2; i other words the subsequece must blow up ad it blows up at most at two poits i. (2) If meets coditio (T), the the above subsequece must blow up at oe poit i. Here coditio (T) is a geometric coditio o. A domai is said to satisfy coditio (T) if (A) is star-shaped with respect to some poit y, i.e. (x y,(x)) > 0 for all x ad (B) ds (x y,(x)) < 2πe where (x) deote the outer ormal of at x. About the locatio of the blow-up poits, we proved i [8] Theorem 1.3. Let be a smooth bouded domai with coditio (T). The for ay sequece v p of v p with p there exists a subsequece of v p, still deoted by v p, such that (1) f := up p up = ( p u p p ) p 1 v p p δ(x 0 ) i the sese of distributio where δ(x 0 ) is the δ fuctio at poit x 0 ad {x 0 } = S. (2) v p G(x,x 0 ) i W 1,q () weakly for ay 1 < q < 2 where G is the Gree s fuctio of o ; furthermore for ay compact subset K of \{x 0 } we have v p G(x,x 0 ) i C 2,α (K). (3) x 0 is a critical poit of fuctio φ where φ(x) = g(x,x) ad g(x,y) = G(x,y) + 1 log x y 2π is the regular part of the Gree s fuctio G. At the ed of [8], we cojectured that coditio (T) is uecessary for Theorem 1.2 (2) ad Theorem 1.3, i.e., least-eergy solutios must develop sigle-peaks regardless of domais. I this paper we shall prove

3 SINGLE-POINT CONDENSATION AND LEAST-ENERGY SOLUTIONS 3 Theorem limif u p L () limsup u p L () e. Theorem 1.5. For ay smooth bouded domai ad a sequece v p of v p with p there exists a subsequece of v p such that the blow-up set S of that subsequece cotais oly oe poit i ; i other words the subsequece must blow up ad it blows up at oe poit. Theorem 1.6. The coclusios i Theorem 1.3 are true without coditio (T). From the above results, we see that whe p gets large, the least-eergy solutios u p look more ad more like a sigle spike. If we cosider a covex domai, the, usig a result of L. Caffarelli ad A. Friedma (Theorem 3.1 of [3]) which implies that φ is strictly covex, φ has strictly positive Hessia ad the oly critical poit of φ is the global miimum, we have Corollary 1.7. For covex domai the coclusios of Theorem 1.5 ad Theorem 1.6 are true for the whole families {u p }, {v p } ad {f p } as p, ad x 0 is the uique critical poit of φ. Remark 1.8. It was show by C.-S. Li [7] that for covex domai the solutio u p of (1.1) miimizig (1.2) is uique. Some techiques i this paper have bee exteded to N dimesioal case where the Laplacia operator is replaced by the N-Laplacia operator. See [9] for the detail. The mixed boudary versio of the two dimesioal problem is also cosidered by the authors i [10]. I this paper we shall prove Theorem 1.4 i sectio 2 ad Theorem 1.5 i sectio 3. We omit the proof of Theorem 1.6 as is idetical with the proof of Theorem 1.3 [8]. (2.1) Defie 2. Proof of Theorem 1.4 c p := if{[ u 2 ] 1/2 : u W 1,2 0 (), u p+1 = 1}. We collect some results from [8] that will lead to the asymptotic behavior of c p. Lemma 2.1. For every t 2 there is D t such that u L t D t t 1/2 u L 2 for all u W 1,2 0 () where is a bouded domai i R 2 ; furthermore Lemma 2.2. Corollary 2.3. lim D t = (8πe) 1/2. t lim c p p 1/2 = (8πe)1/2. lim p u p+1 p = 8πe, lim p u p 2 = 8πe.

4 4 XIAOFENG REN AND JUNCHENG WEI (2.2) We ow defie a importat quatity where (2.3) pν p L 0 = lim 2 e ν p = Notice that the quatity L 0 defied here is differet from the oe i [8]. I deed this differece will lead to Theorem 1.5. We have the followig estimate for L 0. Corollary 2.4. For ay smooth bouded domai of R 2, L 0 4π e. The proofs of these facts ca be foud i sectio 2 of [8]. Now let us start the proof of Theorem 1.4. A uiform lower boud actually exists for ay positive solutios to (1.1). Let λ be the first eigevalue of ad ϕ be a correspodig positive eigefuctio. The if u is ay solutio to (1.2) with the expoet equal to p, we have 0 = u ϕ ϕ u = λ uϕ + u p ϕ. That is (up λu)ϕ = 0. Hece u L () λ 1/(p 1) 1 as p which yields a lower boud for large p. To get a upper boud for {u p }, we use a iteratio argumet. Fix positive α ad ǫ that will be chose small later. Lettig ν = (1 + α)(p + 1), from Lemma 2.1, we have [ u ν p] 1/ν (8πe) 1/2 E (1+α)(p+1) ν 1/2 u p L2 () where lim E (1+α)(p+1) = 1. But from Corollary 2.3, we kow u p p. lim p u p 2 = 8πe. Hece there is P 0 such that for all p > P 0, (2.4) ( u ν p) 2/ν 1 + α + ǫ, or u ν p (1 + α + ǫ) ν/2. We heceforth oly cosider p > P 0. Multiplyig both sides of (1.1) by u 2s 1 p, we get, after itegratig by parts, 2s 1 (2.5) s 2 u s p 2 = up p 1+2s. Usig Lemma 2.1 agai, we deduce [ u νs p ] 1/ν D νs ν 1/2 u s p L2 (); [ u νs p ] 2/ν s 2 C 0 ν 2s 1 u p 1+2s p C 1 νs up p 1+2s

5 SINGLE-POINT CONDENSATION AND LEAST-ENERGY SOLUTIONS 5 where D νs is defied i Lemma 2.1 ad C 0 ad C 1 are costats idepedet of p > P 0. Hece we have (2.6) [ u νs p ] 2/ν C 1 νs up p 1+2s. (2.7) We ow defie two sequeces {s j } ad {M j } by p 1 + 2s 0 = ν, p 1 + 2s j+1 = νs j ; M 0 = (1 + α + ǫ) ν/2, M j+1 = [C 1 νs j M j ] ν/2 where C 1 is the costat i (2.6). From (2.4) ad (2.6), we have (2.8) u νsj 1 p M j. (2.9) Next we claim M j exp[m(α,p,ǫ)νs j 1 ] where m(α,p,ǫ) is a costat depedig o α,p ad lim m(α,p,ǫ) = 1 + α log(1 + α + ǫ). 2α I fact, we ca write dow {s j } explicitly. (2.10) Put Hece Therefore Now we defie {τ j } by s j = 1 ν 2 {(ν 2 )j+1 (ν p 1) + p 1}. σ j = ν 2 log(c 1νs j ), µ j = log M j. µ j+1 = µ j 2 + σ j. σ j = ν 2 {log[ C 1ν ν 2 ] + log[(ν 2 )j+1 (ν p 1) + p 1]} [ν log 2C 1 ν](j + 1). (2.11) τ 0 = µ 0, τ j+1 = 1 2 ντ j + (ν log 2C 1 ν)(j + 1). Clearly µ j τ j. Moreover we have τ j = ( ν 2 )j [µ 0 + 2ν log( ν 2C 1 ν) (ν 2) 2 ] 2 ν 2 [ν log( 2C 1 ν)(j + ν ν 2 )] µ 0 + 2ν log( 2C 1 ν) ν p 1 µ 0 + 2ν log( 2C 1 ν) (ν 2) 1 (ν p 1) ν (ν 1) 2 ν (ν 2) 2 s j 1 ν 2 νs j 1 := m(α,p,ǫ)νs j 1 ν

6 6 XIAOFENG REN AND JUNCHENG WEI where lim m(α,p,ǫ) = 1 + α log(1 + α + ǫ). 2α Remember ν = (1 + α)(p + 1). This proves (2.9). Therefore we get u p L νs j 1() exp[m(α,p,ǫ)]. Sedig j ad the p, we deduce Sedig ǫ,α 0, we deduce lim sup u p L (1 + α + ǫ) 1+α 2α. lim sup u p L e. We iclude a cosequece of Theorem 1.4 here which will be used later. Corollary 2.5. There exist C 1 ad C 2 such that C 1 p u p p C 2 p Proof. The first iequality follows from Theorem 1.4 ad the first limit of Corollary 2.3; the secod iequality follows from the first limit of Corollary 2.3 through a iterpolatio argumet. 3. Proof of Theorem 1.5 The proof is similar to the proof of Theorem 1.2 i [8]. The major differece appears whe we reach (3.7). We first state a boudary estimate lemma. The proof of the lemma is stadard. Oe combies the movig plae method i [5] with a Kelvi trasform. We refer to [4] ad [5] for details. Lemma 3.1. Let u be a solutio of { u + f(u) = 0 i R 2 u = 0, u > 0 i where is bouded, smooth ad f is a smooth fuctio. The there exist a eighborhood ω of ad a costat C both depedig o the geometry of oly such that u L (ω) C u L1 (). Applyig this lemma to v p = u p up p we have the followig uiform boudary estimate. I particular, it implies that {v p } does t blow up o the boudary of ; hece by Corollary 2.5 {u p } has o peak o the boudary of. Lemma 3.2. There exist a costat C ad a eighborhood ω of both depedig o the geometry of oly such that v p C i ω.

7 SINGLE-POINT CONDENSATION AND LEAST-ENERGY SOLUTIONS 7 Proof. Because v p L 1 = 1, combiig the elliptic L p estimate with the duality argumet (see [2] for details), we have that v p is bouded uiformly i W 1,q () for 1 q < 2; hece v p is uiformly bouded i L 1. The usig Lemma 4.1, we obtai the desired result. Lemma 3.3. Let S be the blow-up set defied i (1.5) of a subsequece v of v p. The S is oempty ad there is a small eighborhood ω of which depeds o the geometry of oly such that S ω =. Proof. The secod assertio follows immediately from Lemma 3.2. For the first oe, observe max v (x) C ν p from Theorem 1.4, Corollary 2.5 ad the defiitio of v, (1.3). We quote a iterestig L 1 estimate from [1]. Lemma 3.4. Let u be a solutio of { u = f i u = 0 where is a smooth bouded domai i R 2. We have for 0 < ǫ < 4π (4π ǫ) u(x) exp[ ]dx 4π2 f L 1 ǫ (diam)2. Now recall pν p L 0 = lim 2 e 4π e where ν p = up p. We deote ay sequece u p of u p by u. Because u has property u p up we ca extract a subsequece of u, still deoted by u, so that there is a positive bouded measure µ i M(), the set of all real bouded Borel measures o, such that (3.1) f ϕ ϕdµ for all ϕ C 0 () where = 1 f = νp p 1 v p ad v = u. ν p For ay δ > 0, we call x 0 is a δ-regular poit if there is a fuctio ϕ i C 0 (), 0 ϕ 1, with ϕ = 1 i a eighborhood of x 0 such that (3.2) ϕdµ < 4π L 0 + 2δ.

8 8 XIAOFENG REN AND JUNCHENG WEI We defie Σ(δ) = {x 0 : x 0 is ot a δ-regular poit}. Clearly (3.3) µ(x 0 ) 4π L 0 + 2δ for all x 0 Σ(δ). Our ext lemma plays a cetral role i the proof of Theorem 1.5. It says that smalless of µ at a poit x 0 implies boudedess of v ear x 0. Lemma 3.5. If x 0 is a δ-regular poit, the {v } is bouded i L (B R0 (x 0 )) for some R 0 > 0. Proof. Let x 0 be a δ-regular poit. From the defiitio of δ-regular poits, there exists R 1 > 0 such that f < 4π L 0 + δ. B R1 (x 0) Split v ito two parts v = v 1 + v 2 where v 1 is the solutio of { v1 + f = 0 i B R1 (x 0 ) (3.4) v 1 BR1 (x 0) = 0 ad v 2 solves (3.5) { v2 = 0 i B R1 (x 0 ) v 2 BR1 (x 0) = v BR1 (x 0). From the maximum priciple, v 1, v 2 > 0. By the mea value theorem for harmoic fuctios, we have v 2 L (B R1 /2) C v 2 L1 (B R1 ) C v L1 () C where the last iequality follows as i the proof of Lemma 4.2. So we eed oly to cosider v 1. We first claim that whe is large eough (3.6) for all x. Observe that f (x) exp(l 0 + δ/2)v (x) lim sup u p L (ω) e, lim ( u p ) 1/p = 1 from Theorem 1.4 ad Corollary 2.5. Therefore Let (3.7) The lim sup u p L (ω) ( u p) 1/p e < e. α = u p L (ω) ( u p ) 1/p. lim sup α < e.

9 SINGLE-POINT CONDENSATION AND LEAST-ENERGY SOLUTIONS 9 Cosider fuctio log x/x which is icreasig if x < e. Sice for every x fixig small ǫ, we have for large, Therefore log up(x) ν 1/p u p (x) ν 1/p u (x) α, ν 1/p log α α 1 2 e + ǫ. f (x) exp[ p u p ( 1 νp 1/p ǫ)] = exp[( e 2 e + ǫ)p ν 1 1 p p v ] exp[( 1 2 e + 2ǫ)(limsup p ν p )v ] = exp[(l el 0 ǫ)v ] for large. If we choose ǫ small eough, we have f (x) exp[(l 0 + δ 2 )v (x)] for large. Next we claim that {f } is uiformly bouded i L 1+δ0 (B R1 /2) for δ 0 sufficietly small. Because {v 2 } is uiformly bouded i B R1/2(x 0 ), we see from the previous claim B R1 /2 f 1+δ0 B R1 /2 C exp[(1 + δ 0 )(L δ)v 1 ] C B R1 /2 exp[(1 + δ 0 )(L δ)v ] B R1 /2 exp 4π(1 + δ 0) L0+0.5δ L v 0+δ 1 B R1 /2(x f C 0) with the aid of Lemma 3.3 if we choose δ 0 sufficietly small. So we have proved the claim. Now take B R1/4(x 0 ). We coclude from the weak Haack iequality ( Theorem 8.17, [6] ), v L (B R1 /4(x 0)) C[ v L2 (B R1 /2(x 0)) + f L 1+δ 0(BR1 /2(x 0))] C. Here the boudedess of {v } i L 2 (B R1/2(x 0 )) follows agai from Lemma 3.3. We ow start to prove Theorem 1.5. We first claim S = Σ(δ) for ay δ > 0. Clearly S Σ. I fact, let x 0 Σ; the x 0 is a δ-regular poit. Hece by Lemma 3.5, {v } is bouded i L (B R (x 0 )) for some R, i.e. x 0 S. Coversely suppose x 0 Σ. The we have for every R > 0, passig to a subsequece of {v } if ecessary, (3.8) lim v L (B R(x 0)) =. Otherwise there would be some R 0 > 0 such that v L (B R0 (x 0)) < C for some C idepedet of. The f = ν p 1 v p ( M ) p 1 C p 0 p

10 10 XIAOFENG REN AND JUNCHENG WEI uiformly as o B R0(x 0). The f = B R0 (x 0) B R0 (x 0) ν p 1 v p ǫ 0 < 4π L 0 + 2δ which implies that x 0 is a δ-regular poit, i.e. x 0 Σ. This cotradictio implies (3.8); hece by defiitio (1.4) of S, x 0 S. This completes the proof of our claim. Next we go back to measure µ defied i (3.1). Clearly by (3.3) Hece combiig this with Lemma µ() 4π L 0 + 2δ #Σ(δ) = 4π L 0 + 2δ #S. 1 #S L 0 + 2δ. 4π Applyig Lemma 2.4, we fially coclude that, by choosig δ small, #S = 1. Remark 3.6. The fact that the peak set of {u } is icluded i the blow-up set S of {v } follows easily from the fact that ν 0 as. Because the blow-up set cotais oly oe poit ad the peak set is o-empty from Theorem 1.1, we coclude that the peak set is idetical with the blow-up set. Refereces 1. H. Brezis ad F. Merle, Uiform estimate ad blow-up behavior for solutios of u = V (x)e u i two dimesios, Comm. Part. Diff. Equa. vol. 16, o. 8&9, 1991, H. Brezis ad W. Strauss, Semiliear secod-order elliptic equatios i L 1, J. Math. Soc. Japa, 25 (1973), L. Caffarelli ad A. Friedma, Covexity of solutios of semiliear elliptic equatios, Duke Math. J., 52 (1985), D. G. DeFigueiredo, P. L. Lios ad R. D. Nussbaum, A priori estimates ad existece of positive solutios of semiliear elliptic equatios, J. Math. Pore. Appl. vol. 61, 1982, pp B. Gidas, W.-M. Ni, ad L. Nireberg, Symmetry ad related properties via the maximum priciple, Comm. Math. Phys.(68) o.3, 1979, D. Gilbarg ad S. N. Trudiger, Elliptic Partial Differetial Equatios of Secod Order, Secod Editio, Spriger-Verlag, Berli-Heidelberg-New York-Tokyo, C.-S. Li, Uiqueess of solutios miimizig the fuctioal u 2 /( up+1 ) 2/p+1 i R 2, preprit. 8. X. Re ad J. Wei, O a two dimesioal elliptic problem with large expoet i oliearity, Tra. A.M.S., Vol 343, No. 2, Jue 1994, X. Re ad J. Wei, Coutig peaks of solutios to some quasiliear elliptic equatios with large expoets, J. Diff. Equa., to appear. 10. X. Re ad J. Wei, Asymptotic behavior of eergy solutios to a two dimesioal semiliear problem with mixed boudary coditio, Noliear Aalysis: Theory, Methods, ad Applicatios, to appear. School of Mathematics, Uiversity of Miesota, Mieapolis, MN Curret address: Departmet of Mathematics, Brigham Youg Uiversity, Provo, Utah address: re@math.um.edu re@math.byu.edu School of Mathematics, Uiversity of Miesota, Mieapolis MN Curret address: SISSA, Via Beirut 2-4, Trieste, Italy address: wei@math.um.edu wei@ibm550.sissa.it