NONDEGENERACY OF NONRADIAL SIGN-CHANGING SOLUTIONS TO THE NONLINEAR SCHRÖDINGER EQUATION

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1 NONDEGENERACY OF NONRADIAL SIGN-CHANGING SOLUTIONS TO THE NONLINEAR SCHRÖDINGER EQUATION WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI Abstract We prove that the non-radial sign-changing solutions to the nonlinear Schrödinger equation u u + u p 1 u = in R N, u H 1 (R N constructed by Musso, Pacard, Wei [19] are non-degenerate This provides the first example of a non-degenerate sign-changing solution to the above nonlinear Schrödinger equation with finite energy 1 Introduction statement of main results In this paper, we consider the nonlinear semilinear elliptic equation u u + u p 1 u = in R N, u H 1 (R N, (11 where p satisfies 1 < p < if N = 2, 1 < p < N+2 N 2 if N 3 Equation (11 arises in various models in mathematical physics biology In particular, the study of sting waves for the nonlinear Klein-Gordon or Schrödinger equations reduces to (11 We refer the reader to the papers of Berestyci Lions [2], [3], Bartsch Willem [4] for further references motivation Denote the set of non-zero finite energy solutions to (11 by Σ := {u H 1 (R N : u u + u p 1 u = } If u Σ u >, then the classical result of Gidas, Ni, Nirenberg [11] asserts that u is radially symmetric Indeed, it is nown ([12, 16] that there exists a unique radially symmetric (in fact radially decreasing positive solution for w w + w p = in R N, which tends to as x tends to All of the other positive solutions to (11 belonging to Σ are translations of w Let L be the linearized operator around w, defined by L := 1 + p w p 1 (12 Then, the natural invariance of problem (11 under the group of isometries in R N reduces to the fact that the functions x1 w,, xn w (13 naturally belong to the ernel of the operator L The solution w is non-degenerate in the sense that the L -ernel of the operator L is spanned by the functions given in (13 For further details, see [21] No other example is nown of a non-degenerate solution to (11 in Σ The purpose of this paper is to provide the first example other than w of a non-degenerate solution to (11 in Σ Concerning the existence of other solutions to (11 in Σ, several results are available in the literature Berestyci Lions [2], [3] Struwe [23] have demonstrated the existence of infinitely many radially symmetric sign-changing solutions The proofs of these results are based on the use of variational The research of the second author has been partly supported by Fondecyt Grant Millennium Nucleus Center for Analysis of PDE, NC1317 The research of the third author is partially supported by NSERC of Canada 1

2 2 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI methods The existence of non-radial sign-changing solutions was first proved by Bartsch Willem [4] in dimensions N = 4 N 6 In this case, the result is also proved by means of variational methods, the ey idea is to loo for solutions that are invariant under the action of O(2 O(N 2, in order to recover a certain compactness property Subsequently, the result was generalized by Lorca Ubilla [17] to the N = 5 dimensional case Besides the symmetry property of the solutions, the mentioned results do not provide any other qualitative properties of the solutions A different approach, a different construction, have been developed in [19] [1], where new types of nonradial sign-changing finite-energy solutions to (11 are constructed, a detailed description of these solutions is provided The main purpose of this paper is to prove that the solutions constructed by Musso, Pacard, Wei in [19] are rigid, up to transformations of the equation In other words, these solutions are non-degenerate, in the sense of the definition introduced by Duycaerts, Kenig, Merle [9] To explain the meaning of a non-degenerate solution for a given u Σ, we recall all of the possible invariance of equation (11 We have that equation (11 is invariant under the following two transformations: (1 (translation: If u Σ, then u(x + a Σ a R N ; (2 (rotation: If u Σ, then u(p x Σ, where P O N, O N is the classical orthogonal group If u Σ, then by L u = 1 + p u p 1 (14 we denote the linearized operator around u We define the null space of L u as Z u = {f H 1 (R N : L u f = } (15 If we denote the group of isometries of H 1 (R N generated by the previous two transformations by M, then the elements in Z u generated by the family of transformations M define the following vector space: Z u = span { xj u, 1 j N (x j x x xj u, 1 j < N } (16 A solution u of (11 is non-degenerate in the sense of Duycaerts, Kenig, Merle [9] if Z u = Z u (17 As we already mentioned, the only non-degenerate example of u Σ nown so far is the positive solution w In fact, in this case hence (x j x x xj w =, 1 j < N, Z w = span { xj w, 1 j N } The proof of the non-degeneracy of w relies heavily on the radial symmetry of w For non-radial solutions, the strategy used to prove non-degeneracy in the radial case is no longer applicable Thus, a new strategy is required for non-radially symmetric solutions A similar problem has arisen in the study of non-radial sign-changing finite-energy solutions for the Yamabe type problem 4 u + u N 2 u = in R N, u H 1 (R N, for N 3 In [2], the second third authors of the present paper introduced some new ideas for dealing with non-degeneracy in non-radial sign-changing solutions to the above problem Indeed, they successfully analyzed the non-degeneracy of some non-radial solutions to the Yamabe problem that were previously constructed in [6] For other constructions, we refer the reader to [7] In this paper, we will adopt the idea developed in [2] to analyze the non-degeneracy of the solutions of (11 constructed by Musso, Pacard, Wei [19] The main result of this paper can be stated as follows

3 NONDEGENERACY OF NONRADIAL SOLUTIONS 3 Theorem 11 There exists a sequence of non-radial sign-changing solutions to (11 with arbitrarily large energy, each solution is non-degenerate in the sense of (17 We believe that the non-degeneracy property of the solutions in Theorem 11 can be used to obtain new types of constructions for sign-changing solutions to (11, or related problems in bounded domains with Dirichlet or Neumann boundary conditions We will address this problem in future wor This paper is organized as follows In Section 2, we introduce the solutions constructed by Musso, Pacard, Wei in [19] In Section 3, we setch the main steps, present the proof of Theorem 11 Sections 4 to 8 are devoted to the proof of properties required for the proof of Theorem 11 2 Description of the solutions In this section, we describe the solutions u l constructed in [19], recall some properties that will be useful later To provide the description of the solutions, we introduce some notations The canonical basis of R N will be denoted by e 1 = (1,,,, e 2 = (, 1,,,,, e N = (,,, 1 (21 Let be an integer, assume we are given two positive integers m, n two positive real numbers l, l, which are related by 2 sin π ml = (2n 1 l (22 We shall comment on the possible choices of these parameters later on Consider the regular polygon in R 2 {} R N with edges whose vertices are given by the orbit of the point l y 1 = 2 sin π e 1 R N under the action of the group generated by R Here, R O(2 O(N 2 is the rotation through the angle 2π in the (x 1, x 2 plane By construction, the edges of this polygon have length l We refer to this polygon as the inner polygon We define the outer polygon to be the regular polygon with edges whose vertices are the orbit of the point y m+1 = y 1 + mle 1 under the group generated by R Observe that the distance from y m+1 to the origin is given by ml + l 2 sin, thans to (22, the edges of the outer polygon have length 2n l π By construction, the distance between the points y 1 y m+1 is equal to ml, by y j, for j = 2,, m, we denote the evenly distributed points on the segment between these two points Namely, y j = y 1 + (j 1le 1 for j = 2,, m As mentioned above, the edges of the outer polygon have length 2n l, we evenly distribute points y j, j = m + 2,, m + 2n, along this segment More precisely, if we define then the points y j are given by We also denote Let Π = t = sin π e 1 + cos π e 2 R N, y j = y m+1 + (j m 1 lt for j = m + 2,, m + 2n 1 i= z h = y j for h = 1,, 2n 1, where h = j m 1 ( {R i y j : j = 1,, m + 1} ( {R i z h : h = 1,, 2n 1} (23 Let us introduce the function w to be the unique solution of the following equation: { u u + u p =, u > in R N max x R N u(x = u(,

4 4 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI whose existence properties are obtained in the classical wors ([11, 12, 16] In [19], the authors constructed solutions of (11 that can be viewed as the sum of positive copies of w centered at the points y j, j = 1,, m + 1, together with their images by rotations R i = R R (composition of R, i times for i = 1,, 1, as well as copies of ( 1 h w (hence with the opposite sign centered at the points z h, h = 1,, 2n 1 their images by the rotations R i for i = 1,, 1 More precisely, the solution can be described as follows: 1 ( m+1 u(x U(x := w(x Ry i j + i= j=1 These solutions admit the following invariance: m+2n j=m+2 ( 1 j m 1 w(x Ry i j (24 u(x = u(rx, for R {I 2 } O(N 2, (25 u(r x = u(x u(γx = u(x, (26 where Γ O(2 I(N 2 is the symmetry with respect to the hyperplane x 2 = The numbers m, n, l, l are related by equation (22 by the following equation: Ψ(l = (2 sin π Ψ( l, (27 where Ψ(s is the so-called interaction function which is defined by Ψ(s = w(x sediv(w p (xedx, (28 e R N is any unit vector The definition of Ψ is independent of e The constraint (22 on the parameters m, n, l, l is easy to underst: it is to mae sure that an outer polygon is formed The second constraint (27 is not so easy to see As mentioned in [19], the relation between l l can be understood as a balancing condition, which is a consequence of a conservation law for solutions of (11 Alternatively, it can be understood as a condition that ensures that the approximate solution U is close enough to a genuine solution u of (11 The main theorem in [19] is as follows: Theorem A Let be an integer number with 7 let τ > be a fixed real number Then, there exists a positive number l > such that for all l > l, if l is the solution of (27, m, n are positive integers satisfying (22, m l τ, (29 then (11 admits a sign-changing solution u l that satisfies the symmetry conditions given in (25 (26 Moreover, u l (x = U(x + φ, (21 where U is defined in (24 φ = o(1 as l The energy of u is finite can be exped as where o(1 as l E(u l = (2n + m E(w + o(1,

5 NONDEGENERACY OF NONRADIAL SOLUTIONS 5 An example of a configuration with = 7 edges, m = 7 interior points on any radius, n = 4 for 2n 1 interior points on any edge As mentioned in Remar 12, [19], (29 is a technical condition Observe that, once l is fixed large enough, from (27 we see that l is a function of l which can be exped as l = l + ln(2 sin π + O(1 l, since (log Ψ (s = 1 + N 1 2s + O( 1 s Inserting this information in (22, one gets that 2 2n 1 m = 2 sin π ( 1 ln 2(sin π l 1 + O(l 2, as l The authors in [19] provide examples of possible choices for sequences of m n satisfying the above expansion For instance: for any integer m one can choose an integer n so that 1 2n 1 2 sin π m < 3 Then, if m is sufficiently large, there exists a unique l > l so that (27 (22 are satisfied, c 1 l m c 2 l, for some constants c 1, c 2 Thus Theorem A guarantees the existence of a solution of the form (24 for any such integer m Equation (11 can be rewritten in terms of the function φ in (21 as where φ φ + p U p 1 φ + E + N(φ =, (211 E = U U + U p 1 U (212 N(φ = U + φ p 1 (U + φ U p 1 U p U p 1 φ (213 One has precise control over the size of the error function E when measured in the following weighted norm Let us fix a number 1 < η <, define the weighted norm h = sup ( e η x y 1 h(x, (214 x R N y Π

6 6 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI where Π = 1 i= {Ri y j : j = 1,, m + 2n} (215 is the set of the concentration points In [19], it is proved that there exist l > ξ > such that for l > l, φ satisfies the following estimate (Proposition 41 in [19]: φ Ce 1+ξ 2 l, (216 which gives a considerably smaller bound than U We now define the following functions: π α (x = φ(x for α = 1,, N (217 x α Then, the function π α can be described as follows Proposition 21 The function π α satisfies the following estimates: for some positive constants C ξ that are independent of l π α Ce 1+ξ 2 l, (218 Recall that problem (11 is invariant under the two transformations mentioned in Section 1, translation rotation This invariance will be reflected in the element of the ernel of the linearized operator L(ψ := ψ ψ + p u l p 1 ψ, (219 which is the linearized equation associated to (11 around u l From this point on, we will drop the l in u l, for simplicity Let us now introduce the following 3N 3 functions: z α (x = u(x, for α = 1,, N, (22 x α z N+1 (x = x 1 u(x x 2 u(x (221 x 2 x 1 Observe that z N+1 = θ [u(r θx] θ=, where R θ is the rotation in the x 1 x 2 plane by the angle θ Furthermore, for α = 3,, N, z N+α 1 (x = x 1 z α x α z 1, z 2N+α 3 (x = x 2 z α x α z 2 (222 Observe that the functions defined in (22 are related to the invariance of (11 under translation, while the functions defined in (221 (222 are related to the invariance of (11 under rotations in the (x 1, x 2, (x 1, x α, (x 2, x α planes, respectively The invariance of problem (11 under translation rotation implies that the set Z u (as introduced in (16 associated to the linear operator L introduced in (219 has dimension at least 3N 3, because L(z α =, α = 1,, 3N 3 (223 We will show that these functions are the only bounded elements of the ernel of the operator L In other words, the sign-changing non-radial solutions (21 to problem (11 constructed in [19] are non-degenerate in the sense of [9] 3 Scheme of the proof In this section we describe the main ingredients which constitute the proof of our result Assume that ϕ is a bounded function satisfying L(ϕ =, (31 where L is the linear operator defined by (14 We write our function ϕ as ϕ(x = 3N 3 α=1 a α z α (x + ϕ(x, (32

7 NONDEGENERACY OF NONRADIAL SOLUTIONS 7 where the functions z α (x are defined by (22, (221, (222 respectively, while the constants a α are chosen such that z α ϕ =, α = 1,, 3N 3 (33 Observe that L( ϕ = bounded, then ϕ = Our aim is to show that under the conditions (31-(33, if ϕ is 31 Introduction of the approximate ernels some notation In order to explain our idea, we first introduce some functions These functions Z j,α i correspond to the approximate ernels around each spie For j = 1,, m + 1, i =,, 1, Z i j,1 = R i w(x R i y j, Zi j,2 = R i w(x R i y j,, for α = 3,, N, Z j,α i = w(x R x y i j α Moreover, for j = m + 2,, m + 2n, i =,, 1, we define, for α = 3,, N, Z i j,1 = ( 1 j m 1 t i w(x R i y j, Zi j,2 = ( 1 j m 1 n i w(x R i y j, In the above formulas, we denoted θ i = 2πi Z j,α i = ( 1 j m 1 w(x R x y i j α R i = (cos θ i, sin θ i,, R i = (sin θ i, cos θ i,, t i = ( sin(θ i + π, cos(θ i + π,, n i = (cos(θ i + π, sin(θ i + π, Recall that the solutions constructed in [19] tae the form u = U + φ given in (24-(21, recall further the definition of π α in (217 From this, one can obtain a more precise expression for the real ernels z α mentioned at the end of Section 2 Indeed z 1 (x = u = π 1 + U x 1 x 1 1 ( m+1 = π 1 + (cos θ i Zi j,1 + sin θ i Zi j,2 2n+m j=m+2 i= j=1 ( sin(θi + π Z j,1 i cos(θ i + π Z j,2 i,, for α = 3,, N, z 2 (x = u = π 2 + U x 2 x 2 1 ( m+1 = π 2 + (sin θ i Zi j,1 cos θ i Zi j,2 + 2n+m j=m+2 i= j=1 ( cos(θi + π Z j,1 i + sin(θ i + π Z j,2 i, z α = u 1 = π α + ( x α i= m+2n Z j,α i j=1

8 8 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI Furthermore, z N+1 for α = 3,, N, z N+α 1 U U = x 1 z 2 x 2 z 1 = x 1 π 2 x 2 π 1 + x 1 x 2 x 2 x 1 1 ( m+1 = x 1 π 2 x 2 π 1 + y j ( cos θ i sin θ i w(x R i x 2 x y j 1 + 2n+m j=m+2 i= j=1 ( R i y j n i Zi j,1 R i y j t i Zi j,2 U U = x 1 z α x α z 1 = x 1 π α x α π 1 + x 1 x α x α x 1 1 ( m+1 = x 1 π α x α π 1 + y j cos θ i Zi j,α + 2n+m j=m+2 z 2N+α 3 = x 2 z α x α z 2 i= j=1 ( R i y j n i cos(θ i + π Ri y j t i sin(θ i + π Zi j,α, U U = x 2 π α x α π 2 + x 2 x α x α x 2 1 ( m+1 = x 2 π α x α π 2 + y j sin θ i Zi j,α + 2n+m j=m+2 i= j=1 ( R i y j n i sin(θ i + π + Ri y j t i cos(θ i + π Zi j,α Let us further define the following functions For i =,, 1, Z1,1 i = cos θ i ( w(x Ri y 1 + π 1 x 1 + sin θ i( w(x Ri y 1 + π 2, (34 x 2 Z1,2 i = sin θ i ( w(x Ri y 1 + π 1 x 1 cos θ i( w(x Ri y 1 + π 2, (35 x 2, for α = 3,, N, Z1,α i = w(x Ri y 1 + π α x α (36 Moreover, we define the following functions: Z i j,α = Z i j,α for i =,, 1, j = 2,, 2n + m, α = 1,, N (37 In the following, we will always deal with the ernels in vector form These column vectors simply represent rearrangements of the approximate ernels Z i j,α : Z v,α = (Z 1,α,, Z 1 1,α, Z m+1,α,, Z 1 m+1,α t R 2 This contains the ernels around the vertices of the inner outer polygons Furthermore, define Z i Y 1,α = (Z i 2,α,, Z i m,α t R m 1, Z i Y 2,α = (Z i m+2,α,, Z i 2n+m,α t R 2n 1, which correspond to the spies on the line joining the inner polygon vertex R i y 1 the outer polygon vertex R i y m+1, the spies on the edge joining R i y m+1 R i+1 y m+1 of the outer polygon, respectively Then, we define

9 NONDEGENERACY OF NONRADIAL SOLUTIONS 9 Z α = Z v,α Z Y 1,α Z 1 Y 1,α ZY 2,α Z 1 Y 2,α R (2n+m, Z = Z 1 Z N R N (2n+m 32 Reduction of the main problem As explained in the beginning of this section, our aim is to show that the function ϕ defined by (31-(33 is identically zero, ϕ With the notations introduced in Section 31 in mind, we write our function ϕ as where c α = that c v,α c Y 1,α c 1 Y 1,α c Y 2,α c 1 Y 2,α = Observe that, if we prove that c 1,α ϕ = c (m+2n,α N c α Z α + ϕ (x, α=1, α = 1,, N, are N vectors in R (m+2n defined such Z i j,αϕ = for all α = 1,, N, i =,, 1, j = 1,, m + 2n (38 c α = for all α, ϕ = then we have that ϕ = Hence, our aim is to show that all vectors c α ϕ are zero in the above decomposition This will be a consequence of the following three facts Fact 1: Since L( ϕ =, we have that L(ϕ = N c α L(Z α (39 α=1 Our first result shows that ϕ can be controlled by c α, we have the following a priori estimate of ϕ : The proof is deferred to Section 5 ϕ Ce 1+ξ 2 l N Fact 2 The orthogonality condition (33 taes the form for β = 1,, 3N 3 N α=1 c α α=1 Z α z β = c α (31 ϕ z β, (311

10 1 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI Our second result implies that the above orthogonality condition can be reduced to 3N 3 linear conditions on the vectors c α Let us introduce the following notations: cos θ sin θ cos = R, sin = R, cos(θ i = cos θ 1 are two -dimensional vectors, cos θ i cos θ i are two (m 1-dimensional vectors, cos(θ i + π cos(θ i + π = cos(θ i + π R m 1, sin(θ i = are two (2n 1-dimensional vectors Furthermore, d d l = sin θ 1 R 2n 1, sin(θ i + π = d R l, y i = y i y i sin θ i sin θ i R R m 1, sin(θ i + π sin(θ i + π are constant vectors, where y i denotes the distance from the point y i to the origin For any unit vector e R N, we denote R i R i y 2 e R i y j e = R m 1, R i z 1 e z h e = R i y m e We have the validity of the following R i z 2n 1 e R 2n 1 R 2n 1 Proposition 31 The system (311 reduces to the following 3N 3 linear conditions on the vectors c α : cos sin cos sin cos(θ sin(θ c 1 cos(θ 1 + c 2 sin(θ + π sin(θ 1 = f 1 + O(e (1+ξl 2 L 1 cos(θ + π sin(θ + π cos(θ + π sin cos sin cos sin(θ cos(θ c 1 sin(θ 1 + c 2 cos(θ + π cos(θ 1 = f 2 + O(e (1+ξl 2 L 2 sin(θ + π cos(θ + π sin(θ + π c 1 c N c 1 c N (312 (313

11 NONDEGENERACY OF NONRADIAL SOLUTIONS 11 c 1 R y j R y j R 1 R z h n R 1 R 1 z h n 1 for α = 3,, N, c α c α c α 1 (2n+m = f α + O(e (1+ξl 2 L α c 2 y 1 y m+1 R y j R y j R 1 R z h t R 1 R 1 for α = 3,, N In the above expansions, z h t 1 c 1 c N = f N+1 + O(e (1+ξl y 1 cos y m+1 cos R y j e 1 R 1 y j e 1 = f N+α 1 + O(e (1+ξl R z h e 1 R 1 z h e 1 y 1 sin y m+1 sin R y j e 2 R 1 y j e 2 = f 2N+α 3 + O(e (1+ξl R z h e 2 R 1 z h e 2 f 1 f 3N 3 f 1 f 3N 3 2 L N+α 1 2 L 2N+α 3 (314 2 L N+1 is a fixed vector with l τ ϕ c 1 c N c 1 c N c 1 c N (315 (316, (317 for some positive constant τ that is independent of l Here, L i : R (2n+m R are linear functions whose coefficients are constants uniformly bounded as l The proof is deferred to Section 6 Fact 3: Let us now multiply (39 by Z i j,α, for i =,, 1, j = 1,, m + 2n, α = 1,, N After integrating in R N, we obtain a linear system of (2n+m N equations in the (2n+m N

12 12 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI coefficients c of the form M c 1 c N = r 1 r N with r α = L(ϕ Z v,α L(ϕ Z Y 1,α L(ϕ Z 1 Y 1,α L(ϕ Z Y 2,α L(ϕ Z 1 Y 2,α, (318 where M = ( L(Z t i,α Zs j,β dx is a square matrix of dimension [(2n + m N]2 More detailed information an analysis of the matrix M is provided in Section 4 Our third result concerns the solvability of the above matrix equation We can show the validity of the following statement: Proposition 32 There exists l > C such that for l > l, system (318 is solvable Furthermore, the general solution is ( ( c1 v1 = c 2 v 2 cos sin cos sin cos(θ sin(θ R y j R +s 1 c α cos(θ 1 sin(θ + π sin(θ + π sin sin sin(θ sin(θ 1 cos(θ + π cos(θ + π + s 2 := ( v1 = v α + s α1 1 (2n+m + s α2 sin(θ 1 cos(θ + π cos(θ + π cos cos cos(θ cos(θ 1 sin(θ + π sin(θ + π + s 3 v 2 + s 1 w 1 + s 2 w 1 + s 3 w 3 := v α + s α1 w 4 + s α2 w 5 + s α3 w 6 y j R 1 R z h n R 1 z h n 1 y 1 y m+1 R y j R R 1 y j R 1 R z h t z h t 1 R 1 R 1 y 1 cos y m+1 cos R y j e 1 R 1 y j e 1 + s α3 R z h e 1 R 1 z h e 1 y 1 sin y m+1 sin R y j e 2 R 1 y j e 2 R z h e 2 R 1 z h e 2

13 NONDEGENERACY OF NONRADIAL SOLUTIONS 13 for any s 1, s 2, s 3, s α1, s α2, s α3 R, where the vectors v α are fixed, satisfy for some τ, ξ > The proof is deferred to Section 7 33 Final argument proof of Theorem 11 Let predicted by Proposition 32, given explicitly by ( ( c1 v1 = c 2 v α Cl τ e 1 ξ 2 l ϕ (319 c 1 c N v 2 + s 1 w 1 + s 2 w 2 + s 3 w 3 c α = v α + s α1 w 4 + s α2 w 5 + s α3 w 6, α = 3,, N denote the solutions to (318 Replace the above expressions for c α, α = 1, 2, 3,, N into (312 (317 This leads to a system of (3N 3 non linear conditions on the (3N 3 coefficients s j, s αj for j = 1, 2, 3, α = 3,, N Taing advantage of the explicit form of the vectors w i, i = 1,, 6, one can show that there exists a unique (s 1,, s 3, s 31,, s N3 R 3N 3 for which the above solutions satisfy all 3N 3 conditions of Proposition 31, which furthermore satisfy Hence, there exists a unique solution the estimate (s 1,, s 3, s 31,, s N3 Cl τ ϕ N α=1 c 1 c N to system (318, satisfying conditions (312 (317 c α Cl τ e 1 ξ 2 l ϕ On the other h, from (31 in Fact 1, we conclude that ϕ Ce 1+ξ 2 l N α=1 c α Thus, by combining the above two estimates we conclude that c i j,α =, ϕ =, which implies that for ϕ defined by (31-(33, it holds that ϕ = This proves Theorem 11 4 Analysis of the matrix M This section is devoted to the analysis of the matrix M defined in Section 3 We first derive a simplified form of M Our first observation is that if α is either of the indices {1, 2} β is any of the indices in {3,, N}, then L(Z t i,βz s j,α = for any i, j = 1,, 2n + m, s, t =,, 1 This fact implies that the matrix M has the form ( M1 M = M 2 where M 1 is a matrix of dimension (2 (2n + m 2 M 2 is a matrix of dimension ((N 2 (2n + m 2 (41

14 14 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI Because we can write that L(Zi,αZ s j,β t = ( A B M 1 = B t C L(Z t j,βz s i,α,, (42 where A, B, C are square matrices of dimension ((2n + m 2, with A, C symmetric More precisely, A = ( L(Zi,1Z s j,1 t i,j=1,,2n+m, s,t=,, 1, B = ( L(Zi,1Z s j,2 t i,j=1,,2n+m, s,t=,, 1, C = ( L(Zi,2Z s j,2 t i,j=1,,2n+m, s,t=,, 1 Furthermore, again by symmetry, because L(Z s i,αz t j,β =, if α β, α, β = 3,, N, the matrix M 2 has the form H 3 H 4 M 2 = H N where H α are square matrices of dimension [(2n + m ] 2, each of them is symmetric The matrices H α are defined by H α = ( L(Zi,αZ s j,α t s,t=,, 1 i,j=1,,2n+m for α = 3,, N (44 Thus, given the form of the matrix M as described in (41, (42, (43, system (318 is equivalent to ( ( c1 r1 M 1 =, H c 2 r α c α = r α, for α = 3,, N 2 where the vectors r α are defined in (318 This section is devoted to the analysis of the ernels eigenvalues of the matrices A, B, C, H α The main result of this section is the following solvability condition for the matrix M Proposition 41 Part a There exists l > such that for l > l, the system ( c1 M 1 = c 2 is solvable if ( r1 r 2 ( r1 w 1 = r 2 ( r1 r 2 ( r1 w 2 = r 2 w 3 = Furthermore, the general solution is ( ( c1 v1 = + s c 2 v 1 w 1 + s 2 w 2 + s 3 w 3 (45 2 ( v1 for all s i R, with being a fixed vector such that v 2 ( ( v1 Cl τ e l r1 (46 v 2 r 2 (43

15 NONDEGENERACY OF NONRADIAL SOLUTIONS 15 Part b Let α = 3,, N Then there exists l > such that for any l > l, is solvable if r α w 4 = r α w 5 = r α w 6 = Furthermore, the solution has the form for all s αi R, where v α a fixed vector such that H α (c α = r α (47 c α = v α + s α1 w 4 + s α2 w 5 + s α3 w 6, (48 v α Cl τ e l r α (49 Remar 41 From the statement of the above proposition, because M 1, M 2 are symmetric matrices, one need only show that M 1 has 3 dimensional ernels spanned by w 1, w 2, w 3, while H α has 3 dimensional ernels spanned by w 4, w 5, w 6 Before we prove the above proposition, we first need to introduce some notation For all n 2, we define the n n matrix T n = In practice, the integer n will be equal to m 1 or 2n 1 It is easy to chec that the inverse of T n is the matrix whose entries are given by (41 (T 1 n ij = min(i, j ij n + 1 (411 We define the vectors S S by 1 T n S := R n T n S := R n (412 1 It is simple to chec that S := n n+1 n 1 n+1 2 n+1 1 n+1 We also introduce the following vectors: d L, n = (c,, R n, R n S := 1 n+1 2 n+1 n 1 n+1 n n+1 R n (413 d R, n = (,,, c R n d n = (d, d,, d R n (414 In practice, n = m 1 or 2n 1 We will see below that a circulant matrix will play an important role in our proof We recall the definition of a circulant matrix

16 16 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI A circulant matrix X of dimension has the form x x 1 x 2 x 1 x 1 x x 1 x 2 X = x 1 x x 1, (415 x 1 x 1 x 1 x or equivalently, if x ij, i, j = 1,, are the entries of the matrix X, then x ij = x 1, i j +1 In particular, in order to determine a circulant matrix it is sufficient to now the entries of the first row By X = Cir{(x, x 1,, x 1 }, (416 we denote the above mentioned circulant matrix For the properties of circulant matrices, we refer the reader to [15] The eigenvalues of a circulant matrix X are given by the explicit formula 1 η s = x l e 2πs il, s =,, 1 (417 l=1 with corresponding normalized eigenvectors defined by 1 e 2πs E s = 1 2 e 2πs e 2πs i i2 i( 1 Observe that any circulant matrix X can be diagonalized as where D X is the diagonal matrix P is the matrix defined by X = P D X P t, D X = diag(η, η 1,, η 1 (418 P = (E E 1 E 1 (419 From this point on, we begin to analyze the components of the matrix M, A, B, C, H α, for which explicit expressions are given in Section 8 First, observe that M is a symmetric matrix 41 Analysis of H α proof of part (b of Proposition 41 We first analyze the ernels of the matrix H α First, we denote Ψ 2 ( l Ψ 2 (l = δ 2 2 sin π where Ψ 2 is defined in (82 By dividing both sides of the equation H α (c α = by Ψ 2 (l, we obtain that where H α = Hα Ψ 2(l H α (c α =,

17 NONDEGENERACY OF NONRADIAL SOLUTIONS 17 From the computations regarding H α in Section 7, we now that H α has the form H α,1 H α,2 H α = H α,3 H α,4 H α,5 Hα,2 t Hα,4 t H α,6 + O(e ξl, (42 Hα,5 t H α,7 where 1 δ2 δ 2 sin π 2 sin π δ 2 2 sin 1 π δ2 δ 2 sin π 2 sin π H α,1 = δ 2 2 sin 1 π δ2 δ 2 sin π 1 ( ( H α,5 = δ 2 2 sin π δ 2 2 sin π δ 2 2 sin π H α,2 = H α,3 = H α,4 = 1 L,m 1 m 1 m 1 m 1 1 L,m 1 m 1 m 1 1 L,m 1 δ 2 sin π δ 2 2 sin π δ 2 sin π [(m 1 ] 1 δ 2 sin 1 π 1 1 R,m 1 m 1 m 1 m 1 1 R,m 1 m 1 m 1 1 R,m 1 L,2n 1 2n 1 ( δ 2 δ 2 2 sin π 2 sin π, [(m 1 ],, δ2 sin π δ 2 2 sin π R,2n 1 R,2n 1 ( 2 sin π L,2n 1 2n 1 δ 2n 1 ( 2 δ R,2n 1 ( 2 H α,7 = H α,6 = δ 2 2 sin π T m 1 T m 1 T m 1 2 sin T π 2n 1 δ 2 2 sin T π 2n 1 2 sin π L,2n 1 [(m 1 ]2, δ 2 2 sin π T 2n 1 [(2n 1 ] 2 We want to analyze the eigenvalues of the matrix H α Thus, we assume that, [(2n 1 ] H α (a = (421,

18 18 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI First, by considering the third row of the matrix H α written in the form (42, one can obtain that a i 1 ( T m 1 + O(e ξl (a Y1,i + = O(e ξl a v a i m+1 (T 2n 1 + O(e ξl (a i Y 2 a i m+1 = O(e ξl a v a i+1 m+1 Here, a v corresponds to the unnown variables around the vertices of the inner outer polygons In other words, these equations imply that once a v is fixed, the other unnown variables will be determined From the above two equations, using (413, one has that for i =,, 1, a i 2 = 1 m( (m 1a i 1 + a i m+1 + O(e ξl a v a i m = 1 m( a i 1 + (m 1a i m+1 + O(e ξl a v, ( a i m+2 = 1 2n (2n 1a i m+1 + a i+1 m+1 + O(e ξl a v ( a i m+2n = 1 2n a i m+1 + (2n 1a i+1 m+1 + O(e ξl a v Next, we consider the first second rows of the matrix H α in (42 We can obtain that H α,3 a m+1 a 1 m+1 a 1 m+1 H α,1 + a 1 a 1 1 a 1 1 a m a 1 m + δ2 2 sin π δ2 2 sin π a 2 a 1 2 a 1 2 = O(e ξl a v (a m+2 + a 1 m+2n (a 1 m+2 a m+2n (a 1 m+2 a 2 m+2n (422 (423 a 1 m δ2 2 sin π By using the above two equations (422 (423 for a i 2, a i m, a i m+2, a i m+2n, the above two equations are reduced to a 2 system of 2 unnowns a i 1, a i m+1 for i =,, 1: H α a 1 a 1 1 a m+1 a 1 m+1 H α,i are all circulant matrices with = O(e ξl a v, where Hα = H α,1 = Cir{( 1 m δ 2 sin π, δ 2 = O(e ξl a v ( Hα,1 Hα,2 H t α,2 δ 2 H α,3 (424 2 sin π,,,, 2 sin π }, (425

19 NONDEGENERACY OF NONRADIAL SOLUTIONS 19 H α,2 = Cir{( 1,, }, (426 m H α,3 = Cir{( 1 m + δ 2 2n sin π, δ 2 4n sin π,,,, δ 2 4n sin π } (427 From the above analysis, equation (421 is equivalent to equation (424 above for the 2 variables a v Next, we begin to analyze the matrix H α Eigenvalues of H α,1 : A direct application of (417 gives that the eigenvalues of the matrix H α,1 are given by h 1,i = δ 2 sin π (cos 2πi 1 1 (428 m for i =,, 1 Eigenvalues of H α,3 : The eigenvalues of the matrix H α,3 are given by for i =,, 1 Define Then, simple algebra gives that where We consider the matrix The determinant of D i is given by δ 2 h 3,i = 4n sin π P = (2 2 cos 2πi 1 m (429 ( P P H α = P ( D1 D 2 D 2 D 3 (43 P t, D 1 = diag(h 1,,, h 1, 1, D 2 = diag( 1 m,, 1 m, Det(D i = 2δ 2 n D 3 = diag(h 3,,, h 3, 1 D i = ( h1,i 1 m 1 m h 3,i πi sin2 (1 sin2 πi sin 2 π (431 (1 + O( 1 (432 l One can chec that Det(D i = for i =, 1, 1, Det(D i c n for 2 i 2 From the above analysis, one can see that the matrix H α has at most three ernels, other than zero eigenvalues the eigenvalues will have a lower bound of C l for some τ > Moreover, one can chec τ directly that w 4, w 5, w 6 are in the ernels of H α, so one can obtain part (b of Proposition Analysis of the matrix M 1 proof of part (a of Proposition 41 First, we denote M 1 = 1 Ψ 1 (l M 1, where Ψ 1 is defined in (81, we introduce the following notations: Ψ 2 (l = σ 1 l Ψ 1(l, Ψ 2 ( l = σ 2 l Ψ 1( l (433 Ψ 1 ( l Ψ 1 (l = σ 3 2 sin π (434

20 2 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI for σ 1, σ 2, σ 3 > positive constants In fact, from (85 one can obtain that σ 3 = ( d l dl 1 By the computation in Section 8, we now that M 1 can be written in the following form: M 1 = A 1 11 A 1 12 B11 1 A 3 11 A 2 12 A 3 12 B12 1 A 1,t 12 A 2,t 12 A 1 13 A 3,t 12 A 2 13 B21 1 B 1,t where A 1 11, A 3 11 are circulant matrices: 11 C11 1 C12 1 B 1,t 21 C11 3 C12 2 C12 3 C 1,t 12 C 2,t 12 C13 1 B 1,t 12 C 3,t 12 C13 2 A 1 11 = Cir{(A 1 11,, A 1 11,1,,,, A 1 11, 1}, A 1 11, = 1 σ 3 sin π (sin 2 π + σ 2 π l cos2, A 1 11,1 = A 1 11, 1 = σ 3 2 sin π ( sin 2 π + σ 2 π l cos2, A 3 11 = Cir{( 1 + σ 3 sin π (sin 2 π + σ 2 π l cos2,,, }, 1 L,m 1 m 1 m 1 A 1 m 1 1 L,m 1 m 1 12 =, m 1 1 L,m 1 [(m 1 ] 1 R,m 1 m 1 m 1 A 2 m 1 1 R,m 1 m 1 12 =, m 1 1 R,m 1 [(m 1 ] σ 3 2 L,2n 1 2n 1 σ3 2 R,2n 1 σ3 σ 3 A 3 2 R,2n 1 12 = 2 L,2n 1 2n 1 2n 1 2n 1 2n 1 σ3 σ 3 2 R,2n 1 2 L,2n 1 A 2 13 = A 1 13 = σ 3 T m 1 T m 1 Tm 1 2 sin T π 2n 1 σ 3 2 sin T π 2n 1 σ 3 2 sin T π 2n 1 For the matrix B, we have that B 1 11 is a circulant matrix: B 1 11 = Cir{(, σ 3 cos π 2 [(m 1 ]2,, (435 [(2n 1 ] [(2n 1 ] 2 (1 + σ 2 l,,, σ 3 cos π (1 + σ 2 2 l },,

21 NONDEGENERACY OF NONRADIAL SOLUTIONS 21 B 1 12 = σ 2σ 3 cos π 2l sin π B 1,t 21 = σ 3 cos π 2 sin π 1 L,2n 1 2n 1 1 R,2n 1 1 R,2n 1 1 L,2n 1 2n 1 2n 1 2n 1 2n 1 1 R,2n 1 1 L,2n 1 1 L,2n 1 2n 1 1 R,2n 1 1 R,2n 1 1 L,2n 1 2n 1 2n 1 2n 1 2n 1 1 R,2n 1 1 L,2n 1 [(2n 1 ] [(2n 1 ], The matrices C 1 11, C 3 11 are circulant matrices: C12 3 = ( σ2σ3 C 1 11 = Cir{(C 1 11,, C 1 11,1,,, C 1 11, 1}, C11, 1 = σ 1 l σ 3 sin π (cos 2 π + σ 2 π l sin2, C11,1 1 = C11, 1 1 = σ 3 2 sin π (cos 2 π σ 2 π l sin2, C11 3 = Cir{( σ 1 l + σ 3 sin π C12 1 = C12 2 = (cos 2 π + σ 2 π l sin2,,, } σ 1 l L,m 1 m 1 m 1 σ 1 m 1 l L,m 1 m 1 m 1 σ 1 l L,m 1 σ 1 l R,m 1 m 1 m 1 σ 1 m 1 l R,m 1 m 1 m 1 σ 1 l R,m 1 [(m 1 ] [(m 1 ] R,2n 1 2l L,2n 1 2n 1 ( σ2σ3 2l ( σ2σ3 2l R,2n 1 ( σ2σ3 2l L,2n 1 2n 1 C 2 13 = 2n 1 2n 1 2n 1 ( σ2σ3 2l R,2n 1 ( σ2σ3 2l σ1 l C13 1 T m 1 = σ1 l T m 1 σ1 l T m 1 σ 2σ 3 2l sin T π 2n 1 σ 2σ 3 2l sin T π 2n 1 σ 2σ 3 2l sin π T 2n 1 L,2n 1, [(m 1 ]2, [(2n 1 ] 2 [(2n 1 ] The strategy for dealing with the matrix M 1 is similar to that for H α The main idea is that once the variables for the inner outer vertices are fixed, the other variables will be determined Thus, we will reduce the problem M 1 (a = to a 4 4 matrix equation for the variables around the 2 vertices,

22 22 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI First, by considering the third fourth rows of M1 (a = in the form (435, one can obtain that a i a i 1,1 2,1 ( T m 1 + O(e ξl a i 3,1 a i + = O(e ξl a v, (436 m 1,1 a i m,1 a i m+1,1 T 2n 1 a i m+2,1 a i m+3,1 a i m+2n 1,1 a i m+2n,1 + sin π ( T m 1 + O(e ξl a i m+1,1 a i+1 m+1,1 + cos π a i m+1,2 a i m+1,2 From the seventh eighth rows of the matrix M 1 in (435, we can obtain that a i 2,2 a i a i 1,2 3,2 T 2n 1 a i m+2, a i m+3,2 a i m+2n 1,2 a i m+2n,2 + sin π + a i m 1,2 a i m,2 a i m+1,2 a i+1 m+1,2 a i m+1,2 cos π = O(e ξl a v (437 = O(e ξl a v, (438 a i m+1,1 a i m+1,1 = O(e ξl a v (439 From the above four systems, using (413 one can solve a i Y, 1,j ai Y in terms of a 2,j v for i =,, 1, j = 1, 2 In particular, we can obtain that a i 2,1 = 1 m ((m 1ai 1,1 + a i m+1,1 + O(e ξl a v, (44 a i m,1 = 1 m (ai 1,1 + (m 1a i m+1,1 + O(e ξl a v, a i m+2,1 = sin π 2n ((2n 1ai m+1,1 a i+1 m+1,1 cos π 2n ((2n 1ai m+1,2 + a i+1 m+1,2 + O(e ξl a v a i m+2n,1 = sin π 2n (ai m+1,1 (2n 1a i+1 m+1,1 cos π 2n (ai m+1,2 + (2n 1a i+1 m+1,2 + O(e ξl a v, a i 2,2 = 1 m ((m 1ai 1,2 + a i m+1,2 + O(e ξl a v, a i m,2 = 1 m (ai 1,2 + (m 1a i m+1,2 + O(e ξl a v, a i m+2,2 = sin π 2n ((2n 1ai m+1,2 a i+1 m+1,2 + cos π 2n ((2n 1ai m+1,1 + a i+1 m+1,1 + O(e ξl a v a i m+2n,2 = sin π 2n (ai m+1,2 (2n 1a i+1 m+1,2 + cos π 2n (ai m+1,1 + (2n 1a i+1 m+1,1 + O(e ξl a v (441 (442 (443

23 NONDEGENERACY OF NONRADIAL SOLUTIONS 23 From the first second rows of the equation in (435, one can obtain that A 1 11 a 1,1 a 1 1,1 + a 2,1 a 1 2,1 + σ 3 cos π (1 + σ 2 2 l a 1 1,2 a 1 1,2 a 2 1,2 a 1,2 a 1,2 a 2 1,2 = O(e ξl a v, (444 A 3 11 a m+1,1 a 1 m+1,1 + a m,1 a 1 m,1 + σ 3 2 σ 2σ 3 cos π 2l sin π a m+2,1 a 1 m+2n,1 a 1 m+2,1 a m+2n,1 a 1 m+2,1 a 2 m+2n,1 a m+2,2 + a 1 m+2n,2 a 1 m+2,2 + a m+2n,2 = O(e ξl a v a 1 m+2,2 + a 2 m+2n,2 (445 From the fifth sixth rows of (435, one can obtain that C 1 11 a 1,2 a 1 1,2 + c 1 l a 2,2 a 1 2,2 σ 3 cos π (1 + c 2 2 l a 1 1,1 a 1 1,1 a 2 1,1 a 1,1 a 1,1 a 2 1,1 = O(e ξl a v, (446 C 3 11 a m+1,2 a 1 m+1,2 + σ 1 l a m,2 a 1 m,2 + σ 2σ 3 2l + σ 3 cos π 2 sin π a m+2,2 a 1 m+2n,2 a 1 m+2,2 a m+2n,2 a 1 m+2,2 a 2 m+2n,2 a m+2,1 + a 1 m+2n,1 a 1 m+2,1 + a m+2n,1 = O(e ξl a v a 1 m+2,1 + a 2 m+2n,1 (447 Using the equations for a i 2,j, ai m,j ai m+2,j, ai m+2n,j (44-(443, the above system (444-(447 can be reduced to 4 equations in terms of 4 unnowns a 1,1,, a 1 1,1, a m+1,1,, a 1 m+1,1, a 1,2,, a1,2 1,

24 24 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI a m+1,2,, a 1 m+1,2 : F 11 F 12 F 13 F12 t F 22 F 24 F13 t F 33 F 34 F24 t F34 t F 44 where F ij are circulant matrices given below: The matrix F 11 F 11 is defined by a 1,1 a 1 1,1 a m+1,1 a 1 m+1,1 a = O(e ξl a 1,2 v (448 a 1 1,2 a m+1,2 a 1 m+1,2 F 11 = Cir{(F 11,, F 11,1,,,, F 11, 1 }, (449 where F 11, = 1 m σ 3 sin π (sin 2 π + σ 2 π l cos2 F 11,1 = F 11, 1 = σ 3 2 sin π ( sin 2 π + σ 2 π l cos2 Eigenvalues of F 11 For any l =,, 1, the eigenvalues of F 11 are f 11,l = 1 m σ 3 sin π (sin 2 π + σ 2 π l cos2 + σ 3 sin π ( sin 2 π + σ 2 π 2lπ l cos2 cos The matrix F 12 The matrix F 12 is defined by F 12 = Cir{( 1,,, } m Eigenvalues of F 12 For any l =,, 1, the eigenvalues of F 12 are The matrix F 13 The matrix F 13 is defined by where f 12,l = 1 m F 13 = Cir{(, F 13,1,,,, F 13, 1 }, F 13,1 = σ 3 cos π (1 + σ 2 2 l, F 13, 1 = σ 3 cos π (1 + σ 2 2 l Eigenvalues of F 13 For any l =,, 1, the eigenvalues of F 12 are The matrix F 22 The matrix F 22 is defined by where f 13,l = iσ 3 (1 + σ 2 l cos π 2lπ sin (45 F 22 = Cir{(F 22,, F 22,1,,,, F 22, 1 }, F 22, = 1 m + σ 3 2n sin π (sin 2 π + σ 2 π l cos2

25 NONDEGENERACY OF NONRADIAL SOLUTIONS 25 σ 3 F 22,1 = F 22, 1 = 4n sin π (sin 2 π σ 2 π l cos2 Eigenvalues of F 22 For any l =,, 1, the eigenvalues of F 22 are f 22,l = 1 m + σ 3 2n sin π (sin 2 π + σ 2 π l cos2 + σ 3 2n sin π (sin 2 π σ 2 π 2lπ l cos2 cos (451 The matrix F 24 The matrix F 24 is defined by where F 24 = Cir{(, F 24,1,,,, F 24, 1 }, F 24,1 = σ 3 sin π (1 + σ 2 4n l, F 24, 1 = σ 3 sin π (1 + σ 2 4n l Eigenvalues of F 24 For any l =,, 1, the eigenvalues of F 24 are The matrix F 33 The matrix F 33 is defined by f 24,l = iσ 3 2n (1 + σ 2 l cos π 2lπ sin F 33 = Cir{(F 33,, F 33,1,,,, F 33, 1 }, where F 33, = σ 1 ml σ 3 sin π (cos 2 π + σ 2 π l sin2 F 33,1 = F 33, 1 = σ 3 2 sin π (cos 2 π σ 2 π l sin2 Eigenvalues of F 33 For any l =,, 1, the eigenvalues of F 33 are f 33,l = σ 1 ml σ 3 sin π (cos 2 π + σ 2 π l sin2 + σ 3 sin π (cos 2 π σ 2 π 2lπ l sin2 cos The matrix F 34 The matrix F 34 is defined by F 34 = Cir{( c 1,,, } ml Eigenvalues of F 34 For any l =,, 1, the eigenvalues of F 34 are The matrix F 44 The matrix F 44 is defined by f 34,l = σ 1 ml F 44 = Cir{(F 44,, F 44,1,,,, F 44, 1 }, where F 44, = σ 1 ml + σ 3 2n sin π (cos 2 π + σ 2 π l sin2 F 44,1 = F 44, 1 = σ 3 4n sin π (cos 2 π σ 2 π l sin2 Eigenvalues of F 44 For any l =,, 1, the eigenvalues of F 44 are f 44,l = σ 1 ml + σ 3 2n sin π (cos 2 π + σ 2 π l sin2 σ 3 2n sin π (cos 2 π σ 2 π 2lπ l sin2 cos

26 26 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI The final part of this section is devoted to the analysis of the matrix F 11 F 12 F 13 F = F12 t F 22 F 24 F13 t F 33 F 34 (452 F24 t F34 t F 44 Define P 1 = P P P P where P is defined in (419 Simple algebra gives that F = P 1, D F11 D F12 D F13 D F t 12 D F22 D F24 D F t 13 D F33 D F34 D F t 24 D F t 34 D F44 Pt 1 Here, D X denotes the diagonal matrix of dimension whose entries are given by the eigenvalues of X Let us now introduce the following matrix: where D F = D fi = D f D f1 D f 1 f 11,i f 12,i f 13,i f 12,i f 22,i f 24,i f 13,i f 33,i f 34,i f 24,i f 34,i f 44,i By direct calculation, one can chec that for j =, Det(D f =, D f has only one ernel The other eigenvalues of D f will satisfy λ,i C l for some constant τ C, τ > For j 1, we have that n where Det(D fj =, 2σ 2 3 (1 + d 2 2 (1 ā b j (ā(1 b j + d 2 (1 ā d j ( b j (1 ā b j d2 ā (ā b j 2 d 2 ( b j d2 + ā 2 (1 + d 1 (1 + d 2 ā(1 + (2 + d 1 d 2 ā b j (1 ā(1 + d 2 2 (ā + (ā 1 d 2 (ā( b j 1 + (ā 1 b j d2 = (ā b j 2 n d 2 2σ 2 3ā3 b 3 j (1 ā[( b j c d 2 ] (1 + O(1 l, ā = sin 2 π, b j = sin 2 jπ, d1 = σ 1 l, d2 = σ 2 l From the above computation, we now that for j = 1, 1, Det(D fj =, the matrix D fj has one ernel, all the other eigenvalues have a lower bound C l For j, 1, 1, the matrix D τ fj is non-degenerate, the eigenvalues have a lower bound C l τ From the above analysis, we now that M 1 has three ernels Furthermore, the other eigenvalues of M 1 have a lower bound C l Moreover, we can chec that w τ 1, w 2, w 3 are in the ernels of M1 Thus, we have proved part (a of Proposition 41

27 NONDEGENERACY OF NONRADIAL SOLUTIONS 27 5 Proof of (31 First, we have the following a priori estimate on the linear equation Proposition 51 Assume that h is a function such that h <, where the norm is defined in (214 Let φ be a solution of the following equation: L(φ = h, φzj,α i = for i =,, 1, j = 1,, m + 2n, α = 1,, N (51 Then, for large l, there exists a constant C independent of l such that φ satisfies the following estimate: φ C h (52 Proof This can be proved by contradiction Assume that there exists l n φ n, h n corresponding to (51 such that φ n = 1, h n as n (53 In the following, we omit the index n in the absence of ambiguity Proposition 31 in [19], one can obtain that there exists R i y j such that Following the argument in φ L (B(R i yj,ρ C >, (54 for some fixed C large ρ By employing elliptic estimates together with the Ascoli-Arzela theorem, we can find a sequence R i y j such that φ(x + R i y j converges to φ, which is a solution of φ φ + pw p 1 φ =, satisfies the following orthogonality conditions: φ w x α =, α = 1,, N Thus, φ = This contradicts (54, so this completes the proof Because one can easily verify that L(Z i j,α = p( u p 1 w p 1 (x R i y j Z i j,α + O(e 1+ξ 2 l, L(Z i j,α Ce 1+ξ 2 l (55 for some ξ that is independent of l, which is assumed to be large, where we have applied the estimate (218 Thus, from Proposition 51 the estimate (55, we obtain that ϕ Ce 1+ξ Let us consider (311, with β = 1 That is, 2 l N α=1 6 Proof of Proposition 31 c α (56 N α=1 c α Z α z 1 = ϕ z 1 (61 ϕ z 1 First, we write f 1 = A straightforward computation gives that f w(x ( 1 C(2n + m x 2 1 ϕ l τ ϕ, for a certain constant τ that is independent of l, which is assumed to be large,

28 28 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI where we have assumed that m, n satisfy (29 Second, by direct computation we find for j = 1,, m + 1 that u Zj,1z i 1 = R i w(x R x y i j dx + O(e 1+ξ 2 l 1 = + B l2 (R i yj R N B l2 (R i yj = cos θ i Similarly, one can obtain that Zj,2z i 1 = for α = 3,, N, u x 1 R i w(x R i y j dx u R i w(x R x y i j dx + O(e 1+ξ 2 l 1 ( w 2 dx + O(e (1+ξl 2 x 1 u x 1 R i = sin θ i Zj,αz i 1 = w(x Ry i j dx + O(e 1+ξ 2 l ( w 2 dx + O(e (1+ξl 2, x 1 u w(x R x 1 x y i j =, α by the evenness of u in x α Moreover, for j = m + 2,, m + 2n we have that Zj,1z i 1 = sin(θ i + π ( w 2 + O(e (1+ξl 2, x 1 N α=1 c α Z i j,2z 1 = cos(θ i + π Z α z 1 = c 1 Z i j,αz 1 = for α = 3,, N A direct consequence of the above calculation is that cos cos cos(θ +O(e (1+ξl 2 L ( w 2 + O(e (1+ξl 2, x 1 cos(θ 1 + c 2 sin(θ + π sin(θ + π c 1, c N sin sin sin(θ sin(θ 1 cos(θ + π cos(θ + π where L is a linear function whose coefficients are uniformly bounded in l as l to Thus, (312 follows straightforwardly The proofs of (313 to (317 are similar, left to the reader

29 NONDEGENERACY OF NONRADIAL SOLUTIONS 29 7 Proof of Proposition 32 In this section, we prove Proposition 32 A ey ingredient in the proof is the estimates on the right h side of (318 We have the following result Proposition 71 There exist positive constants C ξ such that for α = 1,, N, it holds that for any sufficiently large l Proof Recall that Then, the estimate follows from r α Ce 1+ξ 2 l ϕ (71 r α = L(ϕ Z v,α L(ϕ Z Y 1,α L(ϕ Z 1 Y 1,α L(ϕ ZY 2,α L(ψ Z 1 Y 2,α L(ϕ Z i j,α Ce 1+ξ 2 l ϕ To prove the above estimate, we fix, for example j = 1, i =, α = 3, we write L(ϕ Z1,3 = L(Z1,3ϕ = L( w(x y 1 ϕ + L( π 3 x 3 ϕ = p( u p 1 w p 1 (x y 1 w(x y 1 ϕ + O(e 1+ξ 2 l ϕ x 3 C w p 2 (x y 1 w(x y 1 w(x z + O(e 1+ξ 2 l ϕ x 3 z Π y1 Ce 1+ξ 2 l ϕ for some ξ > that is independent of l, which is assumed to be large, where we have used the estimate for φ in (218, ie, φ Ce 1+ξ 2 l Thus, we have proved the estimate for α = 3 The other cases can be treated similarly We have now the tools for the following proof Proof of Proposition 32 By Proposition 41, we need only show the following orthogonality conditions: ( r1 r 2 ( r1 w 1 = r 2 ( r1 w 2 = r 2 w 3 = (72 r α w 4 = r α w 5 = r α w 6 = (73

30 3 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI First, recall that L(z 1 = Then, we have that L(z 1 ϕ = L(ϕ z 1 = This gives us precisely the first orthogonality condition in (72 Similarly, from L(z 2 = one can obtain that L(z 2 ϕ = L(ϕ z 2 = This gives us precisely the second orthogonality condition in (72 For α = 3,, N, it follows from L(z α = that L(z α ϕ = L(ϕ z α = This gives us precisely the first orthogonality condition in (73 Second, let us recall that N L(ϕ = c α L(Z α (74 α=1 Thus, the function x L(ϕ (x is invariant under a rotation by the angle 2π in the (x 1, x 2 plane Therefore, we can obtain that 1 m+1 m+2n ( L(ϕ y j Zj,2 i + L(ϕ (Ry i j n i Zj,1 i Ry i j t i Zj,2 i = i= j=1 j=m+2 This gives us the third orthogonality condition in (72 For α = 3,, N, we can obtain that 1 2m+n L(ϕ Zj,αR i y i j e 1 =, i= 1 i= j=1 2m+n j=1 L(ϕ Z i j,αr i y j e 2 = These give the final two orthogonality conditions in (73 By combining the results of Proposition 41 the a priori estimates in (71, we obtain the proof of Proposition 32 8 Some useful computations In this section, we compute the entries of the matrices A, B, C, H α for α = 3,, N We first introduce the following useful functions: Ψ 1 (l = div(w p (xediv(w(x leedx, (81 Ψ 2 (l = div(w p (xe div(w(x lee dx, (82 where e is any unit vector It is simple to chec that this definition is independent of the choice of the unit vector e It is nown that Ψ 1 (l = C N,p,1 e l l N 1 2 (1 + O( 1 (83 l Ψ 2 (l = C N,p,2 e l l N+1 2 (1 + O( 1 l, (84 where C N,p,i > are constants that depend only on p N See, for example, [19, 18] for details

31 NONDEGENERACY OF NONRADIAL SOLUTIONS 31 In fact, one can see from the definitions of Ψ 1 Ψ that Ψ 1(l = 2 sin π Ψ 1( l d l dl (85 By these two definitions, one can easily obtain that pw p 1 a w(xb w(x le = (a e(b eψ 1 (l + (a e (b e Ψ 2 (l (86 Computation of A L(Z1,1Z 1,1dx = p( u p 1 w p 1 (x y 1 ( w(x y 1 2 x 1 = p(p 1 w p 2 (x y 1 (φ + w(x z( w(x y 1 2 x 1 z Π y1 + O(e (1+ξl, where Π y1 denotes the set of closest neighbors of y 1 of Π defined in (23 Recall that φ solves the following equation: where Hence, one has that p(p 1 = = = = = = = φ φ + p U p 1 φ + E + N(φ =, (87 E = U U + U p 1 U N(φ = U + φ p 1 (U + φ U p 1 U p U p 1 φ w p 2 φ( w(x y 1 x 1 2 (pw p 1 (x y 1 φ w(x y 1 x 1 x 1 pw p 1 (x y 1 (φ w(x y 1 x 1 x 1 pw p 1 (x y 1 w(x y 1 x 1 φ x 1 pw p 1 (x y 1 φ 2 w(x y 1 x 2 1 [p( U p 1 w p 1 (x y 1 φ + E + N(φ] 2 w(x y 1 x 2 1 E 2 w(x y 1 x O(e (1+ξl ( U p 1 U w p (x y 1 2 w(x y 1 x 2 1 pw p 1 (x y 1 ( + O(e (1+ξl w(x z 2 w(x y 1 x 2 + O(e (1+ξl z Π y1 1

32 32 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI Taing this into account, we have that L(Z1,1Z 1,1dx = p(p 1 w p 2 ( w(x z( w(x y 1 2 x 1 z Π y1 + pw p 1 (x y 1 ( w(x z 2 w(x y 1 x 2 + O(e (1+ξl z Π y1 1 = (pw p 1 (x y 1 w(x y 1 w(x z + O(e (1+ξl x 1 x 1 z Π y1 = p w p 1 (x y 1 w(x y 1 w(x z + O(e (1+ξl x 1 x 1 z Π y1 By (86, we can obtain that L(Z 1,αZ 1,αdx = [Ψ 1 (l + 2(Ψ 1 ( l sin 2 π + Ψ 2( l cos 2 θ ] Next, we consider L(Z1,1Z 1,1 1 = = (p u p 1 pw p 1 (x y 1 w(x y 1 x 1 R 1 w(x R 1 y 1 pw p 1 (x R 1 y 1 w(x y 1 x 1 R 1 w(x R 1 y 1 + O(e (1+ξl = sin 2 π Ψ 1( l + cos 2 π Ψ 2( l + O(e (1+ξl Similarly, one can obtain that L(Z 1,1Z 1 1,1 = sin 2 π Ψ 1( l + cos 2 π Ψ 2( l + O(e (1+ξl, L(Z 1,1Z j i,1 = O(e (1+ξl for (i, j (1,, (1, 1, (1, 1, (2, Another observation is that L(Zi,1Z s j,1 t = L(Z i,1z t s j,1, where we use the notation Z t s j,1 = Z+t s j,1 if t s < Moreover, for i 2, it holds that 2Ψ 1 (l + O(e (1+ξl, if i = j, s = t, i m, Ψ 1 (l + O(e (1+ξl if j = i 1 or i + 1, s = t i m, 2Ψ 1 ( l + O(e (1+ξl if i = j, s = t, m + 2 i 2n + m, L(Zi,1Z s j,1 t = Ψ 1 ( l + O(e (1+ξl if j = i 1 or i + 1, s = t, m + 2 i 2n + m, [Ψ 1 (l 2(Ψ 1 ( l sin 2 π + Ψ 2( l cos 2 π ] + O(e (1+ξl if i, j = m + 1, s = t, Ψ 1 ( l sin π + O(e (1+ξl if (i, j = (m + 1, m + 2, s = t, Ψ 1 ( l sin π + O(e (1+ξl if (i, j = (m + 1, m + 2n, t = s 1, O(e (1+ξl otherwise

33 NONDEGENERACY OF NONRADIAL SOLUTIONS 33 Computation of C: Similarly, we have that L(Z1,2Z 1,2 = p w p 1 (x y 1 w(x y 1 w(x z + O(e (1+ξl x 2 x 2 z Π y1 [ ( = Ψ 2 (l + 2 Ψ 1 ( l cos 2 π + Ψ 2( l sin 2 π ] + O(e (1+ξl Ψ 1 ( l cos 2 π Ψ 2( l sin 2 π + O(e (1+ξl L(Z1,2Z j i,2 = if (i, j = (1, 1 or (1, 1, O(e (1+ξl otherwise Furthermore, for i 2 one can obtain that 2Ψ 2 (l + O(e (1+ξl, if i = j, s = t, i m, (88 L(Zi,2Z s j,2 t = Ψ 2 (l + O(e (1+ξl if j = i 1 or i + 1, s = t i m, 2Ψ 2 ( l + O(e (1+ξl if i = j, s = t, m + 2 i 2n + m, Ψ 2 ( l + O(e (1+ξl if j = i 1 or i + 1, s = t, m + 2 i 2n + m, [Ψ 2 (l 2(Ψ 1 ( l cos 2 π + Ψ 2( l sin 2 π ] + O(e (1+ξl if i, j = m + 1, s = t, Ψ 2 ( l sin π + O(e (1+ξl if (i, j = (m + 1, m + 2, s = t, Ψ 2 ( l sin π + O(e (1+ξl if (i, j = (m + 1, m + 2n, t = s 1, O(e (1+ξl otherwise Computation of B: Next, we consider L(Zi,1 s Zt j,2 L(Zi,2 s Zt j,1 First, by the symmetry we have that L(Z1,1Z 1,2 = L(Z1,1Z 1,2 1 = pw p 1 (x Ry 1 1 w(x y 1 R 1, w(x R x y = sin π cos π ( Ψ 1 ( l + Ψ 2 ( l + O(e (1+ξl Similarly, we can obtain that L(Z1,1Z 1,2 1 = sin π cos π ( Ψ 1 ( l + Ψ 2 ( l + O(e (1+ξl, L(Zm+1,1Z m+2,2 = L(Z m+1,1z 1 2n+m,2 = Ψ 2( l cos π + O(e (1+ξl, L(Z s i,1z t j,2 = O(e (1+ξl otherwise

34 34 WEIWEI AO, MONICA MUSSO, AND JUNCHENG WEI Similarly, we have the following expansion for L(Zi,2 s Zt j,1 : ( sin π cos π Ψ 1 ( l + Ψ 2 ( l + O(e (1+ξl if i, j = 1, t = s 1, sin π cos π L(Zi,2Z s j,1 t = ( Ψ 1 ( l + Ψ 2 ( l + O(e (1+ξl if i, j = 1, t = s + 1, Ψ 1 ( l cos π + O(e (1+ξl if (i, j = (m + 2, m + 1, s = t or (i, j = (2n + m, m + 1, t = s + 1, O(e (1+ξl otherwise Computation of H α : For the matrix H α for α = 3,, N, the computation is simpler, we directly employ (86 to obtain the following expansion: ( Ψ 2 (l + 2Ψ 2 ( l + O(e (1+ξl if (i, j = (1, 1, s = t, Ψ 2 ( l + O(e (1+ξl if (i, j = (1, 1, t = s 1 or s + 1, ( 2Ψ 2 ( l Ψ 2 (l + O(e (1+ξl if (i, j = (m + 1, m + 1, s = t, L(Zi,αZ s j,α t = 2Ψ 2 (l + O(e (1+ξl if i = j, s = t, 2 i m, Ψ 2 (l + O(e (1+ξl if j = i + 1 or i 1, s = t, 2 i m, 2Ψ 2 ( l + O(e (1+ξl if i = j, s = t, m + 2 i m + 2n, Ψ 2 ( l + O(e (1+ξl if j = i + 1 or i 1, s = t, m + 2 i m + 2n, O(e (1+ξl otherwise References [1] WW Ao, M Musso, F Pacard JC Wei, Solutions without any symmetry for semilinear elliptic problems, J Func Anal, 27 (216, no3, [2] H Berestyci PL Lions, Nonlinear scalar field equations I Existence of a ground state, Arch Rational Mech Anal, 82 (1983, [3] H Berestyci PL Lions, Nonlinear scalar field equations, II, Arch Rat Mech Anal 82 (1981, [4] T Bartsch M Willem, Infinitely many radial solutions of a semilinear elliptic problem on R N, Arch Rational Mech Anal 124 (1993, [5] EN Dancer, New solutions of equations on R n Ann Scuola Norm Sup Pisa Cl Sci (4 3 (22, no 3-4, [6] M del Pino, M Musso, F Pacard A Pistoia, Large energy entire solutions for the Yamabe equation J Differential Equations 251 (211, no 9, [7] M del Pino, M Musso, F Pacard A Pistoia, Torus action on Sn sign-changing solutions for conformally invariant equations Ann Sc Norm Super Pisa Cl Sci (5 12 (213, no 1, [8] M del Pino, M Kowalcyz, F Pacard J Wei, The Toda system multiple-end solutions of autonomous planar elliptic problems, Advances in Mathematics 224 (21, [9] T Duycaerts, C Kenig F Merle, Solutions of the focusing nonradial critical wave equation with the compactness property, Mathematics, 15 (214, [1] M Jleli F Pacard, An end-to-end construction for compactconstant mean curvature surfaces Pacific Journal of Maths, 221 (25, no [11] B Gidas, WM Ni L Nirenberg, Symmetry related properties via the maximun principle, Comm Math Phys 68 (1979, [12] N Kapouleas, Complete constant mean curvature surfaces in Euclidean three space, Ann of Math 131 (199, [13] N Kapouleas, Compact constant mean curvature surfaces, J Differential Geometry, 33 (1991,