Asymptotic non-degeneracy of the solution to the Liouville Gel fand problem in two dimensions

Transcription

1 Comment. Math. Helv ), Commentarii Mathematii Helvetii Swiss Mathematial Soiety Asymptoti non-degeneray of the solution to the Liouville Gel fand problem in two dimensions Tomohio Sato and Taashi Suzui Abstrat. In this paper we study the asymptoti non-degeneray of the solution to the Liouville Gel fand problem v = λv x)e v in, v = 0 on, where R 2 is a smooth bounded domain, Vx) is a positive-valued C ) funtion, and λ>0is a onstant. Mathematis Subjet Classifiation 2000). 35J60. Keywords. Liouville equation, blow-up analysis, Green s funtion.. Introdution The purpose of the present paper is to study the asymptoti non-degeneray of the solution to the Liouville Gel fand problem v = λv x)e v in, v = 0 on, ) where R 2 is a bounded domain with smooth boundary, V = Vx) > 0is a C funtion defined on, and λ>0 is a onstant. We shall extend a result of Gladiali Grossi [5], whih is valid for the homogeneous ase of Vx), based on the following fat []. v = λe v in, v = 0 on 2) Theorem.. If λ,v ) =, 2,...) is a solution sequene for 2) satisfying λ 0, then we have a subsequene denoted by the same symbol) suh that = λ e v πm for some m = 0,, 2,...,+. Aording to this value of m, we have the following. ) If m = 0, then it holds that v 0.

2 354 T. Sato and T. Suzui CMH 2) If 0 <m<+, then the blowup set of v =, 2,...), defined by S = { x 0 there exists x x 0 suh that v x ) + }, is omposed of m-interior points, and v π x 0 S G,x 0) loally uniformly in \ S, where G = Gx,y) denotes the Green s funtion of in with = 0. We have v x)dx x 0 S πδ x 0 dx) in the sense of measure on. Furthermore, it holds that 2 Rx 0) + x Gx 0,x 0 ) = 0 3) x 0 S\{x 0} for eah x 0 S, where Rx) = [ Gx,y) + 2π log x y ] y=x funtion. 3) If m =+, then v + loally uniformly in. is the Robin Gladiali and Grossi [5] are onerned with the ase m =, and study the nondegeneray of λ,v ) for large. From the above theorem, we have S = {x 0 } if m = and this x 0 is a ritial point of the Robin funtion. What they obtained is the following theorem, motivated by the study of the detailed bifuration diagram for 2). Theorem.2. If m = holds in the previous theorem and x 0 S is a non-degenerate ritial point of Rx), then the solution λ,v ) is non-degenerate for large, that is, the linearized operator λ e v in with = 0 is invertible. Theorem., on the other hand, has an extension to ). Although the results of Ma Wei [7] are presented in the mean field formulation, v = λv x)ev Vx)ev in, v = 0 on, it is easy to translate them into the following theorem on ). See also [9].) Theorem.3. All the results stated in Theorem. ontinue to hold for ), provided that and 3) are replaed by = λ Vx)e v and respetively. 2 Rx 0) + x 0 S\{x 0} x Gx 0,x 0 ) + π log Vx 0) = 0, 4)

3 Vol ) Non-degeneray of the solution to the Liouville Gel fand problem 355 In the ase of m = again, equation 4) means that x 0 is a ritial point of Rx) + 4π log Vx). From this point of view, it is natural to extend Theorem.2 as follows. Theorem.4. In Theorem.3, ifm =, Vx) is C 2 near x 0 S, and x 0 is a non-degenerate ritial point of Rx) + 4π log Vx), then the solution λ,v ) is non-degenerate for large, that is, the linearized operator λ Vx)e v in with = 0 is invertible. To prove the above theorem, we follow the argument of [5], namely, the existene of w = w x) =, 2,...) satisfying w = λ Vx)e v w in, w = 0 on, 5) w =, implies a ontradition. The next setion is devoted to examine the validity of the blowup analysis [5] to ), originally developed for 2). In the latter ase, w = v i =, 2) solves the linearized equation w = λ e v w in exept for the boundary ondition). This struture is useful to prove Theorem.2, but obviously does not hold in ). In the final setion, we omplete the proof of Theorem.4, providing new arguments to ompensate this obstrution. 2. Preliminaries In this setion, we onfirm that several assertions for 2) presented in [5] are still valid for ). Heneforth, λ,v ) =, 2,...) is a solution sequene for ) satisfying = λ Vx)e v π, λ 0, 6) and x denotes a maximum point of v : v x ) = v. Then we have x x 0 with S = {x 0 }, and this blowup point x 0 is a ritial point of Rx) + 4π log Vx). The first lemma orresponds to Theorem 6 of [5]. Lemma 2.. There is a onstant C > 0 suh that v e v x ) x) log { + λ Vx )e v x ) x x 2} 2 C 7) for any x and =, 2,...

4 356 T. Sato and T. Suzui CMH Proof. Putting u = v + log λ, we obtain u = Vx)e u in, u = log λ on, e u = O). Passing to a subsequene, we shall show that u x ) + holds. Then, Theorem 0.3 of Y. Y. Li [6] guarantees the existene of C > 0 suh that u e u x ) x) log { + Vx )e u x ) x x 2} 2 C for any x and =, 2,..., or, equivalently, 7). In fat, if u x ) + does not our, then we may assume either u x ) or u x ) R. In the first alternative, we have λ e v 0, whih is impossible by 6), beause there are a,b > 0 suh that a Vx) b x ). In the seond alternative, on the other hand, the sequene {u } is loally uniformly bounded in by Brezis Merle [], while Theorem.3 guarantees u = v +log λ loally uniformly in \ {x 0 }. Again, we have a ontradition, and the proof is omplete. Now we define δ > 0by The next lemma orresponds to Lemma 5 of [5]. Lemma 2.2. It holds that δ 0. δ 2 λ e v x ) =. ) Proof. Inequality 7) reads { v x) v x ) + log + Vx } 2 ) δ 2 x x 2 C for x and =, 2,..., and we have v πg,x 0 ) loally uniformly in \{x 0 }, Vx ) Vx 0 ), and v x ) +. These imply δ 0, beause otherwise we have a ontradition.

5 Vol ) Non-degeneray of the solution to the Liouville Gel fand problem 357 We assume the existene of w = w x) satisfying 5) and derive a ontradition. For this purpose, we put ṽ x) = v x + δ x) v x ), w x) = w x + δ x), Ṽ x) = Vx + δ x), where x for = { x R 2 x + δ x }.Wehave ṽ = Ṽ eṽ, ṽ 0 =ṽ 0) in, λ e v C 2 eṽ = with a onstant C 2 > 0 independent of, and w = Ṽ eṽ w in, w = 0 on, w =. Conerning ṽ, we an apply []. Thus, passing to a subsequene, we obtain ṽ ṽ 0 in C 2,α lo R2 ) for 0 <α<, with ṽ 0 =ṽ 0 x) satisfying ṽ 0 = Vx 0 )eṽ0, ṽ 0 0 =ṽ 0 0) in R 2, eṽ0 < +, R 2 and therefore ṽ 0 x) = log { + Vx 0 ) x 2} 2 by [4]. This implies w w 0 in C 2,α lo R2 ) for a subsequene, with w 0 = w 0 x) satisfying w 0 = Vx 0 )eṽ0 w 0 = w 0. Vx 0 ) { + Vx 0 ) x 2} 2 w 0 in R 2, 9) We shall show w 0 = 0inR 2. In fat, if this is the ase, then it holds that y +, where y denotes a maximum point of w = w x); w y ) = w =. We mae the Kelvin transformation ) ) x x ˆv x) =ṽ x 2, ŵ x) = w x 2,

6 35 T. Sato and T. Suzui CMH and obtain y ŵ = ŵ y 2 ) =, ŵ = x x 4 Ṽ x 2 for large. On the other hand, inequality 7) reads ṽx) + log ) e ˆvŵ in B 0) \{0} for x and =, 2,..., and we have eṽx) This means singularity of ŵ, ŵ = a x)ŵ in B 0) { + Vx ) x 2 } 2 C 0) ) = O uniformly in. x 4 x 4 e ˆv x) = O) uniformly in, and therefore x = 0 is a removable with a = a x) satisfying a L B 0)) = O). Then, the loal ellipti estimate guarantees = ŵ L B /2 0)) C ŵ L 2 B 0)), where the right-hand side onverges to 0 by the dominated onvergene theorem. This is a ontradition and we obtain the proof of Theorem.4. To prove w 0 = 0inR 2, we put = Vx 0 )>0and vx) = w 0 x/ ) in 9). Then, this v = vx) L R 2 ) satisfies v v = { + x 2} in R 2 2 and hene it holds that vx) = i= a i x i + x 2 + b x 2 + x 2 by [2], where a i,b R. Thus, we only have to derive a i = b = 0in a i x i w 0 x) = + x 2 + b x 2 + x 2. i= We note that a i / a i in the formula for vx)) is newly denoted by a i. To show a i = 0, we use the following lemma, proven similarly to 3.3) in [5]. Lemma 2.3. In ase a,a 2 ) = 0, 0), it holds that δ w x) = 2π a j x, x 0 ) + o) ) loally uniformly in x \{x 0 }.

7 Vol ) Non-degeneray of the solution to the Liouville Gel fand problem 359 Proof. In fat, we have w x) = Gx,y)λ Vy)e vy) w y)dy = Gx, x + δ y )Ṽ y )eṽy ) w y )dy = I, x) + I 2, x), where with I, x) = Gx, x + δ y ) f y )dy I 2, x) = Gx, x + δ y ) 64b y 2 + y 2) 3 dy f y) = Ṽ y)eṽy) w y) 64b y 2 + y 2) 3. We have Ṽ y)eṽy) w y) + y 2) 2 i= ) a i y i + y 2 + b y 2 + y 2, or equivalently, f y) f 0 y) = 64 i= a i y i + y 2) 3, loally uniformly in y R 2. ) We have, on the other hand, f y) = O uniformly in =, 2,... by y 4 0), and therefore g y) g 0 y) loally uniformly in y R 2 by the dominated onvergene theorem, where + g y,y 2 ) = f a y +a 2 y 2 a 2+a2 2 a t + a2 2 y a a 2 y 2 a 2 + a2 2,a 2 t a a 2 y a 2y ) 2 a 2 + dt a2 2 for = 0,, 2,... This g, introdued in Lemma 6 of [5], satisfies a g y + a 2 g y 2 = f,

8 360 T. Sato and T. Suzui CMH and therefore it holds that I, x) = Gx, x + δ y )f y )dy = Gx, x + δ y ) = δ a j = δ { = δ {2π g a j y j y )dy x, x + δ y ) g y )dy a j x, x 0 ) } a j x, x 0 ) + o) 6 R 2 } + y 2) 2 dy + o) loally uniformly in x \{x 0 } by the dominated onvergene theorem. To study I 2, x), we note that uy) = log 64 + y 2) 2 satisfies y y e u) + y2 e u) = 2 y 2 and in this ase we obtain I 2, x) = b Gx, x + δ y ) 2 = δ b 2 b { = δ x, x 0 ) 2 y j y 2 + y 2) 3, y j e uy)) y=y dy x, x + δ y ) y j euy ) dy } y R 2 j euy ) dy + o) = o δ ) loally uniformly in x \{x 0 }, again by the dominated onvergene theorem. Thus, the proof of ) is omplete. 3. Proof of Theorem.4 We prove the following lemma, using new arguments.

9 Vol ) Non-degeneray of the solution to the Liouville Gel fand problem 36 Lemma 3.. If Vx) is C 2 near x = x 0 and x 0 is a non-degenerate ritial point of Rx) + 4π log Vx), then it holds that a = a 2 = 0. Proof. We suppose the ontrary, and then obtain ) loally uniformly in x \{x 0 }. We note v = λ Ve v v + λ Ve v log V in and define h i, = h i, x) by h i, = log V λ Ve v in, h i, = 0 on, where i =, 2. Then it follows that w by 5), and therefore we have { w ν v h i, v h i, ) w v = 0 in ) w )} ν v h i, = h i, w. Here and heneforth, ν denotes the outer unit normal vetor on. Sine w = h i, = 0on, the above equation is redued to δ v w ν = δ h i, w = δ h i, w 2) = δ log V λ Ve v w. We have v πg,x 0 ) in C 2,α lo \{x 0}), δ w 2π a j,x 0 ) in C 2,α lo \{x 0}) by Theorem.3 and the ellipti estimate, and therefore the left-hand side of 2) onverges to 6π 2 a j x, x 0 ) 2 G x, x 0 ). ν x Now we apply Lemma 7 of [5]: x, x 0 ) 2 G x, x 0 ) = 2 R x 0 ), 3) ν x 2 x j

10 362 T. Sato and T. Suzui CMH and then obtain v w lim + δ ν = π 2 a j 2 R x j x 0 ). We note here that 3) is shown by the Pohozaev identity [0]. Therefore, if we an show then log V lim + δ λ Ve v w = 2π { 2 R a j x 0 ) + 2 } log V x 0 ) = 0 x j 4π x j a j 2 log V x j x 0 ), 4) follows for i =, 2, and hene a = a 2 = 0 from the assumption. For this purpose, we use the Taylor expansion around x = x,x 2 ) for large and obtain log V x) = log V [ x ) + x x ) + x 2 x 2 ) ] x x 2 log V x ) + R x) x x for x = x,x 2 ) with R x) rx,x ), where r,x ) is uniformly bounded on, and near x 0, rx,x ) = sup 2 log V y) 2 log V x ) y Bx, x x ) x j x. j i,j Therefore, this r,x ) is ontinuous there, satisfying rx,x ) = 0 and onverging to r,x 0 ) uniformly. We shall show that there exists C 3 > 0 suh that 5) δ x x )w x) C 3 6) for any x and =, 2,... Then, we have R x) x x λ Ve v δ w C 3 rx,x )λ Ve v 0 by λ Ve v dx πδ x0 dx) and rx 0,x 0 ) = 0, and therefore the ontribution of the residual term of 5) is negleted in the limit of 2).

11 Vol ) Non-degeneray of the solution to the Liouville Gel fand problem 363 To show 6), we use w x) = I, x) + I 2, x) with δ I, x) = δ I 2, x) = b 2 a j x, x + δ y ) g y )dy, x, x + δ y ) y j euy ) dy. There is C 4 > 0 suh that x, y) y C 4 x y j for any x, y), and therefore δ w x) C 4 a + a 2 + b ) x δ y x g y ) ) y + 2 j e uy ) dy holds true. It is obvious that g y) + y j e uy) C 5 + y 2 ) 3 2 with C 5 > 0 independent of y R 2 and =, 2,..., and hene δ w x) C 4 C 5 a + a 2 + b ) x δ y x + y 2) 2 3 dy. 2 This implies δ δ x ) w x + δ x ) C4 C 5 a + a 2 + b ) x 2 R 2 x y + y 2) 32 dy, butwehave x R 2 x y + y 2) 32 2π dy = dθ x + x + re ıθ 2) 32 dr C with C 6 > 0 independent of x R 2. Hene 6) follows for x and =, 2,... Thus, we have proven that the limit of the right-hand side of 2) is redued to log V { } lim + δ λ Ve v w = lim II0, + II, + II 2,, +

12 364 T. Sato and T. Suzui CMH where II 0, = log V x ) II, = 2 log V x x ) II 2, = 2 log V x 2 x ) λ Ve v δ w, x x ) λ Ve v δ w, x 2 x 2 ) λ Ve v δ w. and First, we have II 0, = log V x ) log V 2 G,x 0 ) = ν x B r x 0 ) δ w = log V x ) x 0 ) 2π a j 2 G,x 0 ) = ν x B r x 0 ) as r 0, where G 0 x, y) = 2π log x y. Then it holds that for x B r x 0 ), and therefore 2 G 0 x, x 0 ) = x j x 0j ν x 2π x x 0 3 B r x 0 ) 2 G 0 ν x,x 0 ) = 0. 2 G ν x,x 0 ) δ w ν 2 G 0 ν x,x 0 ) + o) Thus, we have proven lim + II 0, = 0. Next, we have x l x l ) λ Ve v w = x l x l ) w w = x l x l ) w { x l ν = ν l w x l x l ) w } ν = x l x l ) w ν

13 Vol ) Non-degeneray of the solution to the Liouville Gel fand problem 365 for l =, 2, and this implies II l, = 2 log V x ) x l x l ) δ w x l ν 2 log V x 0 ) 2π a j x l x l x 0l ) 2 G ν x,x 0 ). Here, we have 2 { G x l x 0l ) x, x 0 ) = x l x 0l ) } x, x 0 ) ν x ν x { = x l x 0l ) } x, x 0 ) B r x 0 ) ν x [ + x l x 0l ) ] x, x 0 ) \B r x 0 ) { = x l x 0l ) } 2 G x, x 0 ) + 2 x, x 0 ) B r x 0 ) ν x \B r x 0 ) x l { = x l x 0l ) } x, x 0 ) 2 ν l x, x 0 ) B r x 0 ) ν x B r x 0 ) { = x l x 0l ) } 0 0 x, x 0 ) 2 ν l x, x 0 ) + o) B r x 0 ) ν x B r x 0 ) as r 0, and the first term of the right-hand side is equal to 0 beause { x l x 0l ) } 0 x, x 0 ) = x [ ] l x 0l 0 x, x 0 ) + r 2 G 0 x, x 0 ) = 0 ν x r r in terms of r = x x 0. On the other hand, the seond term is equal to x l x 0l ) ) { x j x 0j l = j), π r 3 = δ jl = 0 l = j), B r x 0 ) and therefore lim II 2 log V l, = 2πa l x 0 ) + x l holds for l =, 2. We obtain 4), and the proof is omplete. One a = a 2 = 0 is obtained, then the proof of b = 0 is similar to [5]. For the sae of ompleteness, we onfirm the following lemma and onlude the proof of Theorem.4.

14 366 T. Sato and T. Suzui CMH Lemma 3.2. Under the assumptions of the previous lemma, it holds that b = 0. Proof. By Lemma 3., we have We assume b = 0 and note the equalities in and also Then we have w x) b x 2 + x 2 in C 2,α lo R2 ). w v = λ Ve v w and v w = λ Ve v v w w v v w ) = λ We also have λ Ve v v w = by 7), and therefore = R 2 Ve v w = λ w v ν v Ṽ eṽṽ w + v λ ) w = 0. ν Ve v v w. 7) Ve v w + x 2 ) 2 log + x 2 ) 2 b x 2 + x 2 dx + v λ Ve v w + o) = πb + v λ Ve v w + o) πb = v ) λ Ve v w + o) ) by 7). We shall show w = o δ ) loally uniformly in \{x 0 } 9) for i =, 2 and v = 2 log λ + 2 log πrx 0) + o). 20)

15 Vol ) Non-degeneray of the solution to the Liouville Gel fand problem 367 In fat, if this is the ase we obtain λ δ 2 by ), and therefore v λ Ve v w w = v ν = o δ log λ ) = o). Then, b = 0 follows from ). Proof of 9). In fat, we have w = λ x, y) Vy)e vy) w y)dy = x, x + δ y )h y )dy with uniformly in and ) h y) = Ṽ y)eṽy) w y) = O y 4 h y) h 0 y) = 64b y 2 + y 2 ) 3 loally uniformly in y R 2. Therefore ζ y) ζ 0 y) loally uniformly in y R 2 for ζ = ζ y) defined in Lemma 6 of [5]: y 2+y2 2 ζ y,y 2 ) = log y 2 + th ty ty 2, dt. y2 2 y 2 + y2 2 y 2 + y2 2 Here we have ) ζ ζ y + y e ζ = h y y 2 and ζ 0 y) = log 32b + y 2 ) 2, and the dominated onvergene theorem guarantees ) w x) = x, x + δ y ζ ζ ) y + y y y 2 = and hene 9). δ = δ { 2 G x, x + δ y ) y j eζ y ) dy e ζ y=y dy 2 G x, x 0 ) 32by } j R 2 + y 2) 2 dy + o) = o δ )

16 36 T. Sato and T. Suzui CMH Proof of 20). We have Gx,y) = and therefore it follows that where 2π log v = v x ) = III, + III 2,, x y + Kx,y) with K C2,α ), III, = λ log x y Vy)e vy) dy, 2π III 2, = λ Kx, y)v y)e vy) dy. We have λ Ve v dx πδ x0 dx), and therefore III 2, = πkx 0,x 0 ) + o) = πrx 0 ) + o). For the first term, on the other hand, we have III, = log δ y Ṽ y 2π )eṽy ) dy = 4π log λ + v ) Ṽ y )eṽy ) dy log y Ṽ y 2π )eṽy ) dy by ), and therefore an asymptoti formula of [3] guarantees Ṽ y )eṽy ) dy = V y)e vy) dy = π + O λ log λ ). Thus we obtain III, = 2 log λ + 2 v ) + O λ log λ )) log y ) 2π R 2 + y 2) 2 dy + o) = 2 log λ + 2 v ) + O λ log λ )) 2 log + o). These results are summarized as v + O λ log λ )) = 2log λ ) + O λ log λ )) + 2 log πrx 0) + o), or 20).

17 Vol ) Non-degeneray of the solution to the Liouville Gel fand problem 369 Referenes [] Brezis, H. and F. Merle, Uniform estimates and blow-up behavior for solutions of u = Vx)e u in two dimensions. Comm. Partial Differential Equations 6 99), Zbl MR 3273 [2] C.-C. Chen and C.-S. Lin, On the symmetry of blowup solutions to a mean field equation. Ann. Inst. H. Poinaré, Analyse Non lineaire 200), Zbl MR 3657 [3] C.-C. Chen and C.-S. Lin, Sharp estimates for solutions to multi-bubbles in ompat Riemann surfaes. Comm. Pure Appl. Math ), Zbl MR 5666 [4] C.-C. Chen and C.-S. Lin, Classifiation of solutions of some nonlinear ellipti equations. Due Math. J ), Zbl MR 247 [5] F. Gladiali and M. Grossi, Some results for the Gel fand problem. Comm. Partial Differential Equations ), Zbl MR [6] Y. Y. Li, Harna type inequality: the method of moving planes. Comm. Math. Phys ), Zbl MR [7] L. Ma, and J. C. Wei, Convergene for a Liouville equation. Comment. Math. Helv ), Zbl MR [] K. Nagasai and T. Suzui, Asymptoti analysis for two-dimensional ellipti eigenvalue problems with exponentially dominated nonlinearities. Asymptoti Analysis 3 990), 73. Zbl MR [9] H. Ohtsua and T. Suzui, Blow-up analysis for Liouville type equation in self-dual gauge field theories. Commun. Contemp. Math ), Zbl 026 MR [0] S. Pohozaev, Eigenfuntions of the equation u + λf u) = 0. Soviet. Math. Dol ), Zbl MR 0924 Reeived September 5, 2004; revised November 29, 2005 Tomohio Sato, Taashi Suzui, Division of Mathematial Siene, Department of System Innovation, Graduate Shool of Engineering Siene, Osaa University, 3 Mahianeyama, Toyonaa, Osaa Japan suzui@sigmath.es.osaa-u.a.jp