NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM

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1 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM MONICA MUSSO AND JUNCHENG WEI Abstract: We prove the existence of a sequence of nondegenerate, in the sense of Duyckaerts-Kenig-Merle [9], nodal nonradial solutions to the critical Yamabe problem Q = Q n Q, Q D, (R n ). This is the first example in the literature of nondegeneracy for nodal nonradial solutions of nonlinear elliptic equations and it is also the only nontrivial example for which the result of Duyckaerts-Kenig-Merle [9] applies.. Introduction In this paper we consider the critical Yamabe problem n(n ) 4 (.) u = u n u, u D, (R n ) 4 where n 3 and D, (R n ) is the completion of C (Rn ) under the norm R n u. If u > Problem (.) is the conformally invariant Yamabe problem. For signchanging u Problem (.) corresponds to the steady state of the energy-critical focusing nonlinear wave equation (.) t u u u n u =, (t, x) R R n. 4 These are classical problems that have attracted the attention of several researchers in order to understand the structure and properties of the solutions to Problems (.) and (.). Denote the set of non-zero finite energy solutions to Problem (.) by { } (.3) Σ := Q D, (R n n(n ) 4 )\{} : Q = Q n Q. 4 This set has been completely characterized in the class of positive solutions to Problem (.) by the classical work of Caffarelli-Gidas-Spruck [5] (see also [, 4, 3]): all positive solutions to (.) are radially symmetric around some point a R n and are of the form (.4) W λ,a (x) = ( λ ) n λ + x a, λ >. The research of the first author has been partly supported by Fondecyt Grant 5. The research of the second author is partially supported by NSERC of Canada.

2 MONICA MUSSO AND JUNCHENG WEI Much less is known in the sign-changing case. A direct application of Pohozaev s identity gives that all sign-changing solutions to Problem (.) are non-radial. The existence of elements of Σ that are nonradial sign-changing, and with arbitrary large energy was first proved by Ding [6] using Ljusternik-Schnirelman category theory. Indeed, via stereographic projection to S n Problem (.) becomes S nv + n(n ) 4 4 ( v n v v) = in S n, (see for instance [3], [4]) and Ding showed the existence of infinitely many critical points to the associated energy functional within functions of the form v(x) = v( x, x ), x = (x, x ) S n R n+ = R k R n+ k, k, where compactness of critical Sobolev s embedding holds, for any n 3. No other qualitative properties are known for the corresponding solutions. Recently more explicit constructions of sign changing solutions to Problem (.) have been obtained by del Pino-Musso-Pacard-Pistoia [7, 8]. However so far only existence is available, and there are no rigidity results on these solutions. The main purpose of this paper is to prove that these solutions are rigid, up to the transformations of the equation. In other words, these solutions are nondegenerate, in the sense of the definition introduced by Duyckaerts-Kenig-Merle in [9]. Following [9], we first find out all possible invariances of the equation (.). Equation (.) is invariant under the following four transformations: () (translation): If Q Σ then Q(x + a) Σ, a R n ; () (dilation): If Q Σ then λ n Q(λx) Σ, λ > ; (3) (orthogonal transformation): If Q Σ then Q(Px) Σ where P O n and O n is the classical orthogonal group; (4) (Kelvin transformation): If Q Σ then x N Q( x x ) Σ. If we denote by M the group of isometries of D, (R n ) generated by the previous four transformations, a result of Duyckaerts-Kenig-Merle [Lemma 3.8,[9]] states that M generates an N parameter family of transformations in a neighborhood of the identity, where the dimension N is given by (.5) N = n + + In other words, if Q Σ we denote L Q := n(n ). n(n + ) 4 Q 4 n the linearized operator around Q. Define the null space of L Q (.6) Z Q = { f D, (R n ) : L Q f = }

3 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 3 The elements in Z Q generated by the family of transformations M define the following vector space ( n)x j Q + x x j Q x j x Q, x j Q, j n, (.7) Z Q = span n (x j xk x k x j )Q, j < k n, Q + x Q. Observe that the dimension of Z Q is at most N, but in principle it could be strictly less than N. For example in the case of the positive solutions Q = W, it turns out that the dimension of Z Q is n + as a consequence of being Q radially symmetric. Indeed, it is known that { } n (.8) Z W = W + x W, x j W, j n. Duyckaerts-Kenig-Merle [9] introduced the following definition of nondegeneracy for a solution of Problem (.): Q Σ is said to be nondegenerate if (.9) Z Q = Z Q. So far the only nondegeneracy example of Q Σ is the positive solution W. The proof of this fact relies heavily on the radial symmetry of W and it is straightforward: In fact since Q = W is radially symmetric (around some point) one can decompose the linearized operator into Fourier modes, getting (.9) as consequence of a simple ode analysis. See also [7]. In the case of nodal (nonradial) solutions this strategy no longer works out. In fact, as far as the authors know, there are no results in the literature on nondegeneracy of nodal nonradial solutions for nonlinear elliptic equations in the whole space. For positive radial solutions there have been many results. We refer to Frank-Lenzmann [], Frank-Lenzmann-Silvestre [3], Kwong [] and the references therein. The knowledge of nondegeneracy is a crucial ingredient to show the soliton resolution for a solution to the energy-critical wave equation (.) with the compactness property obtained by Kenig and Merle in [6, 7]. If the dimension n is 3, 4 or 5, and under the above nondegeneracy assumption, they prove that any non zero such solution is a sum of stationary solutions and solitary waves that are Lorentz transforms of the former. See also Duyckaerts, Kenig and Merle [, ]. Nondegeneracy also plays a vital role in the study of Type II blow-up solutions of (.). We refer to Krieger, Schlag and Tataru [], Rodnianski and Sterbenz [6] and the references therein. The main result of this paper can be stated as follows: Main Result: There exists a sequence of nodal solutions to (.), with arbitrary large energy, such that they are nondegenerate in the sense of (.9). Now let us be more precise. Let (.) f (t) = γ t p t, for t R, and p = n + n.

4 4 MONICA MUSSO AND JUNCHENG WEI The constant γ > is chosen for normalization purposes to be n(n ) γ =. 4 In [7], del Pino, Musso, Pacard and Pistoia showed that Problem (.) u + f (u) = in R n, admits a sequence of entire non radial sign changing solutions with finite energy. To give a first description of these solutions, let us introduce some notations. Fix an integer k. For any integer l =,..., k, we define angles θ l and vectors n l, t l by (.) θ l = π k (l ), n l = (cos θ l, sin θ l, ), t l = ( sin θ l, cos θ l, ). Here stands for the zero vector in R n. Notice that θ =, n = (,, ), and t = (,, ). In [7] it was proved that there exists k such that for all integer k > k there exists a solution u k to (.) that can be described as follows (.3) u k (x) = U (x) + ϕ(x) where (.4) U (x) = U(x) U j (x), while ϕ is smaller than U. The functions U and U j are positive solutions to (.), respectively defined as ( ) n (.5) U(x) = + x, U j (x) = µ n k U(µ k (x ξ j)). For any integer k large, the parameters µ k > and the k points ξ l, l =,..., k are given by (.6) ( cos θ l ) µ n n k = ( + O( k ) ), for k l> in particular, as k, we have Furthermore µ k k if n 4, µ k k log k if n = 3. (.7) ξ l = j= µ (n l, ). The functions U, U j and U are invariant under rotation of angle π k in the x, x plane, namely (.8) U(e π k x, x ) = U( x, x ), x = (x, x 3 ), x = (x 3,..., x n ). They are even in the x j -coordinates, for any j =,..., n (.9) U(x,..., x j,..., x n ) = U(x,..., x j,..., x n ), j =,..., n

5 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 5 and they respect invariance under Kelvin s transform: (.) U(x) = x n U( x x). In (.3) the function ϕ is a small function when compared with U. We will further describe the function u, and in particular the function ϕ in Section. Let us just mention that ϕ satisfies all the symmetry properties (.8), (.9) and (.). Recall that Problem (.) is invariant under the four transformations mentioned before: translation, dilation, rotation and Kelvin transformation. These invariances will be reflected in the element of the kernel of the linear operator (.) L(φ) := φ + f (u k )φ = φ + pγ u k p u φ which is the linearized equation associated to (.) around u k. From now on, for simplicity we will drop the label k in u k, so that u will denote the solution to Problem (.) described in (.3). Let us introduce the following set of 3n functions (.) z (x) = n u(x) + u(x) x, (.3) z α (x) = u(x), x α for α =,..., n, and (.4) z n+ (x) = x u(x) + x u(x) x x where u is the solution to (.) described in (.3). Observe that z n+ is given by z n+ (x) = θ [u(r θx)] θ= where R θ is the rotation in the x, x plane of angle θ. Furthermore, (.5) z n+ (x) = x z (x) + x z (x), z n+3 (x) = x z (x) + x z (x) for l = 3,..., n (.6) z n+l+ (x) = x l z (x) + x z l (x), u n+l (x) = x l z (x) + x z l (x). The functions defined in (.5) are related to the invariance of Problem (.) under Kelvin transformation, while the functions defined in (.6) are related to the invariance under rotation in the (x, x l ) plane and in the (x, x l ) plane respectively. The invariance of Problem (.) under scaling, translation, rotation and Kelvin transformation gives that the set Z Q (introduced in (.7)) associated to the linear operator L introduced in (.) has dimension at least 3n, since (.7) L(z α ) =, α =,..., 3n. We shall show that these functions are the only bounded elements of the kernel of the operator L. In other words, the sign changing solutions (.3) to Problem (.) constructed in [7] are non degenerate in the sense of Duyckaerts-Kenig-Merle [9].

6 6 MONICA MUSSO AND JUNCHENG WEI To state our result, we introduce the following function: For any positive integer i, we define cos(l x) sin(l x) P i (x) = l i and Q i (x) = l i. l= Up to a normalization constant, when n is even, P n and Q n are related to the Fourier series of the Bernoulli polynomial B n (x), and when n is odd P n and Q n are related to the Fourier series of the Euler polynomial E n (x). We refer to [] for further details. We now define (.8) g(x) = which can be rewritten as Observe that j= l= cos( jx) j n, x π g(x) = P n () P n (x). g (x) = Q n (x), Theorem.. Assume that (.9) g (x) < n (g (x)) n g(x) Then all bounded solutions to the equation L(φ) = g (x) = P n (x). x (, π). are a linear combination of the functions z α (x), for α =,..., 3n. When n = 3, condition (.9) is satisfied. Indeed, in this case we observe that g (x) = ln( sin x ). Thus, if we call ρ(x) = g (x)g(x) (g (x)), we get ρ (x) = g (x)g(x) = cot( x )g(x) <. Since ρ() =, condition (.9) is satisfied. When n = 4, let us check the condition (.9): let x = πt, t (, ). Using the explicit formula for the Bernoulli polynomial B 4 we find that (.3) g(t) = t ( t) and hence (.9) is reduced to showing (.3) t t + < 8 3 ( + t), t (, ) which is trivial to verify. In general we believe that condition (.9) should be true for any dimension n 4. In fact, we have checked (.9) numerically, up to dimension n 48. Nevertheless, let us mention that even if (.9) fails, our result is still valid for a subsequence u k j, k j +, of solutions (.3) to Problem (.). Indeed, also in this case, our proof can still go through by choosing a subsequence k j + in order to avoid the resonance. We end this section with some remarks.

7 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 7 First: very few results are known on sign-changing solutions to the Yamabe problem. In the critical exponent case and n = 3 the topology of lower energy level sets was analyzed in Bahri-Chanillo [3] and Bahri-Xu [4]. For the construction of sign-changing bubbling solutions we refer to Hebey-Vaugon [5], Robert-Vetois [8, 9], Vaira [3] and the references therein. We believe that the non-degeneracy property established in Theorem. may be used to obtain new type of constructions for sign changing bubbling solutions. Second: as far as we know the kernels due to the Kelvin transform (i.e. x j z + x z j ) were first used by Korevaar-Mazzeo-Pacard-Schoen [8] and Mazzeo-Pacard ([3]) in the construction of isolated singularities for Yamabe problem by using a gluing procedure. An interesting question is to determine if and how the nondegenerate sign-changing solutions can used in gluing methods. Third: for the sign-changing solutions considered in this paper, the dimension of the kernel equals 3n which is strictly less than N = n + + n(n ). An open question is whether or not there are sign-changing solutions whose dimension of kernel equals N. Acknowledgements: The authors express their deep thanks to Professors M. del Pino and F. Robert for stimulating discussions. We thank Professor C. Kenig for communicating his unpublished result [9].. Description of the solutions In this section we describe the solutions u k in (.3), recalling some properties that have already been established in [7], and adding some further properties that will be useful for later purpose. In terms of the function ϕ in the decomposition (.3), equation (.) gets rewritten as (.) ϕ + pγ U p ϕ + E + γn( ϕ) = where E is defined by (.) E = U + f (U ) and N(ϕ) = U + ϕ p (U + ϕ) U p U p U p ϕ. One has a precise control of the size of the function E when measured for instance in the following norm. Let us fix a number q, with n < q < n, and consider the weighted L q norm n n+ (.3) h = ( + y ) q h L q (R n ). In [7] it is proved that there exist an integer k and a positive constant C such that for all k k the following estimates hold true (.4) E Ck n q if n 4, E C log k if n = 3.

8 8 MONICA MUSSO AND JUNCHENG WEI To be more precise, we have estimates for the -norm of the error term E first in the exterior region k j= { y ξ j > k }, and also in the interior regions { y ξ j < k }, for any j =,..., k. Here > is a positive and small constant, independent of k. In the exterior region. We have if n 4, while if n = 3. ( + y ) n+ n q E(y) L q ( k j= { y ξ j > k }) Ck n q n n+ ( + y ) q E(y) L q ( k j= { y ξ j > k }) C log k In the interior regions. Now, let y ξ j < k for some j {,..., k} fixed. It is convenient to measure the error after a change of scale. Define We have Ẽ j (y) := µ n+ E(ξ j + µy), y < µk. n n+ ( + y ) q Ẽ j (y) L q ( y ξ j < µk ) n Ck q if n 4 and n n+ ( + y ) q Ẽ j (y) L q ( y ξ j < µk ) C k log k We refer the readers to [7]. if n = 3. The function ϕ in (.3) can be further decomposed. Let us introduce some cutoff functions ζ j to be defined as follows. Let ζ(s) be a smooth function such that ζ(s) = for s < and ζ(s) = for s >. We also let ζ (s) = ζ(s). Then we set ζ( k y (y ξ y ) ) if y >, ζ j (y) = ζ( k y ξ ) if y, in such a way that The function ϕ has the form (.5) ϕ = ζ j (y) = ζ j (y/ y ). ϕ j + ψ. j= In the decomposition (.5) the functions ϕ j, for j >, are defined in terms of ϕ (.6) ϕ j (ȳ, y ) = ϕ (e π j k i ȳ, y ), j =,..., k. Each function ϕ j, j =,..., k, is constructed to be a solution in the whole R n to the problem (.7) ϕ j + pγ U p ζ j ϕ j + ζ j [ pγ U p ψ + E + γn( ϕ j + Σ i j ϕ i + ψ)] =,

9 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 9 while ψ solves in R n ψ + pγu p ψ + [ pγ ( U p U p )( Σ k j= ζ j) + pγu p Σ k j= ζ j ] ψ (.8) + pγ U p j ( ζ j ) ϕ j + ( Σ k j= ζ j) ( E + γn(σ k j= ϕ j + ψ) ) =. Define now ϕ (y) = µ n ϕ (µy + ξ ). Then ϕ solves the equation ϕ + f (U)ϕ + χ (ξ + µy)µ n+ E(ξ + µy) (.9) +γµ n+ N(ϕ )(ξ + µy) = in R n where N(ϕ ) = p( U p ζ U p ) ϕ + ζ [p U p Ψ(ϕ ) (.) +N( ϕ + ϕ j + Ψ(ϕ ))] In [7] it is shown that the following estimate on the function ψ holds true: (.) ψ n Ck n q if n 4, ψ n C log k where (.) ϕ n := ( + y n )ϕ L (R n ). On the other hand, if we rescale and translate the function ϕ (.3) ϕ (y) = µ n ϕ (ξ + µy) we have the validity of the following estimate for ϕ (.4) ϕ n Ck n q if n 4, ϕ n Furthermore, we have j C k log k if n = 3, if n = 3. (.5) N(ϕ ) Ck n q if n 4, N(ϕ ) C(k log k) if n = 3, see (.). Let us now define the following functions (.6) π α (y) = y α ϕ(y), for α =,..., n; π (y) = n ϕ(y) + ϕ(y) y. In the above formula ϕ is the function defined in (.3) and described in (.5). Observe that the function π is even in each of its variables, namely π (y,..., y j,..., y n ) = π (y,..., y j,..., y n ) j =,..., n, while π α, for α =,..., n is odd in the y α variable, while it is even in all the other variables. Furthermore, all functions π α are invariant under rotation of π k in the first two coordinates, namely they satisfy (.8). The functions π α can be further described, as follows.

10 MONICA MUSSO AND JUNCHENG WEI Proposition.. The functions π α can be decomposed into (.7) π α (y) = π α, j (y) + ˆπ α (y) where j= π α, j (y) = π α, (e π k j iȳ, y ). Furthermore, there exists a positive constant C so that if n 4, and ˆπ n Ck n q, ˆπ j n Ck n q, j =,..., k, ˆπ n C log k, ˆπ j n C, j =,..., k, log k if n = 3. Furthermore, if we denote π α, (y) = µ n π α, (ξ + µy), then if n 4, and if n = 3. π, n Ck n q, π α, n Ck n q, α =,..., n π, n C k log k, π α, n C C k log k, α =,..., 3 The proof of this result can be obtained using similar arguments as the ones used in [7]. We leave the details to the reader. 3. Scheme of the proof Let φ be a bounded function satisfying L(φ) =, where L is the linear operator defined in (.). We write our function φ as (3.) φ(x) = 3n α= a α z α (x) + φ(x) where the functions z α (x) are defined in (.), (.3), (.4) (.5), (.6) respectively, while the constants a α are chosen so that (3.) u p z α φ =, α =,..., 3n. Observe that L( φ) =. Our aim is to show that, if φ is bounded, then φ. For this purpose, recall that ( u(x) = U(x) U j (x) + ϕ(x), with U(x) = and j= U j (x) = µ n k U(µ k (x ξ j)). + x ) n

11 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM We introduce the following functions (3.3) Z (x) = n U(x) + U(x) x, and (3.4) Z α (x) = U(x), for α =,..., n. x α Moreover, for any l =,..., k, we define (3.5) Z l (x) = n U l(x) + U l (x) (x ξ l ). Observe that, as a consequence of (.) and (.3), we have that [ z (x) = Z (x) Z,l (x) + µ cos θ l U l (x) x l= ] + µ sin θ l U l (x) + π (x), x where π is defined in (.6). Define, for l =,..., k, [ ] (3.6) Z l (x) = µ cos θ l U l (x) + sin θ l U l (x) x x (3.7) Z l (x) = where θ l = π k [ ] µ sin θ l U l (x) + cos θ l U l (x) x x (l ). Furthermore, for any l =,..., k, (3.8) Z α,l (x) = x α U(x), for α = 3,..., n. Thus, we can write (3.9) z (x) = Z (x) [ Z,l (x) + Z,l (x) ] + π (x), l= (3.) (3.) z (x) = Z (x) = Z (x) z (x) = Z (x) = Z (x) l= l= l= l= x U l (x) + π (x) [cos θ l Z l (x) sin θ l Z,l (x)] µ x U (x) + π (x) [sin θ l Z l (x) + cos θ l Z,l (x)] µ + π (x) + π (x)

12 MONICA MUSSO AND JUNCHENG WEI and, for α = 3,..., n, (3.) z α (x) = Z α (x) Z α,l + π α (x) l= Furthermore (3.3) z n+ (x) = Z l (x) + x π (x) x π (x) l= (3.4) z n+ (x) = l= µ cos θ l Z l (x) l= x π (x) + x π (x) µ cos θ l Z l (x) (3.5) z n+3 (x) = and, for α = 3,..., n, l= (3.6) z n+α+ (x) = (3.7) z n+α (x) = Let µ sin θ l Z l (x) l= µ sin θ l Z l (x) x π (x) + x π (x) µ cos θ l Z αl (x) + x π α (x) l= µ sin θ l Z αl (x) + x π α (x). (3.8) Z α (x) = Z α (x) + π α (x), α =,..., n, and introduce the (k + )-dimensional vector functions Z α, (x) Z α (x) Π α (x) = Z α (x) for α =,,..., n... Z αk (x) c c For a given real vector c = c R k+, we write.. c k c Π α (x) = l= c l Z αl (x). l=

13 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 3 With this in mind, we write our function φ as n (3.9) φ(x) = c α Π α (x) + φ (x) α= c α c where c α = α, α =,,..., n, are (n + ) vectors in R... k+ defined so that Observe that c αk U p l (x)z αl (x)φ (x) dx =, for all l =,,..., k, α =,..., n. (3.) c α = for all α and φ = φ. Hence, our purpose is to show that all vector c α are zero vectors and that φ. This will be consequence of the following three facts. Fact. The orthogonality conditions (3.) take the form n n (3.) c α Π α u p z β = c αl Z αl u p z β = α= α= l= φ u p z β for β =,..., 3n. Equation (3.) is a system of (n + ) linear equations (β =,..., 3n ) in the (n + ) (k + ) variables c αl. Let us introduce the following three vectors in R k cos θ (3.) k =, cos = sin θ, sin = cos θ k sin θ k Let us write c α = [ cα, c α ], with c α, R, c α R k, α =,,..., n, and c c, c =.. Rn(k+), ĉ =.. Rn+ c n c n, We have the validity of the following Proposition 3.. The system (3.) reduces to the following 3n linear conditions of the vectors c α : [ ] [ ] (3.3) c + c = t k + Θ k L ( c) + Θ ˆL k (ĉ), k [ ] [ ] (3.4) c + c cos = t sin + Θ k L ( c) + Θ ˆL k (ĉ),

14 4 MONICA MUSSO AND JUNCHENG WEI [ (3.5) c sin ] + c [ ] cos = t + Θ k L ( c) + Θ k ˆL (ĉ), for α = 3,..., n [ ] (3.6) c α = t α + Θ k L α( c) + Θ ˆL k α (ĉ), k [ ] (3.7) c = t n+ + Θ k L n+( c) + Θ ˆL k n+ (ĉ), k [ (3.8) c [ (3.9) c cos sin ] [ c ] [ c cos sin ] = t n+ + Θ k L n+( c) + Θ ˆL k n+ (ĉ), ] = t n+3 + Θ k L n+3( c) + Θ ˆL k n+3 (ĉ), for α = 3,..., n, [ ] (3.3) c α = t cos n+α+ + Θ k L n+α+( c) + Θ ˆL k n+α+ (ĉ), [ ] (3.3) c α = t sin n+α + Θ k L n+α ( c) + Θ ˆL k n+α (ĉ), t t In the above expansions, is a fixed vector with... t n t t C φ... t n and L j : R k(n+) R 3n, ˆL j : R n R 3n are linear functions, whose coefficients are constants uniformly bounded as k. The number q, with n < q < n, is the one already fixed in (.3). Furthermore, Θ k and Θ k denote quantities which can be described respectively as and Θ k = k n q O(), if n 4, Θ k = (k log k) O(), if n = 3, Θ k = k n q O(), if n 4, Θ k = (log k) O(), if n = 3, where O() stands for a quantity which is uniformly bounded as k. We shall prove (3.3) (3.3) in Section 8.

15 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 5 Fact. Since L( φ) =, we have that (3.3) n c α L(Π α (x)) = α= Let φ = φ + k l= φ l and for any l =,..., k Furthermore, let and define where L(φ ) = L(φ l ) = φ l n α= l= c αl L(Z α,l ) = L(φ ) n c α L(Z α, ) α= n c αl L(Z α,l ). α= (y) = µ n φ l (µy + ξ l ), (3.33) φ = φ n + φ l n where the n is defined in (.). A first consequence of (3.3) is that there exists a positive constant C such that n (3.34) φ Cµ c α l= α= for all k large. We postpone the proof of (3.34) to Section 9. Fact 3. Let us now multiply (3.3) against Z βl, for β =,..., n and l =,,..., k. After integrating in R n we get a linear system of (n + ) (k + ) equations in the (n + ) (k + ) constants c α j of the form c r L(φ )Z R n α, c (3.35) M r = L(φ, with r.... α = )Z R n α,.. c n r n L(φ )Z R n α,k Observe first that relation (3.9) together with the fact that L(z α ) = for all α =,..., n, allow us to say that the vectors r α have the form k+ (3.36) row (r ) = [row l (r ) + row l (r )] l= (3.37) row (r ) = µ k+ [cos θ l row l (r ) sin θ l row l (r )], l=

16 6 MONICA MUSSO AND JUNCHENG WEI (3.38) row (r ) = µ k+ [sin θ l row l (r ) + cos θ l row l (r )] l= k+ (3.39) row (r α ) = row l (r α ) for all α = 3,..., n. l= Here with row l we denote the l-th row. The matrix M in (3.35) is a square matrix of dimension [(n + ) (k + )]. The entries of M are numbers of the form L(Z αl )Z β j dy R n for α, β =,..., n and l, j =,,..., k. A first observation is that, if α is any of the indeces {,, }, and β is any of the index in {3,..., n}, then by symmetry the above integrals are zero, namely L(Z αl )Z β j dy = for any l, j =,..., k R n This fact implies that the matrix M has the form [ ] M (3.4) M = M where M is a square matrix of dimension (3 (k + )) and M is a square matrix of dimension [(n ) (k + )]. Since L(Z αl )Z β j dy = R n L(Z β j )Z αl dy R n for α, β =,..., n and l, j =,,..., k, we can write Ā B C (3.4) M = B F D C T D T Ḡ where Ā, B, C, D, F and Ḡ are square matrices of dimension (k + ), with Ā, F and Ḡ symmetric. More precisely, ( ) ( ) (3.4) Ā = L(Z i )Z j, F = L(Z i )Z j, ( (3.43) Ḡ = and ( (3.44) C = L(Z i )Z j ) L(Z i )Z j ) i, j=,,...,k i, j=,,...,k i, j=,,...,k (, B = (, D = L(Z i )Z j ) L(Z i )Z j ) i, j=,,...,k i, j=,,...,k i, j=,,...,k Furthermore, again by symmetry, since L(Z αi )Z β j dx =, if α β, α, β = 3,..., n,

17 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 7 the matrix M has the form (3.45) M = H 3 H H n H n where H j are square matrices of dimension (k+), and each of them is symmetric. The matrices H α are defined by ( ) (3.46) H α = L(Z αi )Z α j, α = 3,..., n. i, j=,,...,k Thus, given the form of the matrix M as described in (3.4), (3.4) and (3.45), system (3.35) is equivalent to (3.47) M c c c = r r r where the vectors r α are defined in (3.47)., H α c α = r α for α = 3,..., n, Observe that system (3.47) impose (n + ) (k + ) linear conditions on the (n + ) (k + ) constants c α j. We shall show that 3n equations in (3.47) are linearly dependent. Thus in reality system (3.47) reduce to only (n+) (k +) 3n linearly independent conditions on the (n + ) (k + ) constants c α j. We shall also show that system (3.47) is solvable. Indeed we have the validity of the following Proposition 3.. There exist k and C such that, for all k > k System (3.47) is solvable. Furthermore, the solution has the form c v k c = v + s + s cos + s c v k µ 3 sin µ sin cos µ µ cos sin +s 4 + s 5 + s cos 6 sin k and [ ] [ ] [ ] c α = v α + s α + s α + s k cos α3, α = 3,..., n sin for any s,..., s 6, s α, s α, s α3 R, where the vectors v α are fixed vectors with v α C φ, α =,,..., n.

18 8 MONICA MUSSO AND JUNCHENG WEI Conditions (3.3) (3.3) guarantees that the solution c α to (3.47) is indeed unique. Furthermore, we shall show that there exists a positive constant C such that n (3.48) c α C φ. α= Here denotes the euclidean norm in R k. Estimates (3.48) combined with (3.34) gives that (3.49) c α = α =,..., n. Replacing equation (3.49) into (3.34) we finally get (3.), namely c α = for all α and φ. Scheme of the paper: In Section 4 we discuss and simplify system (3.47). In Section 5 we establish an invertibility theory for solving (3.47). Section 6 is devoted to prove Proposition 3.. In Section 7 we prove Theorem.. Section 8 is devoted to the proof of Proposition 3., while Section 9 is devoted to the proof of (3.34). Section is devoted to the detailed proofs of several computations. 4. A first simplification of the system (3.47) Let us consider system (3.47) and let us fix α {3,..., n}. Recall that the function z α defined in (.3) satisfies L(z α ) =. Hence, by (3.9), (3.8) and (3.46) we have that k+ row ( H α ) = row l ( H α ). l= [ ] This implies that kernel( H α ) and thus that the system H α (c α ) = r α is [ k ] solvable only if r α =. On the other hand, this last solvability condition is k satisfied as consequence of (3.39). Thus H α c α = r α is solvable. Arguing similarly, we get that and row k+ (M ) = row k+3 (M ) = k+ row (M ) = row l (M ) + µ µ l= k+ l=k+3 cos θ l row k++l (M ) l= sin θ l row k++l (M ) + l= row l (M ), sin θ l row k+3+l (M ), l= cos θ l row k+3+l (M ). l=

19 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 9 This implies that the vectors k w =, w = k and thus that the system M cos µ sin µ c c c = r r r, w = sin µ cos µ is solvable only if r r r kernel(m ) w j =, for j =,,. On the other hand, this last solvability condition is satisfied as consequence of (3.36), (3.37) and (3.38). We thus conclude that system (3.47) is solvable and the solution has the form c c (4.) c = + tw c + sw + rw for all t, s, r R c c α and, if α = 3,..., n (4.) c α = [ ] [ ] + t c α k for all t R In (4.)-(4.), c α for α =,..., n, are (n + ) vectors in R k, respectively given by c α c (4.3) c α = α.... These vectors correspond to solutions of the systems c r (4.4) N c = r, H α [ c α ] = r α for α = 3,..., n. c r In the above formula r α for α =,..., n, are (n + ) vectors in R k, respectively given by c αk L(φ )Z r α = R n α,... L(φ )Z R n α,k

20 MONICA MUSSO AND JUNCHENG WEI In (4.4) the matrix N is defined by A B C (4.5) N := B T F D C T D T G where A, B, C, D, F, G are k k matrices whose entrances are given respectively by ( ) ( ) (4.6) A = L(Z i )Z j, F = L(Z i )Z j, ( (4.7) G = and ( (4.8) C = L(Z i )Z j ) L(Z i )Z j ) i, j=,...,k i, j=,...,k i, j=,...,k (, B = (, D = L(Z i )Z j ) L(Z i )Z j ) i, j=,...,k i, j=,...,k Furthermore, in (4.4) the matrix H α is defined by ( ) (4.9) H α = L(Z αi )Z α j, α = 3,..., n. i, j=,...,k i, j=,...,k The rest of this section is devoted to compute explicitely the entrances of the matrices A, B, C, D, F, G, H α and their eigenvalues. We start with the following observation: all matrices A, B, C, D, F, G and H α in (4.4) are circulant matrices of dimension k k. For properties of circulant matrices, we refer to [9]. A circulant matrix X of dimension k k has the form x x x k x k x k x x x k... x X = k x x......, x x x k x or equivalently, if x i j, i, j =,..., k are the entrances of the matrix X, then x i, j = x, i j +. In particular, in order to know a circulant matrix it is enough to know the entrances of its first row. The eigenvalues of a circulant matrix X are given by the explicit formula k (4.) m = l= x l e π m k i l, m =,..., k,

21 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM and with corresponding normalized eigenvectors defined by (4.) E m = k e π k m e π k m i e π k m i... i (k ) m =,..., k. Observe that any circulant matrix X can be diagonalized where D X is the diagonal matrix X = P D X P T (4.) D X = diag(,,..., k ) and P is the k k invertible matrix defined by (4.3) P = [ E E... Ek ]. The matrices A, B, C, D, F, G and H α are circulant as a consequence of the invariance under rotation of an angle π k in the (x, x )-plane of the functions Z α j. This is trivial in the case of Z l and Z α,l for all α = 3,..., n. On the other hand, if we denote by R j the rotation in the (x, x ) plane of angle π k ( j ), then we get Z, j (x) = U j (x) ξ j = µ n R j (y ξ ) U( ) R j ξ µ = µ n R j U( R j(y ξ ) ) ξ, x = R j y. µ Thus, for instance (F) j j = L(Z j )Z j = L(Z )Z = (F), j =,..., k and, after a rotation of an angle of π k ( h j + ), (F) h j = L(Z h )Z j = L(Z )Z ( j h+) = (F) ( j h +) In a similar way one can show that Z, j (x) = µ n R j U( R j(y ξ ) ) ξ µ, x = R jy. With this in mind, it is straightforward to show that also the matrices B, C, D and G are circulant. A second observation we want to make is that while A, B, F, G, H α are symmetric C, D are anti-symmetric.

22 MONICA MUSSO AND JUNCHENG WEI The fact that A, F, G and H α are symmetric follows directly from their definition. On the other hand, we have thus Furthermore, and thus Z j (x) = µ n R j U(R j(y ξ k j+ ) ) ξ k j+, x = R j y µ B, j = L(Z, )Z, j = L(Z, )Z,k j+ = B,k j+. Z j (x) = µ n R j U(R j(y ξ k j+ ) ) ( ξ k j+ ), x = R j y µ C, j = L(Z, )Z, j = L(Z, )Z,k j+ = C,k j+, and D, j = L(Z, )Z, j = L(Z, )Z,k j+ = D,k j+, for j. Combining this property with the property of being circulant, we get that B is symmetric, while C and D are anti-symmetric. Let us now introduce the following positive number ( ) (n ) (4.4) Ξ = p γ y U p Z (y) dy. Next we describe the entrances of the matrices A, F, G, B, C, D and H α, together with their eigenvalues. We refer the reader to Section for the detailed proof of the following expansions. With O() we denotes a quantity which is uniformly bounded, as k. The matrix A. The matrix A = (A i j ) i, j=,...,k defined by A i j = L(Z i )Z j R n is symmetric. We have (4.5) A = k n µ n O() and for any integer l >, (4.6) A l = Ξ (n ) ( cos θ l ) n where O() is bounded as k. µn + µ n k n O(),

23 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 3 Eigenvalues for A: A direct application of (4.) gives that the eigenvalues of the matrix A are given by a m = n Ξ µ n cos(mθ l ) ( + O( k ) l> ( cos θ l ) ) n ( = Ξ ā m µ n + O( ) (4.7) k ) for m =,,..., k, where (4.8) ā m = n where g is the function defined in (.8). k n ( π) n g ( π k m) The matrix F. The matrix F = (F i j ) i, j=,...,k defined by F i j = L(Z i )Z j R n is symmetric. We have cos θ l (4.9) F = Ξ ( cos θ l ) n µ n + O(µ n ) and, for any l > l> n (4.) F l = Ξ cos θ l n ( cos θ l ) n where O() is bounded as k. µn + O(µ n ) Eigenvalues for F. For any m =,..., k, the eigenvalues of F are (4.) f m = Ξ f m µ n. where (4.) f m = + l> l> cos θ l ( cos θ l ) n n cos θ l n cos mθ ( cos θ l ) n l l> ( + O( k ) ). The matrix G. The matrix G = (G i j ) i, j=,...,k defined by G i j = L(Z i )Z j R n is symmetric. We have n (4.3) G = Ξ cos θ l + n ( cos θ l ) n µ n + µ n O()

24 4 MONICA MUSSO AND JUNCHENG WEI and, for l >, n (4.4) G l = Ξ cos θ l + n ( cos θ l ) n Again O() is bounded as k. µn + O(µ n ) Eigenvalues for G. The eigenvalues of G are given by ( n g m = Ξ µ n cos θ l + n ) ( cos mθl ) l> ( cos θ l ) n ( + O( k ) ) ( = Ξḡ m µ n + O( ) (4.5) k ), for m =,..., k where (4.6) ḡ m = ( π) (n ) g(π n k m) see (.8) for the definition of g. The matrix B. The matrix B = (B i j ) i, j=,...,k defined by B i j = L(Z i )Z j R n is symmetric. We have (4.7) B = µ n k n O() and, for any l >, (4.8) B l = Ξ k n n ( cos θ l ) n Eigenvalues for B. For any m =,..., k b m = Ξ µ n n cos mθ l (4.9) with (4.3) l> = Ξ b m µ n ( + O( k ) ) b m = n see (.8) for the definition of g. µn + µ n k n O(). ( cos θ l ) n k n ( π) n g ( π k m) The matrix C. The matrix C = (C i j ) i, j=,...,k defined by C i j = L(Z i )Z j R n is anti symmetric. We have (4.3) C = k n µ n O() ( + O( k ) )

25 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 5 and, for l >, n (4.3) C l = Ξ sin θ l ( cos θ l ) n µn + k n µ n O(). Eigenvalues for C. For any m =,..., k c m = Ξ i µ n n sin θ l sin mθ l l> ( cos θ l ) n ( + O( k ) ) ( = Ξ i c m µ n + O( ) (4.33) k ) where (4.34) c m = n see (.8) for the definition of g. k n ( π) n g ( π k m) The matrix D. The matrix D = (D i j ) i, j=,...,k D i j = L(Z i )Z j R n is anti symmetric. We have (4.35) D = k n µ n O() and, for l >, n (4.36) D l = Ξ sin θ l ( cos θ l ) n µn 3 + k n µ n O(). Eigenvalues for D. For any m =,..., k d m = i Ξ µ n n sin θ l sin mθ l l> ( cos θ l ) n ( + O( k ) (4.37) ) ( = i Ξ d m µ n + O( ) (4.38) k ) with d m = n see (.8) for the definition of g. k n ( π) n g ( π k m) The matrix H α, for α = 3,..., n. Fix α = 3. The other dimensions can be treated in the same way. The matrix H 3 = (H 3,i j ) i, j=,...,k defined by H 3,i j = L(Z 3i )Z 3 j R n

26 6 MONICA MUSSO AND JUNCHENG WEI is symmetric. We have (4.39) H 3, = Ξ µ n cos θ l ( cos θ l ) n + O(µ n ) and, for l >, [ ] (4.4) H 3,l = Ξ µ n + O(µ n ( cos θ l ) n ). l> Eigenvalues for H 3. For any m =,..., k (4.4) h 3,m = Ξ h 3,m µ n where cos θ l + cos mθ l h 3,m = ( cos θ l ) n l> 5. Solving a linear system. ( + O( k ) ). This section is devoted to solve system (4.4), namely c s N c = s, H α [ c α ] = s α for α = 3,..., n. c s s for a given right hand side s R3k, and s α R k, where N is the matrix defined s in (4.5) and H α are the matrices defined in (4.9). Let (5.) Υ = ( π) n pγ n Ξ, where Ξ is defined in (4.4). We have the validity of the following Proposition 5.. Part a. There exist k and C > such that, for all k > k, System c s (5.) N c = s c s is solvable if (5.3) s k = ( s + s ) cos = ( s + s ) sin =. Furthermore, the solutions of System (5.) has the form c w cos sin (5.4) c = w + t + t cos + t 3 sin c w k

27 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 7 w for all t, t, t 3 R, and with w a fixed vector such that w w s (5.5) w C w k n µ n s. s Part b. Let α = 3,..., n. There exist k and C such that, for any k > k, system (5.6) H α [ c α ] = s α is solvable only if (5.7) s α cos = s α sin =. Furthermore, the solutions of System (5.6) has the form (5.8) c α = w α + t cos + t sin for all t, t R, and with [ w α ] a fixed vector such that (5.9) [ w α ] C k n µ n [ s α ]. Proof. Part a. Define P P = P P where P is defined in (4.3), a simple algebra gives that where N = PDP T D A D B D C D = D B D F D D. D C D D D G Here D X denotes the diagonal matrix of dimension k k whose entrances are given by the eigenvalues of X. For instance D A = diag (a, a,..., a k ) where a j are the eigenvalues of the matrix A, defined in (4.7). Using the change of variables ȳ c s h (5.) ȳ = PT c ; s = P h, c s h ȳ

28 8 MONICA MUSSO AND JUNCHENG WEI with ȳ α = y α, y α,... y α,k, hα = is equivalent to solving h α, h α,... h α,k (5.) D Furthermore, observe that R k, α =,,, one sees that solving c s N c = s c s ȳ ȳ ȳ h = h. h (5.) ȳ α = c α, and hα = s α, α =,,. Let us now introduce the matrix D... D D = D k where for any m =,..., k, D m is the 3 3 matrix given by a m b m c m ā m b m i c m (5.3) D m = b m f m d m = Ξ µn b m f m i d m c m d m g m i c m i d m ḡ m where a m, b m, c m, f m, g m, d m are the eigenvalues of the matrices A, B, C, F, G and D respectively. In the above formula we have used the computation for the eigenvalues a m, b m, c m, d m, f m and g m that we obtained in (4.7), (4.9), (4.33), (4.37), (4.) and (4.5). An easy argument implies that system (5.) can be re written in the form (5.4) D m y,m+ y,m+ y,m+ = h,m+ h,m+ h,m+ m =,,..., k. Taking into account that ā m = b m and c m = d m, a direct algebraic manipulation of the system gives that (5.4) reduces to the simplified system b m i c m y,m+ y,m+ h (5.5) f m + b m y,m+ =,m+ i c m ḡ Ξµ n h,m+ + h,m+. m h,m+ Let, for any m =,..., k, y,m+ (5.6) l m := ( b m + f m ) [ḡm b m + c m], being l m the determinant of the above matrix. We have the following cases

29 Case. and NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 9 If m =, we have that ḡ = c = and so l =. Furthermore, b = n k n ( π) n g () ( + O( k ) ) kn f + b = ( π) n g () ( + O( k ) ). We conclude that System (5.5) for m = is solvable if h = and there exists a positive constant C, independent of k, such that the solution has the form y, ŷ, y, = ŷ, + t y, ŷ, for any t R and for a fixed vector ŷ, ŷ, ŷ, ŷ, ŷ, ŷ, with C µ n k n h, h, h, Case. If m =, we have that f + b =. By symmetry, for m = k we also have f k + b k =. Furthermore b = b k = n k n ( π) n g () ( + O( k ) ), and. k n ḡ = ḡ k = (n ) ( π) n g () ( + O( k ) ), k n c = c k = (n ) ( π) n g () ( + O( k ) ). We conclude that System (5.5) for m = is solvable if h + h = and there exists a positive constant C, independent of k, such that the solution has the form y, ŷ, y, = ŷ, + t y, ŷ,

30 3 MONICA MUSSO AND JUNCHENG WEI for any t R and for a fixed vector ŷ, ŷ, ŷ, ŷ, ŷ, ŷ, with C µ n k n h, h, h, On the other hand, when m = k System (5.5) is solvable if h,k + h,k = and there exists a positive constant C, independent of k, such that the solution has the form y,k ŷ,k y,k = ŷ,k + t y,k ŷ,k for any t R and for a fixed vector Case 3. and In particular ŷ,k ŷ,k ŷ,k ŷ,k ŷ,k ŷ,k with C µ n k n h,k h,k h,k.. Let now m be,, k. In this case we have b m = n f m + b m = k n ( π) n g ( π k m) k n ( π) n g ( π k m) k n ḡ m = (n ) ( π) g(π n k m) c m = n k n ( π) n g ( π k m) ( + O( k ) ), ( + O( k ) ), ( + O( k ) ), ( + O( k ) ). l m = n k 3n ( π) 3n g ( π m ) [ (n )g( πk m)g ( πk m) + (n )(g ( πk ] ( m)) + O( k ) ) Thus under condition (.9), we have that l m < m =,..., k.

31 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 3 Hence, for all m,, k, System (5.5) is uniquely solvable and there exists a positive constant C, independent of k, such that the solution ŷ, ŷ, ŷ, C µ n k n h, h, h,. ŷ, ŷ, ŷ, satisfies Going back to the original variables, and applying a fixed point argument for contraction mappings we get the validity of Part a of Proposition 5.. Part b. Fix α = 3,..., n. We have where P is defined in (4.3), and H α = PD α P T D α = diag ( h α,, h α,,..., h α,k ) where h α, j are the eigenvalues of the matrix H α, defined in (4.4). Using the change of variables ȳ α = P T c α and s α = P T h α, we have to solve D α y α = h α. Recall that, for any m =,..., k where h α,m = Ξ h α,m µ n cos θ l + cos mθ l h α,m = ( cos θ l ) n If m = or m = k, we have that k l> cos θ l +cos mθ l l> ( cos θ l ) n ( + O( k ) ). only if h α, = h α,k =. On the other hand we have h α, = Ξµ n k n ( ( + O( k ) π) ) n and for m =,..., k h α,m = Ξµ n k n ( ( π) g(π n k m) + O( k ) ) =, so the system is solvable Going back to the original variables, we get the validity of Part b, and this concludes the proof of Proposition 5..

32 3 MONICA MUSSO AND JUNCHENG WEI 6. Proof of Proposition 3. A key ingredient to prove Proposition 3. is the estimates on the right hand sides of sistems (4.4). We have Proposition 6.. There exists a positive constant C such that, for any α =,,..., n, (6.) r α C µ n φ for any k sufficiently large. Proof. We prove (6.), only for α =. Recall that L(φ )Z R n r =.... L(φ )Z R n k Then estimate (6.) will follows from (6.) L(φ )Z j C µ n φ, R n for any j =,..., k. To prove (6.), we fix j = and we write L(φ )Z dx = L(Z )φ R n R n = L(Z )φ + L(Z )φ R n \ B(ξ j, k +σ ) j= B(ξ j, k +σ ) where and σ are small positive numbers, independent of k. We start to estimate B(ξ, k +σ ) L(Z )φ. We have L(Z ) = [ f (u) f (U )] Z. As we have already observed very close to ξ, U (x) = O(µ n ) and so in B(ξ, the function U dominates globally the other terms, provided is chosen small enough. Thus, after the change of variable x = ξ + µy, L(Z )φ B(ξ, k +σ ) C f (U) Υ(y) Z (y)[µ n φ (ξ + µy) ] dy B(, k +σ ) µ C φ f (U) Υ(y) Z (y) dy B(, k +σ ) µ where Υ(y) = µ n U(ξ + µy) + A direct consequence of (.) is then that L(Z )φ B(ξ, k +σ ) U(y + µ (ξ ξ l )) l Cµ n φ. k +σ )

33 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 33 Let now j and consider B(ξ j, k +σ ) L(Z )φ. In this case, after the change of variables x = ξ j + µy, we get L(Z )φ B(ξ j, k +σ ) C φ (ξ j + µy)] U p Z (y + µ (ξ ξ j ))[µ n B(, µk +σ ) C φ ( R n U p ( + y ) n where we used (.5). Thus we estimate L(Z )φ j> B(ξ j, k +σ ) ) µ n ( cos θ j ) n Cµ n φ. Finally, in the exterior region R n \ B(ξ j, ) we can estimate k +σ L(Z )φ R n \ B(ξ j, k +σ ) U p C φ R n \ B(ξ j, k +σ ) ( + y ) n Z (y) dy Cµ n φ. Thus we have proven (6.) for α =. The other cases can be treated similarly. We have now the tools for the Proof of Proposition 3.. System (3.47) is solvable only if the following orthogonality conditions are satisfied: r k r (6.3) r r = r r k cos = r r µ sin =, r µ sin cos µ µ (6.4) cos sin r r r r = r = r cos = sin r r r k and [ ] [ ] (6.5) r α = r α k cos [ = r α sin ] = α = 3,..., n

34 34 MONICA MUSSO AND JUNCHENG WEI L(φ )Z We recall that r α = R n α,... As we already mentioned at the beginning of L(φ )Z R n α,k Section 4, the orthogonality conditions (6.3) are satisfied as consequence of (3.36), (3.37) and (3.38). Similarly, the first orthogonality condition in (6.5) is satisfied as consequence of (3.39). Let us recall from (3.3) that L(φ ) = n α= l= c αl L(Z α,l ). Thus the function x L(φ )(x) is invariant under rotation of angle π k (x, x )-plane. Thus = l= and, for all α = 3,..., n, ( cos θ l L(φ )Z αl (x) dx = l= thus r α cos =, and similarly = sin θ l l= L(φ )Z l (x) dx = r k L(φ )Z α (x) dx) cos θ l =, l= L(φ )Z αl (x) dx = r α sin in the namely the first orthogonality condition in (6.4) and the remaining orthogonality conditions in (6.5) are satisfied. Let us check that also the last two orthogonality conditions in (6.4) are verified. Observe that L(φ )(x) = x n L(φ )( x ). The remaining orthogonality conditions in (6.4) are consequence of the x following Lemma 6.. Let h be a function in R n such that h(y) = y n h( y ). Then y ( (6.6) µ Uµ (x ξ l ) ) h(y) dy = ξ l U µ (x ξ l )h(y) dx R n µ R n We postpone the proof of the above Lemma to the end of this Section. Combining the result of Proposition 5. and the a-priori estimates in Proposition 6., a direct application of a fixed point theorem for contraction mapping readily gives the proof of Proposition 3.. We conclude this section with

35 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 35 Proof of Lemma 6.. Proof of (6.6). Assume l =. Define I(t) = ω µ (y tξ )h(y) dy where ω µ (y t ξ ) = µ n y t ξ U( ). R n µ We have (6.7) d dt I(t) = ω µ (y t ξ ) ξ h(y) dy, R n and ( ) d dt I(t) = ω µ (y ξ ) ξ h(y) dy. t= R n On the other hand, using the change of variables y = x, we have x I(t) = ω µ ( x R n x t ξ )h( x x ) x n dx = ω µ ( x R n x t ξ )h(x) x n dx = where µ(t) = Observe that µ() = µ, p() = ξ, d dt µ(t) = Hence d dt I(t) = d dt µ(t) This gives = ω µ (x p)h(x) dx R n µ µ + t ξ, p(t) = t µ + t ξ ξ. tµ µ + t ξ, d dt p(t) = R n [ µ + t ξ t ξ ] µ + t ξ ξ. µ ω µ(x p)h(x) dx d dt p(t) ω µ (x p)h(x) dx. R n ( ) d dt I(t) = µ ξ t= R n µ ω µ(x ξ )h(x) dx (6.8) ( ξ ) ω µ (x ξ ) ξ h(x) dx. R n From (6.7) and (6.8) we conclude with the validity of (6.6). If l > in (6.6), the same arguments hold true. The thus conclude with the proof of the Lemma.

36 36 MONICA MUSSO AND JUNCHENG WEI 7. Final argument. c c c Let be the solution to (3.47) predicted by Proposition 3., given by c 3... c n c v k c = v + s + s cos + s c v k µ 3 sin µ sin cos µ µ cos +s 4 + s 5 + s cos 6 k and [ ] [ ] [ c α = v α + s α + s α + s k cos α3 sin A direct computation shows that there exists a unique sin sin ], α = 3,..., n (s,..., s 6, s 3,, s 3,, s 3,3,..., s n,, s n,, s n,3 ) Rn for which the above solution satisfies all the n conditions of Proposition 3.. Furthermore, one can see that (s,..., s 6, s 3,, s 3,, s 3,3,..., s n,, s n,, s n,3 ) C µ φ. c c c Hence, there exists a unique solution to systems (4.4), satisfying estimates c 3... c n in Proposition 3.. Furthermore, one has c c c C φ c 3... c n

37 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 37 for some positive constant C independent of k. On the other hand, from (3.34) we conclude that c c (7.) φ Cµ c c 3... c n where again C denotes a positive constant, independent of k. Thus we conclude that c α, j =, for all α =,,..., n, j =,..., k. Plugging this information into (7.), we conclude that φ and this proves Theorem.. 8. Proof of Proposition 3. We will give the proof of Proposition 3. when dimension n 4. The estimates for dimension n = 3 can be obtained with similar arguments. The key ingredient to prove Proposition 3. are the folllowing estimates u p Z α,l Z = U p Z dy + O(µ n ) if α =, l = (8.) (8.) (8.3) (8.4) u p Z α,l Z β = u p Z α,l Z, j = u p Z α,l Z β, j = = O(µ n ) otherwise U p Z = O(µ n ) otherwise U p Z = O(µ n ) otherwise U p Z = O(µ n ) otherwise dy + O(µ n ) if α = β, l = dy + O(µ n ) if α =, l = j dy + O(µ n ) if α = β, l = j We prove (8.3). Let > be a small number, fixed independently from k. We write u p Z αl Z j = u p Z αl Z l + u p Z αl Z, j B(ξ l, k ) R n \B(ξ l, k ) = i + i.

38 38 MONICA MUSSO AND JUNCHENG WEI We claim that the main term is i. Performing the change of variable x = ξ l + µy, we get i = u p (ξ l + µy)z α (y)z (y)dy B(, µk ( ) ) = U p Z + O((µk)n ) if α = = if α. On the other hand, to estimate i, we write i = R n \ u p Z αl Z, j + u p Z αl Z, j = i + i k j= B(ξ j, k ) B(ξ j, k ) The first integral can be estimated as follows i C R n \ k j= B(ξ j, k ) µ n+ j l x ξ l n ( + x ) while the second integral can be estimated by i C j l B(ξ j, k ) µ n n dx Cµ n+ x ξ l n u p Z j dx Cµ n where again C denotes an arbitrary positive constant, independent of k. This concludes the proof of (8.3). The proofs of (8.), (8.) and (8.4) are similar, and left to the reader. Now we claim that (8.5) U p Z = U p Z = n 4 n (n ) Γ( n ) Γ(n + ). The proof of identity (8.5) is postponed to the end of this section. Let us now consider (3.) with β =, that is n α= l= c αl Z αl u p z = φ u p z. First we write t = U p Z φ u p z. A straightforward computation gives that t C φ, for a certain constant C independent from k. Second, we observe

39 NONDEGENERACY OF NONRADIAL NODAL SOLUTIONS TO YAMABE PROBLEM 39 that, direct consequence of (8.) (8.4), of (3.9) and Proposition. is that n α= l= c αl Z αl u p z = c [c l l= U p Z U p Z c l U p Z c c + O(k n c q )L( ) + O(k n c q ) ˆL( ) c n c n where L and ˆL are linear function, whose coefficients are uniformly bounded in k, as k. Here we have used the fact that there exists a positive constant C independent of k such that u p Z αl π (x) dx C ˆπ n and u p Z αl π (x) dx C ˆπ n, together with the result in Proposition.. The condition (3.3) follows readily. The proof of (3.4) (3.3) is similar to that performed above, and we leave it to the reader. We conclude this section with the proof of (8.5). Using the definition of Z and Z, we have that U p Z = a (n ) x n n ( + x dx ) n+ and U p Z = a (n ) n 4 ( x ) ( + x dx, ) n+ for a certain positive number a n that depends only on n. Using the formula we get (8.6) and (8.7) ( r ) q q+α q α Γ( )Γ( + r dr = ) r+α Γ(q) n ( + x ) n+ dx = ( n + )Γ( n ), Γ(n + ) x ( + x ) n+ dx = ( n ) Γ( n ) Γ(n + ). ]

40 4 MONICA MUSSO AND JUNCHENG WEI Replacing (8.6), (8.7) in U p Z and U p Z we obtain n U p Z U p Z = (n ) a Γ( n ) n Γ(n + ) [ 4 (n + ) + n 4 ] 4 (n + ) thus (8.5) is proven. We start with the following Proposition 9.. Let 9. Proof of (3.34). L (ϕ) = ϕ + pγu p ϕ + a(y)ϕ in R n. =, Assume that a L n (R n ). Assume furthermore that h is a function in R n with h L n n+ (R n ) bounded and such that y n h( y y) = ±h(y). Then there exists a positive constant C such that any solution ϕ to (9.) L (ϕ) = h satisfies ϕ n C h. Proof. Since a L n (R n ) and U p = O( + y 4 ), the operator L is a compact perturbation of the Laplace operator in the space D, (R n ). Thus standard argument gives that ϕ L (R n ) + ϕ n C h L n (R n ) L n+ n (R n ) where the last inequality is a direct consequence of Holder inequality. Being ϕ a weak solution to (9.), local elliptic estimates yields D ϕ L q (B ) + Dϕ L q (B ) + ϕ L (B ) C h L n n+ (R n ). Consider now the Kelvin s transform of ϕ, ˆϕ(y) = y n ϕ( y y). This function satisfies (9.) ˆϕ + pu p ˆϕ + y 4 a( y y) ˆϕ = ĥ in R n \ {} where ĥ(y) = y n h( y y). We observe that ĥ L q ( y <) = y n+ n q h L q ( y > C h L n n+ (R n ), y 4 a( y y) n L ( y <) = a n L ( y > ) and ˆϕ L (R n ) + ˆϕ n C h L n (R n ) L n+ n. (R n ) Applying then elliptic estimates to (9.), we get D ˆϕ L q (B ) + D ˆϕ L q (B ) + ˆϕ L (B ) CC h L n n+ (R n ). This concludes the proof of the proposition since ˆϕ L (B ) = ϕ L (R n \B ).