2 JUNCHENG WEI Lin, Ni and Takagi rst in [11] established the existence of leastenergy solutions and Ni and Takagi in [13] and [14] showed that for su

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1 UNIQUENESS AND EIGENVALUE ESTIMATES OF BOUNDARY SPIKE SOLUTIONS JUNCHENG WEI Abstract. We study the properties of single boundary spike solutions for the following singularly perturbed problem 2 u? u + u p 0 in ; u > 0 in and u 0 on It is known that at a nondegenerate critical point of the mean curvature function H(P ), there exists a single boundary spike solution. In this paper, we show that the single boundary spike solution is unique and moreover it has exactly (N? 1) small eigenvalues. We obtain the exact asymptotics of the small eigenvalues in terms of H(P ). 1. Introduction Of concern are properties of boundary spike solutions of the following singularly perturbed Neumann problem 2 u? u + u p 0 in ; u > 0 in and u 0 on (1.1) where > 0 is a small parameter, is a smooth bounded domain in R N, (x) is the outer normal at x 2, and 1 < p < N +2 N?2 for N 3 and 1 < p < +1 for N 1; 2. Problem (1.1) is a typical singular perturbation problem which arises in many applied branches of applied science. It can be considered as the stationary equation to the shadow system of the Gierer-Meinhardt system (see [3]) which models biological pattern formation. It is also known as the steady state problem for a chemotactic aggregation model with logarithmic sensitivity by Keller-Segel [8], see e.g. [11] Mathematics Subject Classication. Primary 35B40, 35B45; Secondary 35J40. Key words and phrases. Spike Layer Solutions, Uniqueness, Stability, Eigenvalues, Nonlinear Elliptic Equations. 1

2 2 JUNCHENG WEI Lin, Ni and Takagi rst in [11] established the existence of leastenergy solutions and Ni and Takagi in [13] and [14] showed that for suciently small the least-energy solution has only one local maximum point P and P 2 (therefore the least energy solutions have a boundary spike layer). Moreover, H(P )! max P 2 H(P ) as! 0, where H(P ) is the mean curvature of P at. In [15] Ni and Takagi constructed multiple spike solutions with spikes on the boundary in an axially symmetric domains. In [16], the author constructed boundary spike solutions in general domains. To state the results, we need to introduce some notation. It is known that the solution of the following problem 8 < : w? w + w p 0; in R N w > 0; w(z)! 0 as jzj! 1 w(0) max z2r N w(z) is radial ([4]) and unique ([7]). We denote this solution as w. Let Let u 2 H 1 () and J (u) 2 2 J(w) 1 2 jruj R N (jrwj 2 + w 2 )? 1 p + 1 (1.2) u 2? 1 u p+1 : (1.3) p + 1 R N w p+1 : (1.4) Family of solutions u of (1.2) are called level-1/2 if lim!0?n J (u ) 1 J(w). Certainly least energy solutions are level-1/2. 2 In [16], the following result was proved. Theorem A: If u is a solution of (1.2) and lim!0?n J (u ) 1 J(w). 2 Then for suciently small u has only one local (hence global ) maximum point P and P 2. Moreover, r P H(P )! 0 as! 0 where r P is the tangential derivative at P. On the other hand, let P 0 2. Suppose that P 0 is a nondegenerate critical point of H(P ). Then for suciently small there exists a solution u to (1.2) such that?n J (u )! 1 2 J(w) and u has only one local maximum point P and P 2. Moreover P! P 0.

3 BOUNDARY SPIKE LAYER 3 Remark: In [1] and [19], multiple boundary spike solutions are constructed at multiple nondegenerate critical points of H(P ). On the other hand, single and multiple boundary spike solutions with degenerate critical points of H(P ) are studied in [2], [5], [9], [15], [18], etc. In this paper, we study the properties of u constructed in Theorem A. In particular, we establish the uniqueness of u and the eigenvalue estimates of the associated linearized operator. These properties are essential in the study of existence and stability of boundary spike solutions for the corresponding reaction-diusion system (such as Gierer- Meinhardt system and Keller-Segel system). See [12], [17] for progress in this direction. The rst result concerns the uniqueness of u. Theorem 1.1. Suppose that P 0 is a nondegenerate critical point of the mean curvature function H(P ). Then for suciently small, if there are two families of level- 1 2 solutions u ;1 and u ;2 of (1.2) such that P 1! P 0 ; P 2! P 0 where u ;1 (P 1 ) max P 2 u (P ); u ;2 max P 2 u ;2 (P ), then we have P 1 P 2 ; u ;1 u ;2 : The second result concerns the eigenvalue estimates associated with the linearized operator at u L : 2? 1 + pu p?1 : We rst note the following result. Lemma 1.2. The following eigenvalue problem? + pw p?1 ; in R N 2 H 1 (R N ) (1.5) admits the following set of eigenvalues 1 > 0; 2 ::: N +1 0; N +2 < 0; ::: (1.6) Proof: Since w is the least energy solution of (1.2), then by a similar argument as in the proof of Theorem 2.12 of [10], we can show that

4 4 JUNCHENG WEI there is only one positive eigenvalue which is simple. The multiplicity of the eigenvalue 0 is N. See Lemma C in [14]. Next we introduce an important denition: for each P 2, we set w ;P to be the unique solution of 2 u? u + w p ( x?p ) 0 in ; u 0 on : (1.7) Then we have Theorem 1.3. The following eigenvalue problem 2? + pu p?1 in 0 on (1.8) admits the following set of eigenvalues o(1); 2 o(1); :::; N o(1); j j+1 + o(1); j N + 1: Moreover, j+1! c 0 j ; j 1; :::; N? 1 (1.9) 2 where j are the eigenvalues of G(P 0 ) : (r 2 P0 H(P 0 )) and c 0 N? 1 N + 1 R R R N + (w 0 ) 2 z N dz R N + ( w z 1 ) 2 dz : Furthermore the eigenfunction corresponding to j+1 ; j 1; :::; N? 1 is given by the following: j (a j + o(1)) w ;P j (P ) where P is the local maximum point of u, a j is the eigenvector corresponding to j, namely G(P 0 )a j j a j ; j 1; :::; N? 1: From Theorem 1.1, u is an isolated critical point of J. From Theorem 1.3, L is invertible. Hence u is nondegenerate. We obtain Corollary 1.4. u is an isolated nondegenerate critical point of J.

5 BOUNDARY SPIKE LAYER 5 The basic idea of our proof is to transform problems in H 1 () into a nite dimensional problem. This is done by the so-called Liapunov- Schmidt reduction procedure. For each P 2, we may assume that ; j j 1; :::; N? 1 are the (P ) (N? 1)?tangential derivatives. To avoid clumsy notations, sometimes we use j to denote. j (P ) It is easy to see that if u is a level- 1 2 solution, then we have ku (x)? w ;P (x)k L 1 ()! 0 (1.10) where P is the unique local maximum point of u. and Set Let K ;P f j w ;P jj 1; :::; N? 1g H 2 () C ;P f j w ;P jj 1; :::; N? 1g L 2 () fp 2 jjp? P 0 j g where is so small such that P 0 is the only critical point of H(P ) in. For each P 2 we can nd a unique v ;P 2 K? ;P such that We then dene S (w ;P + v ;P ) 2 C ;P K (P ) J (w ;P + v ;P ) :! R A key observation is the following fact: Lemma 1.5. u w ;P + v ;P is a critical point of J if and only if P is a critical point of K in. By the above Lemma, the uniqueness problem is reduced to counting the number of critical points of K in. Thus we need to compute j (P K ) (P ) and 2 i (P ) j (P K ) (P ). This is the main estimate. Theorems 1.3 can be proved by using the estimates. Finally we remark that all the theorems above are true if we replace u p in equation (1.1) by general nonlinearities f(u).

6 6 JUNCHENG WEI The organization of our paper is as follows. In Section 2, we analyze the projection of w in H 1 (). Then we set up the technical framework for the problem and reduce the problem into a nite dimensional problem in Section 3. Section 4 is devoted to the proofs of Theorems 1.1. In Section 5, we prove Theorem 1.3. Section 6 contains some important estimates for 2 K (P ). Finally in Appendix we prove an estimate left in Section 6. In this paper we denote various generic constants by C. We use O(A); o(a) to mean jo(a)j CjAj; o(a)jaj! 0 as jaj! 0, respectively. Whenever we have repeated indices, we mean summation that index from 1 to N? 1 unless otherwise specied. Acknowledgments. This research is supported by an Earmarked Grant from RGC of Hong Kong. 2. Preliminaries and Analysis of Projections In this section, we introduce a nice approximate function (see (2.4) ) and derive the asymptotic expansion of the function as well as its tangential derivatives. Most of the materials in this section are taken from [16]. We rst transform the boundary. Let P 2. Since is smooth, we can nd a R 0 > 0; : B 0 (R 0 )! R a smooth function such that (0) 0; r 0 x (0) 0 and 1 \ B(R 0 ) f(x 0 ; x N ) 2 B(R 0 )j x N? P N > (x 0? P 0 )g! 1 \ B(R 0 ) f(x 0 ; x N ) 2 B(R 0 )j x N? P N (x 0? P 0 ) where x 0 (x 1 ; :::; x ) and B(R 0 ) fx 2 R N j jxj < R 0 g; B 0 (R 0 ) fx 0 2 R j jx 0 j < R 0 g: Note that H(P 0 ) 0 x (0). By Taylor's expansion, we can assume that (a) 1 2 ij(0)a i a j ijk(0)a i a j a k + O(jaj 4 )

7 for a 2 R small, where BOUNDARY SPIKE LAYER 7 i x i ; ij 2 ; etc: x i x j For z 2, let (z) denote the unit outward normal at z, denote the normal derivative, ( 1 (z); :::; (z)) denote the (N? 1) linearly independent tangent vectors and ( (N? 1) tangential derivatives at z. For x 2! 1, we have (x) 1 p 1 + jrx 0j 2 i (x) 1 (z) ; ::; 1 p 1 + jrx 0j 2 (r x 0(x0? P 0 );?1); j i (x) (0; :::; 1; ::; 0;? x j ) the corresponding (z) x N xn?p N (x 0?P 0 ) ; x i (x 0? P 0 )); i 1; :::; N? 1; + i (x 0? P 0 ) j xn?p x i x N (x 0?P 0 ); i 1; :::; N? 1: N Let w be the unique solution of (1.2) and w ;P be the unique solution of (1.11). By the Maximum Principle, w ;P > 0. For each u; v 2 H 1 (), we dene hu; vi?n ( 2 ru rv + uv): We denote hu; ui as kuk 2. We now analyze w ;P. To this end, we set h ;P (x) w( x? P )? w ;P : We introduce the following functions. Let v ij (y) be the unique solution of ( v? v 0 in R N + ; v 2 H 1 (R N + ) v y N? 1 w 0 2 jyj y iy j on R N + : (2.1) Set v 1 (x) : ij ij (P )v ij ( x? P ):

8 8 JUNCHENG WEI Let (a) be a function such that (a) 1 for a 2 B(P; R 0 ), (a) 0 2 for a 2 B(P; R 0 ) c. Set y 0 x 0? P 0 ; y N x N? P N? (x 0? P 0 ) h ;P (x) ij ij (P )v ij ( x? P )(x? P ) + 2 Then by Proposition 2.1 of [16] we have Lemma 2.1. k 1k C: 1(x): We next analyze j (P w ) ;P(x). By the coordinate system we choose, we can assume that Then P j h ;P (x) satises v j (P ) P j : 2 v? v 0 in ; x? P P j w on : Let w j kl be the unique solution of ( v? v 0 on R N + ; v 2 H 1 (R N + ) v y N 1 w 00 2 jyj? w0 2 jyj y 3 j y k y l on R N + : (2.2) Then we have (see Proposition 2.2 of [16]) Proposition 2.2. w j (P )? w ;P j (P ) [ kl jkl (P )w j? kl (x P )?2 k jk (P )v jk ( x? P )](x?p )+ 2(x) where w j kl ; v jk are dened above and k 2k C:

9 BOUNDARY SPIKE LAYER 9 3. Liapunov-Schmidt Reduction In this section, we solve problem (1.2) in appropriate kernel and cokernel and then reduce our problem to nite dimensional problem. Since the procedure is now standard, we will omit most of the details. See Section 3 of [6]. We rst introduce some notation. Set f(u) : u p ; P 0 0; ;P : fyjy + P 2 g; : ;P0 : Let H 2 ( ) be the Hilbert space dened by H( 2 ) Dene u 2 H 2 ( ) u 0 on S (u) u? u + f(u) for u 2 H( 2 ): Then solving equation (1.1) is equivalent to S (u) 0; u 2 H( 2 ): Fix P 2. To study (1.1) we rst consider the linearized operator ~L : (z) 7! (z)? (z) + f 0 (w ;P )(z); H( 2 )! L 2 ( ): It is easy to see (integration by parts) that the cokernel of L ~ coincides with its kernel. Choose approximate cokernel and kernel as K ;P span w;p j (P ) C ;P span w;p j (P ) j 1; : : : ; N? 1 j 1; : : : ; N? 1 L 2 ( ): : ; H 2 ( ); Let ;P denote the projection in L 2 ( ) onto C? ;P. Our goal in this section is to show that the equation ;P S (P ;P w + ;P ) 0 has a unique solution ;P 2 K? ;P is C 1 in P. if is small enough. Moreover ;P

10 10 JUNCHENG WEI We recall the following three propositions in [6]. Proposition 3.1. Let L ;P ;P L ~. There exist positive constants ; such that for all 2 (0; ) and P 2 kl ;P k L 2 ( ) kk H 2 ( ) (3.1) for all 2 K?. ;P Proposition 3.2. For all 2 (0; ) and P 2 the map L ;P ;P L ~ : K? ;P! C? ;P is surjective. Proposition 3.3. There exists > 0 such that for any 0 < < and P 2 there exists a unique v ;P 2 K? ;P satisfying S (w ;P + v ;P ) 2 C? ;P and kv ;P k H 2 ( ) C (3.2) In fact v ;P is actually smooth in P. Lemma 3.4. Let v ;P be dened by Proposition 3.3. Then v ;P 2 C 1 in P. Proof: See Lemma 3.5 of [6]. In fact, we have the following asymptotic expansion for ;P. For the proof, please see Proposition 3.4 of [18]. Proposition 3.5. where v ;P (x) ( 0 ( x? P )(x? P )) + 2 ;P (x) (3.3) k ;Pk C and 0 is the unique solution of 0? 0 + f 0 (w) 0? f 0 (w)v 1 0; in R N + ; 0 0 on R N + y N

11 BOUNDARY SPIKE LAYER 11 0 is orthogonal to the kernel of L 0 (3.4) where v 1 P kl kl(0)v kl (y) and L 0? 1 + f 0 (w) : H 2 (R N + ) : fu 2 H 2 (R N + )j u y N 0 on R N + g! L 2 (R N + ). Finally, we now dene We have K (P ) J (w ;P + v ;P ) :! R: Proposition 3.6. P 2 is a critical point of K if and only if u w ;P + v ;P is a critical point of J. 4. uniqueness of u In this section, we prove Theorem 1.1. Let P 0 2 be such that (i) i (P H(P 0 ) 0) 0; i 1; :::; N? 1, (ii) G(P 0 ) ( 2 H(P 0 )) is nonsingular. Let us introduce the set? fp 2 jjp? P 0 j < g: Set ~K (P ) N f 1 I(w) + BH(P )g 2 where B :? N? 1 (w 0 (jzj)) 2 z N dz: (4.1) N + 1 R N + The crucial estimate is the following: Proposition 4.1. For suciently small, we have K (P ) is of C 2 in?. Moreover, we have (1) K (P ) ~ K (P ) + o( N +1 ); (2) K (P )? ~ K (P ) o( N +1 ) uniformly for P 2?; (3) If P is a critical point of K (P ), then we have 2 K (P )? 2 ~ K (P ) o( N +1 ):

12 12 JUNCHENG WEI The proof of the above proposition will be delayed until the end of this section. Let us now use it to prove the uniqueness of u. Proof of Theorem 1.1: By Proposition 3.6, we just need to prove that K (P ) has only one critical point in?. We prove it in the following steps. Step 1. deg(k (P );?; 0) (?1) k where k is the number of negative eigenvalues of G(P 0 ). Proof: This follows from (2) of Proposition 4.1 and a continuous argument. In fact, since deg( ~ K (P );?; 0) (?1) k : By (2) both K (P ) and ~ K (P ) has no critical points on? and a continuous deformation argument shows that K (P ) has the same degree as ~ K (P ) on?. Step 2: At each critical point P of K (P ), we have for suciently small. deg(k ; B (P ) \?; 0) (?1) k ; This follows from (3) of Proposition 4.1. From Step 2, we deduce that K (P ) has only nite number of critical points in?, say k number of critical points. By the properties of the degree, we have By Step 1, k 1. Theorem 1.1 is thus proved. deg(k (P );?; 0) k (?1) k In the rest of this section, we shall prove Proposition 4.1. To this end, we rst estimate v ;P ; j v ;P, where v ;P 3.3. Set j v ;P ;j + k1 jk k w ;P ; ;j 2 K? ;P : is dened by Proposition

13 BOUNDARY SPIKE LAYER 13 We rst estimate jk. Since v ;P? l w ;P ; l 1; :::; N? 1, we have Hence k1 jk < l w ;P ; k w ;P >? < v ;P ; 2 jl w ;P > : jk < 0 ; 2 ik w > < jw; j w > +O( 2 ) where 0 is given by Proposition 3.5. It remains now to estimate ;j. Note that by Proposition 3.3, we have S (w ;P + v ;P ) for some constants j (P ); j 1; :::; N? 1. Since by Lemma 4.1 of [18], we obtain Hence we have ;j satises j j w ;P S (w ;P + ;P ) j w ;P O( N +1 ) j (P ) O( 3 ); j 1; :::; N? 1: S 0 (w ;P +v ;P )( ;j )+S 0 (w ;P +v ;P )( jk k w ;P + j w ;P )? Thus we have obtained Lemma 4.2. For suciently small, we have ;j 0( x?p )? + O() x j where 0 is the function dened in Proposition 3.5. Finally we prove Proposition 4.1. Proof of Proposition 4.1: (1) follows from Lemma 4.1 of [16]. We now prove (2) of Proposition 4.1 as follows: K j (P ) < w ;P + v ;P ; j w ;P >? K l1 f(w ;P + v ;P ) j w ;P l 2 lj w ;P 2 C ;P :

14 14 JUNCHENG WEI + < w ;P + v ;P ; j v ;P >? ~ K j (P ) + O(N +2 ) + ~ K j (P ) + O(N +2 ) + f(w ;P + v ;P ) j v ;P S (w ;P + v ;P ) j v ;P k1 k k w ;P j v ;P ~ K j (P ) + O(N +2 ): The proof of (3) is very complicated and thus is left to Section eigenvalue estimates In this section, we shall study eigenvalue estimates for and prove Theorem 1.3. L : 2? 1 + f 0 (u ) By Section 4, u is unique. Let ( ; ) be a pair such that L in 0 on : (5.1) We may assume that k k 1. We assume that! 0. Then after a scaling and limiting process, we have ~ (y) (P + y)! 0 where 0 is a solution of v? v + f 0 (w)v 0 in R N + v y N 0 on R N + : Hence there exists a j such that We now decompose as where? K ;P. 0 Since k k 1, we have k k C. a j w y j : a ;j j w ;P +

15 Note that satises L + Therefore we have BOUNDARY SPIKE LAYER 15 i1 a ;i [f 0 (u ) i w ;P? f 0 (w) i w] (5.2) i1 a ;i i w ;P + O(( + ) ja ;j j): Moreover by comparison with Lemma 4.2 we have Hence a ;j ( j v ;P ) + O(j j 2 a ;j ( j (w ;P + v ;P )) + O(j j 2 Multiplying (5.2) by k (w ;P + v ;P ) we obtain a ;j N ja ;j j) ja ;j j): [L (( j (w ;P + v ;P )) + O(j j 2 ja ;j j)) k (w ;P + v ;P )] Thus we have?a ;j 2 [( j (w ;P + v ;P )) k (w ;P + v ;P )] + +O(j j 2 ja ;j j): K (P ) a ;k j k (A + o(1)) + O(j 2 j2 where A R R N + ( w y 1 ) 2 dy. Therefore we have 2! 0 0 where 0 satises G(P 0 )~a 0 ~a Hence 0 is an eigenvalue of G(P 0 ), where 0? R R N + B ( w y 1 ) 2 ja ;j j):

16 16 JUNCHENG WEI On the other hand, we can construct a solution ( ;j ; ;j ) by putting ;j j (w ;P + v ;P ) + 2 ;j; ;j 2 K? ;P ; ;j 2 ( j + j ): By using Proposition 3.3 and the implicit function theorem, we have that there exists ( ;j ; ;j ) satisfying equation (3.1). 6. Estimate of 2 i (P ) j (P ) K In this section, we compute 2 i (P ) j (P K ) proof of (3) of Proposition 4.1. and therefore nish the Let P : P be a critical point of K (P ) in, then we have Let We rst have kkj (P ) 0; j 1; :::; N? 1: D j j (P )? J : x j : D i [ru r j (w ;P +v ;P )+u j (w ;P +v ;P )?f(w ;P +v ;P )( j (w ;P +v ;P ))] o( N +1 ) (6.1) The proof of (6.1) requires some work. Please see Appendix. We can now estimate 2 ij K (P ). By denition, we have 2 i (P ) j (P ) K < i (w ;P +v ;P ); j (w ;P +v ;P ) >? + < w ;P + v ;P ; 2 ij(w ;P + v ;P ) >? f 0 (w ;P +v ;P ) i (w ;P +v ;P ) j (w ;P +v ;P ) f(w ;P + v ;P ) 2 ij(w ;P + v ;P ) i [ru r j (w ;P +v ;P )+u j (w ;P +v ;P )?f(w ;P +v ;P )( j (w ;P +v ;P ))] x i [ru r j (w ;P +v ;P )+u j (w ;P +v ;P )?f(w ;P +v ;P )( j (w ;P +v ;P ))]

17 + BOUNDARY SPIKE LAYER 17 D i [ru r j (w ;P +v ;P )+u j (w ;P +v ;P )?f(w ;P +v ;P )( j (w ;P +v ;P ))] [ru r j (w ;P +v ;P )+u j (w ;P +v ;P )?f(w ;P +v ;P ) j (w ;P +v ;P )] i (x)dx+j? j [ 1 2 (2 jru j 2 + u 2 )? F (w ;P + v ;P )] i (x)dx + o( N +1 ) x j [ 1 2 (2 jru j 2 + u 2 )? F (w ;P + v ;P ))] i (x)dx + o( N +1 ) x j [ 1 2 (2 jru j 2 + u 2 )? F (w ;P + v ;P ))] i (x)dx + o( N +1 ) [ 1 2 (2 jru j 2 + u 2 )? F (w ;P + v ;P )] i;j (x)dx + o( N +1 ) I 1 + I 2 where I 1 and I 2 are dened at the last equality. We rst compute I 1. I 1 [( 1 p y j 2 [jrwj2 +w 2 ]?F (w)) i (y)] 1 + jrj 2 dy+o( N +1 ) R? R [( 1 p 2 [jrwj2 +w 2 ]?F (w)) i (y)] 1 + jrj 2 dy+o( N +1 ) x j? N +1 R [( 1 2 [jrwj2 + w 2 ]? F (w)) it y t ] jm ml y l dy + O( N + 2 )? N +1 ik (0) km (0) mj (0) [( 1 R 2 [jrwj2 +w 2 ]?F (w))]y 2 1 dy+o(n +2 ): To estimate I 2, we note that 1 i;j (x) p ( rr j ij? i 1 + jrj jrj ) 2 Hence i;j (x)? ij (0) N ijm (0)x m ijmk(0)x m x k? ik (0) jt (0) tm (0)x m x k + O(jxj 3 ): By Pohozaev Identity, we have In particular [ jru j u2? F (u )] j dx 0; j 1; :::; N: [ jru j u2? F (u )] N dx 0:

18 18 JUNCHENG WEI Since N?1p 1 + jrj2, we have I 2? [ jru j u2? F (u )][ i;j (x)? ij (0) N ]dx + O(e? )? [ jru j u2? F (u )] ijm (0)x m? 1 [ jru j u2? F (u )] ijmk (0)x m x k + [ jru j u2? F (u )] tm (0) ik (0) jt (0)x m x k + O( N +2 ) N +1 f? 1 2 ijmk(0) R ( 1 2 jrwj w2? F (w))y m y k g + N +1 f tm (0) ik (0) jt (0) ( 1 R 2 jrwj w2?f (w))y m y k g+o( N +2 ) Combining the estimates for I 1 and I 2 together we obtain I 1 + I 2? 1 2 ijmk(0) N +1 R N ( 1 2 jrwj w2? F (w))y m y k + O( N + 2 )? 1 2 H ij(0) N +1 R N ( 1 2 jrwj w2? F (w))y 2 1 dy + O(N + 2 ) by Lemma 3.3 of [13]. N +1 BH ij (0) + O( N +2 ) Appendix: Estimates of J In this appendix, we give the proof for the estimate for J in Section 6. These estimates are long and straightforward. We shall omit most of the details. Let P : P be a critical point of K (P ). Hence we have kki (P ) 0; i 1; :::; N? 1:

19 Recall We then have Estimate A: D j w ;P? BOUNDARY SPIKE LAYER 19 D j where v kl is dened in Section 2. Proof: By direct computations. Estimate B: Proof: ij j (P )? x j : klj (P )v kl ( x? P ) + O( 2 ) < D i (w ;P + v ;P ); j (w ;P + v ;P ) >? D i f(w ;P + v ;P ) j (w ;P + v ;P ) o( N +1 ): Lef t Hand [f 0 (w) j wd i (w ;P +v ;P )?f 0 (w ;P +v ;P )D i (w ;P +v ;P ) j (w ;P +v ;P )] o( N +1 ) since the rst term in D i (w ;P + v ;P ) is an even function. Appendix C: D j i (P ) w ;P kl ijkl (P )w j kl (x? P )? 2 k kij (P )v jk + O() where w j kl is dened in Section 2. Proof: By direct computations. Estimate D: Proof: < (w ;P + v ;P ); D i j (w ;P + v ;P ) >? f(w ;P + v ;P )D i j (w ;P + v ;P ) o( N +1 ): lef t Hand [f(w)? f(w ;P + v ;P )]D i j (w ;P + v ;P ) Note that the rst expansion of f(w)?f(w ;P +v ;P ) is an even function while by Estimate C the rst expansion of D i j (w ;P + v ;P ) consists of

20 20 JUNCHENG WEI an odd function and an even function. For the odd function part, it is of O( N +2 ). For the even function part, we will obtain N +1 kki (P ) + o( N +1 ) o( N +1 ). Combining Estimates B and D we obtain the estimate for J. References [1] P. Bates, E.N. Dancer and Shi, Multi-spike stationary solutions of the Cahn- Hilliard equation in higher-dimension and instability, preprint. [2] M. del Pino, P. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal., to appear. [3] A. Gierer and H. Meinhardt, A theory of biological pattern formation. Kybernetik (Berlin) 12(1972), [4] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in R n, Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Studies 7A (1981), Academic Press, New York, [5] C. Gui, Multi-peak solutions for a semilinear Neumann problem, Duke Math. J. 84 (1996), [6] C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincare Anal. Non Lineaire, to appear. [7] M. K. Kwong, Uniqueness of positive solutions of u? u + u p 0 in R n, Arch. Rational Mech. Anal. 105 (1989), [8] K. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), [9] Y.-Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. in PDE. 23(1998), [10] C.-S. Lin and W.-M. Ni, On the diusion coecient of a semilinear Neumann problem, Calculus of variations and partial dierential equations (Trento, 1986) 160{174, Lecture Notes in Math., 1340, Springer, Berlin-New York, [11] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Di. Eqns 72 (1988), [12] W.-M. Ni, Diusion, cross-diusion, and their spike-layer steady states, Notices of Amer. Math. Soc. 45(1998), [13] W.-M. Ni and I. Takagi, On the shape of least energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math. 41(1991), [14] W.-M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), [15] W.-M. Ni and I. Takagi, Point-condensation generated by a reaction-diusion system in axially symmetric domains, Japan J. Industrial Appl. Math. 12(1995), [16] J. Wei, On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem, J. Di. Eqns., 134 (1997),

21 BOUNDARY SPIKE LAYER 21 [17] J. Wei, On single interior spike solutions of Gierer-Meinhardt system: uniqueness and spectrum estimates, Europ. J. Appl. Math., to appear. [18] J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincare Anal. Non Lineaire 15(1998), [19] J. Wei and M. Winter, Multiple boundary spike solutions for a wide class of singular perturbation problems, J. London Math. Soc., to appear. Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong address: weimath.cuhk.edu.hk