Entire radial and nonradial solutions for systems with critical growth
aa r X i v : . [ m a t h . A P ] D ec ENTIRE RADIAL AND NONRADIAL SOLUTIONS FOR SYSTEMSWITH CRITICAL GROWTH
FRANCESCA GLADIALI, MASSIMO GROSSI, AND CHRISTOPHE TROESTLER
Abstract.
In this paper we establish existence of radial and nonradial solutions tothe system − ∆ u = F ( u , u ) in R N , − ∆ u = F ( u , u ) in R N ,u > , u > in R N ,u , u ∈ D , ( R N ) , where F , F are nonlinearities with critical behavior. Introduction
The aim of this paper is to prove existence of radial and nonradial solutions to somenonlinear systems − ∆ u = F ( u , u ) in R N , − ∆ u = F ( u , u ) in R N ,u > , u > in R N ,u , u ∈ D , ( R N ) , where F , F are nonlinearities with critical behavior in the Sobolev sense, N > and D , ( R N ) = (cid:8) u ∈ L ∗ ( R N ) such that |∇ u | ∈ L ( R N ) (cid:9) with ∗ = NN − . A commonfeature of the systems that we will study is their invariance by translations and dilations .Papers on existence or qualitative properties of solutions to systems with critical growthin R N are very few, due to the lack of compactness given by the Talenti bubbles and thedifficulties arising for the lack of good variational methods. The first example of systemwhich we consider is given by − ∆ u = αu ∗ − + (1 − α ) u N − u NN − in R N , − ∆ u = αu ∗ − + (1 − α ) u N − u NN − in R N ,u > , u > , u , u ∈ D , ( R N ) , (1.1) Mathematics Subject Classification.
Primary 35J47, 35B33, 35B32; Secondary 35B09, 35B08.
Key words and phrases.
Non-cooperative system of PDEs, non-radial solutions, critical exponent,critical hyperbola, global bifurcation, entire solutions.The first author is partially supported by GNAMPA. The first two authors are supported by PRIN-2012-grant “Variational and perturbative aspects of nonlinear differential problems”. The third authoris partially supported by the project “Existence and asymptotic behavior of solutions to systems ofsemilinear elliptic partial differential equations” (T.1110.14) of the
Fonds de la Recherche FondamentaleCollective , Belgium. where N > and α is a real parameter. This system, also known as Gross-Pitaevskii,arises in many physical contexts such as nonlinear optics and the Hartree-Fock theory, see[M] for its derivation, and it is very studied mainly in the cubic case, which correspondsto the critical case in R or on bounded domains where the cubic exponent is subcriticalin R . It is coupled when − α = 0 and cooperative when − α > . Physically, thiscondition means the attractive interaction of the states u and u , while − α < meansthe repulsive interaction between them. Note that System (1.1) has a gradient structurewith the energy functional E ( u , u ) = 12 Z R N |∇ u | + |∇ u | − N − N Z R N α (cid:0) u ∗ + u ∗ (cid:1) + (1 − α ) (cid:16) u NN − u NN − (cid:17) even if it is not so easy to apply variational methods to find solutions. System (1.1) wasalready considered in [GLW] where the existence of infinitely many nontrivial solutionsis obtained using a perturbation argument.Another particular case of (1.1) is the following generalization of the system consideredby O. Druet, E. Hebey [DH], namely − ∆ u = h(cid:0) αu + (1 − α ) u (cid:1) i N − u in R N , − ∆ u = h(cid:0) (1 − α ) u + αu (cid:1) i N − u in R N ,u > , u > , u , u ∈ D , ( R N ) . (1.2)In [DH] the case of α = was studied and the stability of solutions on manifolds wasconsidered. Further, the radial symmetry and uniqueness of the solutions in R N isproved.We also mention the paper [CSW] where the radial symmetry of solutions is provedfor a particular critical nonlinearity.The starting point of our study is the paper [GGT] where we studied the existence ofradial solutions for the k × k system of equations − ∆ u i = k X j =1 a ij u ∗ − j in R N ,u i > in R N ,u i ∈ D , ( R N ) , (1.3)for i = 1 , . . . , k , where N > and the matrix A := ( a ij ) i,j =1 ,...,k is symmetric andsatisfies k X j =1 a ij = 1 for any i = 1 , . . . , k. (1.4)Note that the case k = 2 and A = (cid:0) (cid:1) is known in the literature as nonlinearitybelonging to the critical hyperbola . NTIRE RADIAL AND NONRADIAL SOLUTIONS FOR SYSTEMS WITH CRITICAL GROWTH 3
Under the assumption (1.4) it is straightforward that system (1.3) always admits thetrivial solutions u = · · · = u k = U δ,y ( x ) := (cid:2) N ( N − δ (cid:3) N − ( δ + | x − y | ) N − for any δ > and y ∈ R N . To simplify the notation, let U ( x ) := U , ( x ) = [ N ( N − N − (1 + | x | ) N − . (1.5)A careful study of the linearized system of (1.3) at this trivial solution allows us to provethe existence of nontrivial radial solutions when the eigenvalues of the matrix A reachsome specific values using bifurcation theory.Note that System (1.3) does not have a variational structure and indeed our methodsdo not require it.Even if the existence of radial solutions to some of the previous examples (1.1)–(1.3) isa new result, the main interest is the existence of nonradial ones. Nonradial solutions maybe found mainly for noncooperative systems where the lack of the maximum principlecan give a symmetry breaking of the solutions. Indeed, in [DH] and [CSW], the radialsymmetry of the solutions is proved in a particular cooperative case.In this paper we want to purse several goals. First, we want to introduce a newsetting which allows us to consider Systems (1.1)–(1.3) jointly. Indeed all these problemsadmit the trivial solutions u = u = U δ,y ( x ) which is the starting point to apply thebifurcation theory like in [GGT]. A general treatment of these problems is possible sincewe significantly improve the final part of the paper [GGT] showing that the Lagrangemultiplier introduced to “kill” the direction of dilation invariance coming from the criticalSobolev exponent is indeed a natural constraint if we allow some invariance (Kelvininvariance) on the solutions. This lets us switch from a local bifurcation result in [GGT]to a global one.This invariance is a good tool to overcome the degeneracy of critical problems in R N which are invariant under dilation and can also be applied to the result in [DGG], wherea Pohozaev identity gives the result only locally.Another technical problem arises since our nonlinearities in general are not C atzero. This problem was already noticed by [GLW] and indeed their existence results aregiven in dimension where they are able to define and to invert the linearized operatorassociated to their system. To overcome this problem we use a different functional settingthat allows us to work only with positive values of u and u . Observe that the functionalsetting of our operator is a delicate part of the proof.Secondly we continue the study in [GGT] and we address to the existence of nonradialsolutions to (1.3) using in a tricky way some even and odd symmetries. Obviously oursolutions cannot be invariant with respect to odd symmetries since we are looking forpositive ones. But we can introduce a suitable setting (see Eq. 2.9) in which we canmake use of this invariance. This is a new aspect that has never been investigated beforeand that can shed light on how solutions of systems of this type are.This use of the symmetries is the key point that allows us to distinguish between radialand nonradial solutions. GLADIALI, GROSSI, AND TROESTLER
A crucial step of our method is the characterization of the kernel of the linearizedoperator associated to our systems. Actually, in [GGT], we find radial solutions usingthe classical
Crandall-Rabinowitz Theorem which requires a one dimensional kernel. Thisis achieved by restricting the problem to radially symmetric functions and “killing” thedirection of scale invariance.Considering also nonradial functions the dimension of the kernel increases dramaticallyand it becomes very hard to control it. Moreover it is not clear whether the solutionobtained considering this new kernel is nonradial . As said before, the use of suitable evenand odd symmetries is significant and allows us to prove that in many cases the kernelcontains only nonradial functions and it is odd dimensional. To exploit them, we needsome invariance on the operator associated to our problem. This invariance naturallyappears in the case of a × system while it not clear whether it applies in the generalcase of more equations as (1.3). For this reason we focus hereafter on the case × andwe believe that a further study is needed to understand the general case. To compute thedimension of the kernel in these symmetric spaces we need a classification of symmetricspherical harmonics in S N and indeed this is part of Section 4 and 5.Finally we also give an asymptotic expansion of the solutions near the bifurcationpoint so as to better understand them. In this way we can distinguish different nonradialsolutions by their symmetries and expansions.2. Statement of the main results
Let us introduce our abstract setting. We consider − ∆ u = F ( α, u , u ) in R N , − ∆ u = F ( α, u , u ) in R N ,u > , u > , u , u ∈ D , ( R N ) , (2.1)where the F i satisfy the following assumptions: for all α ∈ R and for i = 1 , ,(F1) the derivatives ∂ α F i , ∂ u F i and ∂ αu F i of the map F i : R × (0 , + ∞ ) → R : ( α, u ) F i ( α, u ) exist and are continuous;(F2) for all α ∈ R , there exists a neighborhood A of α and a constant C such that,for all α ∈ A and ( u , u ) ∈ (0 , + ∞ ) , | ∂ u F i ( α, u , u ) | C ( u ∗ − + u ∗ − ) and | ∂ αu F i ( α, u , u ) | C ( u ∗ − + u ∗ − ) ;(F3) F i ( α, ,
1) = 1 ;(F4) F i ( α, λu , λu ) = λ ∗ − F i ( α, u , u ) for all λ > and ( u , u ) ∈ (0 , + ∞ ) ;(F5) F ( α, u , u ) = F ( α, u , u ) for all ( u , u ) ∈ (0 , + ∞ ) ;(F6) for all α , ∂ α β ( α ) > where β ( α ) := ∂ u F ( α, , − ∂ u F ( α, , .By (F3) it is straightforward that System (2.1) admits, for any α ∈ R , the trivial solution ( u , u ) = ( U, U ) and (F4) says that our system is scale invariant. Further, in view ofEq. (2.1), it is also translation invariant. NTIRE RADIAL AND NONRADIAL SOLUTIONS FOR SYSTEMS WITH CRITICAL GROWTH 5
This generalization encompasses the following
Schrodinger system − ∆ u = αu ∗ − + (1 − α ) u p u ∗ − − p in R N , − ∆ u = (1 − α ) u ∗ − − p u p + αu ∗ − in R N ,u > , u > , u , u ∈ D , ( R N ) , (2.2)with p < ∗ − and α is a real parameter. When p = 0 System (2.2) becomes − ∆ u = αu ∗ − + (1 − α ) u ∗ − in R N , − ∆ u = (1 − α ) u ∗ − + αu ∗ − in R N ,u > , u > , u , u ∈ D , ( R N ) , (2.3)while for p = N − we get System (1.1). Moreover System (2.1) includes System (1.2).Our first result is the generalization of the local radial bifurcation result obtained in[GGT] for (2.3) to a global one for System (2.1). An important role in our results willbe played by the Jacobi polynomials P ( β,γ ) j that we introduce now. They are defined as P ( β,γ ) m ( ξ ) = m X s =0 (cid:18) m + βs (cid:19)(cid:18) m + γm − s (cid:19) (cid:18) ξ − (cid:19) m − s (cid:18) ξ + 12 (cid:19) s (2.4)for m ∈ N , β, γ ∈ R + and ξ ∈ R . Theorem 2.1.
Assume (F1) – (F6) . The point ( α ∗ , U, U ) is a radial bifurcation pointfrom the curve of trivial solutions ( α, U, U ) to System (2.1) if α ∗ satisfies β ( α ∗ ) = (2 n + N )(2 n + N − N ( N − (2.5) for some n ∈ N , where β is defined in (F6) . More precisely there exists a continuouslydifferentiable curve defined for ε small enough ( − ε , ε ) → R × (cid:0) D , rad ( R N ) (cid:1) : ε (cid:0) α ( ε ) , u ( ε ) , u ( ε ) (cid:1) passing through ( α ∗ , U, U ) , i.e., (cid:0) α (0) , u (0) , u (0) (cid:1) = ( α ∗ , U, U ) , such that, for all ε ∈ ( − ε , ε ) , ( u ( ε ) , u ( ε )) is a radial solution to (2.1) with α = α ( ε ) . Moreover, ( u ( ε ) = U + εW n ( | x | ) + εφ ,ε ( | x | ) ,u ( ε ) = U − εW n ( | x | ) + εφ ,ε ( | x | ) , (2.6) with W n being the function W n ( | x | ) := 1(1 + | x | ) N − P ( N − , N − ) n (cid:18) − | x | | x | (cid:19) (2.7) where φ ,ε , φ ,ε are functions uniformly bounded in D , ( R N ) with respect to ε ∈ ( − ε , ε ) ,and such that φ i, = 0 for i = 1 , . Finally the bifurcation is global and the Rabinowitzalternative holds. The values α ∗ in (2.5) are all of those for which the linearized system at the trivialsolution ( U, U ) is non-invertible showing that condition (2.5) is also necessary. GLADIALI, GROSSI, AND TROESTLER
Corollary 2.2.
For any n ∈ N , let α ∗ n = (2 n + N − n + N )2 N ( N +2 − p ( N − + N +22( N +2 − p ( N − − p ( N − N +2 − p ( N − in (2.2) , (2 n + N − n + N )2 N + N − N in (1.1) , (2 n + N − n + N )+ N (6 − N )8 N in (1.2) . (2.8) Then ( α ∗ , U, U ) is a radial bifurcation point of Systems (2.2) , (1.1) and (1.2) from itscurve of trivial solutions ( α, U, U ) if α ∗ = α ∗ n for some n ∈ N . Moreover, the expansionaround the bifurcation point given by Theorem 2.1 holds and the curve is global.Remark . An interesting fact is that in (2.2) the exponent p does not enter in arelevant way in the proof of the previous results and indeed the solutions we find have,near a bifurcation point, the same expansion for every value of p . In this way we havea path of solutions connecting (2.3) with (1.1) showing that these solutions are not duethe variational structure of (2.3).The next step is to find nonradial solutions. In [GL] was proved that in the cooperativecase (i.e., when − α > ), System (2.2) admits only radial solutions. Note that, for all n > , − α ∗ n < − α ∗ = 0 where α ∗ n is defined by (2.8). Then α ∗ n are good “candidates”to find nonradial solutions. Moreover, at each value α ∗ n the linearized system possessesmany nonradial solutions and the kernel becomes richer and richer as n → ∞ (seeProposition 3.1). However, one technical problem in looking for nonradial solutions isthat the kernel of the linearized problem at a degeneracy point always contains the radialfunction W n defined by (2.7). So our aim becomes to choose a suitable subspace of thekernel in which W n does not lies. This will be done by using in a tricky way some odd-symmetries. It is possible indeed to apply such symmetries to a linear combination ofthe components u , u even if the solutions we are interested in are positive.Here is our basic idea: if one writes ( u = U + z + z u = U + z − z (2.9)then the system satisfied by z , z admits solutions obtained by imposing the followingsymmetries on ( z , z ) : ∀ ( x ′ , x N ) ∈ R N , z ( x ′ , x N ) = z ( | x ′ | , − x N ) and z ( x ′ , x N ) = − z ( | x ′ | , − x N ) , (2.10)(more general symmetries will be imposed later; see Section 4.2 for more details). The crucial remark is that the new system in ( z , z ) obtained by (2.9) is invariant for thesymmetries in (2.10) (see (3.1)–(3.5)). This use of odd symmetries is unclear if weconsidered directly System (2.1).In order to state our first nonradial bifurcation result, we use in R N the sphericalcoordinates ( r, ϕ, θ , . . . , θ N − ) ∈ [0 , + ∞ ) × [0 , π ) × [0 , π ) N − . We have Theorem 2.4.
Assume (F1) – (F6) and let α ∗ n be the unique solution to (2.5) for some n ∈ N . The point ( α ∗ n , U, U ) is a nonradial bifurcation point for the curve of trivialsolutions ( α, U, U ) to System (2.1) when n mod 4 ∈ { , } . More precisely, there exist a NTIRE RADIAL AND NONRADIAL SOLUTIONS FOR SYSTEMS WITH CRITICAL GROWTH 7 continuum C of nonradial solutions ( u , u ) to System (2.1) , bifurcating from ( α ∗ n , U, U ) ;the bifurcation is global and the Rabinowitz alternative holds. Finally for any sequenceof solutions ( α k , u ,k , u ,k ) → ( α ∗ n , U, U ) , we have that (up to a subsequence) ( u ,k = U + ε k Z n ( x ) + o ( ε k ) ,u ,k = U − ε k Z n ( x ) + o ( ε k ) , (2.11) as k → ∞ where ε k = k z ,k k X → (see (2.9) and (3.13) ) and Z n is the function Z n ( x ) = n X h =1 , h odd a h r h (1 + r ) h + N − P ( h + N − , h + N − ) n − h (cid:18) − r r (cid:19) P ( N − , N − ) h (cos θ N − ) (2.12) for some coefficients a h ∈ R . Observe that the functions P ( N − , N − ) h (cos θ N − ) are the spherical harmonics that are O ( N − -invariant. Corollary 2.5.
Let n ∈ N and α ∗ n as defined in Corollary 2.2. Then the same claims ofTheorem 2.4 hold for Systems (2.2) and (1.2) . It is possible to prove a similar result using more symmetries. Here we ask the followingones: ∀ x = ( x ′ , x N − m +1 , . . . , x N ) ∈ R N − m × R m , z ( x ) = z ( | x ′ | , ± x N − m +1 , . . . , ± x N ) , and z ( x ′ , x N − m +1 , x N ) = − z ( | x ′ | , − x N − m +1 , . . . , x N ) , · · · z ( x ′ , x N − m +1 , x N ) = − z ( | x ′ | , x N − m +1 , . . . , − x N ) (cid:9) . Imposing these symmetries on the functions z , z defined in (2.9), we get the followingresult: Theorem 2.6.
Let m N and let α ∗ n be the unique solution to (2.5) for some n > m . Suppose that (cid:18) m + (cid:4) n − m (cid:5) m (cid:19) is an odd integer. (2.13) Then for any m there exists a continuum C m of nonradial solutions that satisfies Sys-tem (2.1) , bifurcating from ( α ∗ n , U, U ) and the bifurcation is global and the Rabinowitzalternative holds. Moreover the continua C m are distinct and we have that, up to asubsequence, ( u , u ) has the same expansion as in (2.11) where Z n ( x ) = n X h =1 a h r h (1 + r ) h + N − P ( h + N − ,h + N − ) n − h (cid:18) − r r (cid:19) Y h ( θ ) (2.14) and the spherical harmonics Y h ( θ ) are O ( N − m ) invariant and odd in the last m variables. Corollary 2.7.
Let n ∈ N and α ∗ n as defined in Corollary 2.2. Then the same claims ofTheorem 2.6 hold for System (2.2) and (1.2) . For the reader’s convenience, we state the previous theorem when m = 2 . GLADIALI, GROSSI, AND TROESTLER
Corollary 2.8.
Let m = 2 in Theorem 2.6. Then if n mod 8 ∈ { , , , } (2.15) the claim of Theorem 2.6 holds and Z n in this case is given by Z n ( x ) = n X h =1 a h r h (1 + r ) h + N − P ( h + N − ,h + N − ) n − h (cid:18) − r r (cid:19) Y h ( θ ) (2.16) for some coefficients a h ∈ R , where Y h ( θ ) are spherical harmonics which are O ( N − invariant and are odd with respect to x N and to x N − . We conclude by giving one more existence result which produces a nonradial solutionsfor every value of n . These solutions are found imposing an odd symmetry with respectto an angle in spherical coordinates and also a periodicity assumption. They are differentfrom the previous ones since they have a different expansion. Theorem 2.9.
Assume (F1) – (F6) and α ∗ n be the unique solution to (2.5) for some n ∈ N . Then for any n ∈ N , n > , there exists a continuum D n of nonradial solutionsto System (2.1) , bifurcating from ( α ∗ n , U, U ) . When ε is small enough this continuum isa continuously differentiable curve ( − ε , ε ) → R × (cid:0) D , rad ( R N ) (cid:1) : ε (cid:0) α ( ε ) , u ( ε ) , u ( ε ) (cid:1) passing through ( α ∗ n , U, U ) , i.e., (cid:0) α (0) , u (0) , u (0) (cid:1) = ( α ∗ n , U, U ) , such that, for all ε ∈ ( − ε , ε ) , ( u ( ε ) , u ( ε )) is a nonradial solution to (2.1) with α = α ( ε ) . Moreover, ( u ( ε ) = U + εZ n ( x ) + εφ ,ε ( x ) ,u ( ε ) = U − εZ n ( x ) + εφ ,ε ( x ) , with Z n ( r, ϕ, Θ) = a r n (1 + r ) n + N − sin( nϕ )(sin θ ) n · · · (sin θ N − ) n , a ∈ R , (2.17) (here we use the spherical coordinates ( r, ϕ, Θ) = ( r, ϕ, θ , . . . , θ N − ) in R N ). Moreoverthe bifurcation is global and the Rabinowitz alternative holds.Remark . Note that the function Y n ( ϕ, Θ) = sin( nϕ )(sin θ ) n · · · (sin θ N − ) n is theunique spherical harmonic of order n which is odd and periodic of period πn with respectto the angle ϕ . Moreover, in Cartesian coordinates we have that Y n ( x ) = ℑ m( x + ix ) n . Corollary 2.11.
Let n ∈ N and α ∗ n as defined in Corollary 2.2. Then the same claimsof Theorem 2.9 hold for System (2.2) and (1.2) .Remark . It is difficult to give a formula with the exact number of solutions whichtakes in account all the previous theorems. Here we describe a particular case: choose n = 4 in (2.5) and N > then we have the existence of at least five solutions bifurcatingby ( U, U ) as follows: i ) one radial solution (Theorem 2.1), ii ) one nonradial solution with z even in all the coordinates and z odd with respectto x N − and x N and even in other coordinates (Corollary 2.8), NTIRE RADIAL AND NONRADIAL SOLUTIONS FOR SYSTEMS WITH CRITICAL GROWTH 9 iii ) one nonradial solution with z odd with respect to x N − , . . . , x N and even inother coordinates (Theorem 2.6 with m = 3 ), iv ) one nonradial solution in R N with N > with z odd with respect to x N − , . . . ,x N and even in other coordinates (Theorem 2.6 with m = 4 ), v ) one nonradial solution where z and z are periodic of period π with respect tothe angle ϕ and z is odd in ϕ (Theorem 2.9).In the following table, which does not pretend to be exhaustive, we show the number ofsolutions bifurcating from ( U, U ) arising from Theorems 2.1–2.9. N = 3 N = 4 N = 5 n = 2 4 4 4 n = 3 4 4 4 n = 4 4 5 5 n = 5 4 5 6 n = 6 3 4 5 n = 7 2 3 3 Remark . Note that the our results for System (1.1) hold for any dimension N > , extending some recent results of [GLW]. Finally, as observed in [GLW], when thedimension N > , System (1.1) becomes linear or sublinear in some of its components andthis fact produces problem in defining and estimating the linearization. In some sense,we can say that the bifurcation theory suits well this problem. We remark moreover thatthe solutions founded in [GLW] are always different from ours since their expansion is ofthe following type u = U + εφ and u = P k U δ k ,y k + εφ .The paper is organized as follows: in Section 3 we recall some preliminaries andintroduce the functional setting to find the nonradial solution. In Section 4 we definethe symmetric spaces and prove Theorems 2.1, 2.4 and 2.6. In Section 5 we proveTheorem 2.9. 3. Preliminary results and the functional setting
To study System (2.1), we perform the following change of variables ( z = u + u − U,z = u − u , (3.1)that turns (2.1) into the system − ∆ z = f ( | x | , z , z ) in R N , − ∆ z = f ( | x | , z , z ) in R N ,z , z ∈ D , ( R N ) , (3.2) where f ( | x | , z , z ) := F (cid:16) α, U + z + z , U + z − z (cid:17) + F (cid:16) α, U + z + z , U + z − z (cid:17) − U ∗ − , (3.3) f ( | x | , z , z ) := F (cid:16) α, U + z + z , U + z − z (cid:17) − F (cid:16) α, U + z + z , U + z − z (cid:17) . (3.4)One important feature in looking for nonradial solutions is that, using (F5), this changeof variables gives the following invariance: f ( | x | , z , − z ) = f ( | x | , z , z ) ,f ( | x | , z , − z ) = − f ( | x | , z , z ) . (3.5)Solutions to (2.1) are zeros of the operator T ( α, z , z ) := z − ( − ∆) − (cid:0) f ( | x | , z , z ) (cid:1) z − ( − ∆) − (cid:0) f ( | x | , z , z ) (cid:1)! . Clearly, T ( α, ,
0) = (0 , for all α ∈ R (thanks to (F3) and (F4)). A necessary conditionfor the bifurcation is that the linearized operator ∂ z T ( α, , is not invertible. Thiscorresponds to study the system: − ∆ w = ∂f ∂z ( | x | , , w + ∂f ∂z ( | x | , , w in R N , − ∆ w = ∂f ∂z ( | x | , , w + ∂f ∂z ( | x | , , w in R N ,w , w ∈ D , ( R N ) . (3.6)A simple computation shows ∂f ∂z ( α, ,
0) = 12 (cid:20) ∂F ∂u ( α, U, U ) + ∂F ∂u ( α, U, U ) + ∂F ∂u ( α, U, U ) + ∂F ∂u ( α, U, U ) (cid:21) ,∂f ∂z ( α, ,
0) = 12 (cid:20) ∂F ∂u ( α, U, U ) − ∂F ∂u ( α, U, U ) + ∂F ∂u ( α, U, U ) − ∂F ∂u ( α, U, U ) (cid:21) , and a very similar expression holds for ∂f ∂z i ( α, , for i = 1 , . First observe that from(F5) we get ∂F ∂u ( α, U, U ) = ∂F ∂u ( α, U, U ) and ∂F ∂u ( α, U, U ) = ∂F ∂u ( α, U, U ) . Then, differentiating (F4) with respect to λ we get ( ∂ u F + ∂ u F )( α, U, U ) = (2 ∗ − U ∗ − F ( α, ,
1) = N + 2 N − U N − . Moreover, using again (F4): ∂ u j F i ( α, λu , λu ) = λ ∗ − ∂ u j F i ( α, u , u ) for i = 1 , and j = 1 , , NTIRE RADIAL AND NONRADIAL SOLUTIONS FOR SYSTEMS WITH CRITICAL GROWTH 11 and in particular ∂ u j F i ( α, U, U ) = U ∗ − ∂ u j F i ( α, , . Putting together all these remarks, it is straightforward that system (3.6) becomes − ∆ w = N +2 N − U N − w in R N , − ∆ w = β ( α ) U N − w in R N ,w , w ∈ D , ( R N ) , (3.7)with β ( α ) defined in (F6).System (3.7) is degenerate for any α , since the problem is invariant by translationsand dilations. Indeed, it is well known that the first equation admits the solutions W ( x ) := −| x | (1+ | x | ) N/ and W i ( x ) = ∂U∂x i for i = 1 , . . . , N . The second equation instead hassolutions if and only if β ( α ) is an eigenvalue of the linearized equation of the classicalcritical problem at the standard bubble U . Using the classification of the eigenvaluesand eigenfunctions in [GGT, Theorem 1.1], one gets that the second equation admitsnontrivial solutions if and only if β ( α ) = λ n N +2 N − with λ n := (2 n + N − n + N ) N ( N +2) for some n ∈ N . So we have the following classification result for (3.7). Proposition 3.1.
Let β n be given by β n := (2 n + N )(2 n + N − N ( N − . (3.8)i) When β ( α ) = β n for all n ∈ N , all solutions to (3.7) are given by ( w , w ) = N X i =1 a i ∂U∂x i + bW, ! (3.9) for some real constants a , . . . , a N , b , where W is the radial function defined by W ( x ) := 1 d (cid:18) x · ∇ U + N − U (cid:19) = 1 − | x | (1 + | x | ) N/ (3.10) with d := N ( N − / ( N − ( N +2) / . ii) When β ( α ) = β n for some n ∈ N , all solutions to (3.7) are given by ( w , w ) = N X i =1 a i ∂U∂x i + bW, n X k =0 A k W n,k ( r ) Y k ( θ ) ! (3.11) for some real constants a , . . . , a N , b, A , . . . , A n , where W n,k are W n,k ( r ) := r k (1 + r ) k + N − P ( k + N − , k + N − ) n − k (cid:18) − r r (cid:19) (3.12) for k = 0 , . . . , n . Here, as usual, Y k ( θ ) denotes a spherical harmonic related tothe eigenvalue k ( k + N − and P ( a,b ) j are the Jacobi polynomials. In [GGT] we restricted to the radial functions and since the kernel of the secondequation in (3.7) at the values β n is one dimensional, Crandall-Rabinowitz’ Theoremallowed us to prove the bifurcation result. In the nonradial setting, the kernel of thesecond equation in (3.7) is very rich. We prove a bifurcation result using the LeraySchauder degree, when this kernel has an odd dimension.Of course, in this case, we need some compactness of the operator T . Since weseek positive solutions to System (2.1) and the maximum principle does not apply, thestandard space D , ( R N ) does not seem to be the best one. For this reason we use asuitable weighted functional space. Set D := n u ∈ L ∞ ( R N ) (cid:12)(cid:12)(cid:12) sup x ∈ R N | u ( x ) | U ( x ) < + ∞ o endowed with the norm k u k D := sup x ∈ R N | u ( x ) | U ( x ) and define X := D , ( R N ) ∩ D. (3.13)Then X is a Banach space when equipped with the norm k u k X := max {k u k , , k u k D } where k u k , = ( R R N |∇ u | ) / is the classical norm in D , ( R N ) . Definition 3.2.
Let us denote by X the space X := (cid:8) ( z , z ) ∈ X (cid:12)(cid:12) ∃ δ > , | z | (2 − δ ) U + z (cid:9) and define the operator T : R × X → X × X as T ( α, z , z ) := z − ( − ∆) − (cid:0) f ( | x | , z , z ) (cid:1) z − ( − ∆) − (cid:0) f ( | x | , z , z ) (cid:1)! . (3.14)Note that if ( z , z ) ∈ X , both quantities U + z + z and U + z − z are positive sothat F i ( α, U + z + z , U + z − z ) are well defined on R N and C . Moreover, X is an opensubset of X .The zeros of the operator T correspond to the solutions to System (2.1). As saidbefore, Problem (2.1) is degenerate for any α . To overcome this degeneracy we will usesome symmetry and invariance properties. The solutions we will find will inherit thesymmetry and the invariance. To overcome the degeneracy of the first equation in (3.7),which is due to the scale invariance of the problem, we use the Kelvin transform k ( z ) of z , namely k ( z )( x ) := 1 | x | N − z (cid:18) x | x | (cid:19) (3.15)and we denote by X ± k ⊆ X the subset of functions in X which are invariant (up to thesign) by a Kelvin transform, i.e. X + k := { z ∈ X | k ( z ) = z } and X − k := { z ∈ X | k ( z ) = − z } . (3.16)Observe that U ∈ X + k , W ∈ X − k and, using the fact that the Jacobi polynomials P ( a,b ) j are even if j is even and odd if j is odd, an easy computation shows that W n,k ∈ X + k if n − k is even while W n,k ∈ X − k if n − k is odd . NTIRE RADIAL AND NONRADIAL SOLUTIONS FOR SYSTEMS WITH CRITICAL GROWTH 13
First we prove some properties of the operator T . Lemma 3.3.
The operator T given by (3.14) is well defined and continuous from R × X to X . Moreover, ∂ α T , ∂ z T and ∂ αz T exist and are continuous. Finally, T maps R × (cid:0) X ∩ ( X + k × X ± k ) (cid:1) to X + k × X ± k .Proof. First notice that, (F4) implies lim λ → F i ( α, λu , λu ) = 0 . Thus, using (F2), onegets | F i ( α, u , u ) | = (cid:12)(cid:12)(cid:12)(cid:12) Z ∂ λ (cid:0) F i ( α, λu , λu ) (cid:1) d λ (cid:12)(cid:12)(cid:12)(cid:12) Z (cid:12)(cid:12) ∂ u F i ( α, λu , λu )[ u ] + ∂ u F i ( α, λu , λu )[ u ] (cid:12)(cid:12) d λ C ( u ∗ − + u ∗ − )( u + u ) C ( u ∗ − + u ∗ − ) . (3.17)(Different occurrences of C may denote different constants.) Given that z , z and U belong to X , (3.17) implies that (cid:12)(cid:12) F i (cid:0) α, U + z + z , U + z − z (cid:1)(cid:12)(cid:12) CU ∗ − and thus, using(3.3) and (3.4), | f i ( | x | , z , z ) | CU ∗ − for i = 1 , . Then f i ( | x | , z , z ) belong to L NN +2 ( R N ) and there exists a unique g i ∈ D , ( R N ) for i = 1 , such that g i is a weak solution to − ∆ g i = f i ( | x | , z , z ) in R N . (3.18)The solution g i enjoys the following representation: g i ( x ) = 1 ω N ( N − Z R N | x − y | N − f i ( | y | , z , z ) d y where ω N is the area of the unit sphere in R N . This implies | g i ( x ) | C Z R N | x − y | N − U ∗ − ( y ) d y = CU ( x ) and g i ∈ X showing that T is well defined from X to X × X .Next we have to show that the operator T maps Kelvin invariant (up to a sign) func-tions into functions that are Kelvin invariant (with the same sign). It is enough to showthat (cid:0) ( − ∆) − ( f ( | x | , z , z )) , ( − ∆) − ( f ( | x | , z , z )) (cid:1) maps X ∩ ( X + k × X ± k ) into X + k × X ± k .Assume ( z , z ) ∈ X ∩ ( X + k × X ± k ) and let, as before, g i = ( − ∆) − ( f i ( | x | , z , z )) . Then g i ∈ X is a weak solution to (3.18) and letting e g i := k ( g i ) , the Kelvin transform of g i wehave that e g i weakly solves − ∆ e g i = − | x | N +2 ∆ g i (cid:16) x | x | (cid:17) = 1 | x | N +2 f i (cid:18) x | x | , z (cid:16) x | x | (cid:17) , z (cid:16) x | x | (cid:17)(cid:19) An easy consequence of (F4) is that | x | N +2 F i (cid:18) α, (cid:16) U + z + z (cid:17)(cid:16) x | x | (cid:17) , (cid:16) U + z − z (cid:17)(cid:16) x | x | (cid:17)(cid:19) = F i (cid:16) α, k ( U ) + k ( z ) + k ( z )2 , k ( U ) + k ( z ) − k ( z )2 (cid:17) . This, together with the fact that U and z are Kelvin invariant while z is Kelvin invariantup to a sign (depending which space X ± k we are dealing with) shows that | x | N +2 f i (cid:18) x | x | , z (cid:16) x | x | (cid:17) , z (cid:16) x | x | (cid:17)(cid:19) = f i (cid:0) | x | , z ( x ) , ± z ( x ) (cid:1) where ± depends on the space X ± k we consider. Then, using (3.5), it follows that | x | N +2 f (cid:18) | x | , z (cid:16) x | x | (cid:17) , z (cid:16) x | x | (cid:17)(cid:19) = f (cid:0) | x | , z ( x ) , z ( x ) (cid:1) while | x | N +2 f (cid:18) x | x | , z (cid:16) x | x | (cid:17) , z (cid:16) x | x | (cid:17)(cid:19) = ± f (cid:0) | x | , z ( x ) , z ( x ) (cid:1) . This implies that e g weakly solves − ∆ e g = f ( | x | , z ( x ) , z ( x )) and e g solves − ∆ e g = ± f ( | x | , z ( x ) , z ( x )) . The uniqueness of solutions in D , ( R N ) then implies e g = g and e g = ± g which shows that g ∈ X + k and g ∈ X ± k . This concludes the first part of theproof.Let us now prove the continuity of T on R × X . Let α n → α in R and ( z ,n , z ,n ) → ( z , z ) in X as n → ∞ , and set g i,n := ( − ∆) − f i,n where f i,n ( x ) := f i ( | x | , z ,n , z ,n ) with α = α n . Since z i,n → z i in D , ( R N ) , the convergence also holds in L ∗ ( R N ) . Using (3.17) andLebesgue’s dominated convergence theorem and its converse, one deduces that f i,n → f i in L NN +2 . Therefore g i,n → g i in D , and T ( α n , z n ) → T ( α, z ) in D , . Now let us showthe convergence in D . We have that | g i,n ( x ) − g i ( x ) | U ( x ) ω N ( N − U ( x ) Z R N | x − y | N − | f i,n ( y ) − f i ( y ) | U ( y ) ∗ − U ( y ) ∗ − d y C sup y ∈ R N | f i,n ( y ) − f i ( y ) | U ( y ) ∗ − . (3.19)Moreover, using (F4), one gets | f i,n ( y ) − f i ( y ) | U ( y ) ∗ − X i =1 (cid:12)(cid:12)(cid:12) F i (cid:16) α n , z ,n + z ,n U , z ,n − z ,n U (cid:17) − F i (cid:16) α, z + z U , z − z U (cid:17)(cid:12)(cid:12)(cid:12) NTIRE RADIAL AND NONRADIAL SOLUTIONS FOR SYSTEMS WITH CRITICAL GROWTH 15
Thanks to the convergence in D , z j,n /U → z j /U uniformly for j = 1 , . Thus z ,n ± z ,n U → z ± z U uniformly on R N . The continuity of the maps F i then imply thatboth terms of the sum converge uniformly to .The existence and continuity of the derivatives is proved in a similar way. (cid:3) Next we show a compactness result for the operator ( z , z ) (cid:0) ( − ∆) − f , ( − ∆) − f (cid:1) .Here we need some decay estimates on solutions of a semilinear elliptic equation. Lemma 3.4 ([ST]) . If < p < N and h is a non negative, radial function belonging to L ( R N ) , then Z R N h ( y ) | x − y | p d y = O (cid:18) | x | p (cid:19) as | x | → + ∞ . Now we can prove our compactness result:
Lemma 3.5.
For all α , the operator M ( z , z ) := (cid:0) ( − ∆) − f ( | x | , z , z ) , ( − ∆) − f ( | x | , z , z ) (cid:1) (3.20) is compact from X to X .Proof. 1. From Lemma 3.3, we have that M : X → X is continuous. Now let ( z n ) =( z ,n , z ,n ) be a bounded sequence in X and let us prove that, up to a subsequence, g n := M ( z n ) converges strongly to some g ∈ X × X . On one hand, since ( z n ) is boundedin D , × D , , going if necessary to a subsequence, one can assume that ( z n ) convergesweakly to some z = ( z , z ) in D , × D , and z n → z almost everywhere. On the otherhand, ( k z n k D × D ) is also bounded which means that | z i,n | CU where C is independentof i and n and so, using (3.17), | f i ( | x | , z n ) | CU ∗ − . Lebesgue’s dominated convergencetheorem then implies that f i ( | x | , z n ) converges strongly to f i ( | x | , z ) in L NN +2 for i = 1 , .From the continuity of ( − ∆) − : L NN +2 → D , , one concludes that g n → g in D , × D , .The inequality | z i,n | CU also implies | g i,n ( x ) | C Z R N | x − y | N − | f i ( z n ( y )) | d y C Z R N U ∗ − ( y ) | x − y | N − d y = C U ( x ) , and passing to the limit yields g i ∈ D . It remains to show that k g n − g k D × D → . First, Hölder’s inequality allows to getthe estimate: | g i,n ( x ) − g i ( x ) | C Z R N | x − y | N − (cid:12)(cid:12) f i ( | y | , z n ( y )) − f i ( | y | , z ( y )) (cid:12)(cid:12) d y = C Z R N U ∗ − − ε ( y ) | x − y | N − (cid:12)(cid:12) f i ( | y | , z n ( y )) − f i ( | y | , z ( y )) (cid:12)(cid:12) U ∗ − − ε ( y ) d y C Z R N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U N +2 N − − ε ( y ) | x − y | N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) qq − q − q Z R N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) f i ( | y | , z n ( y )) − f i ( | y | , z ( y )) (cid:12)(cid:12) U ∗ − ( y ) U ε ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q q where ε > will be chosen small and q > large such that εq = 2 ∗ . Note that (3.17)implies | f i ( | y | , z ( y )) | C ( U + | z | + | z | ) ∗ − + CU ∗ − and so the ratio in the rightintegral is bounded on R N . Thus the integrand of the right integral is bounded by C q U εq ( y ) CU ∗ ( y ) ∈ L ( R N ) where C is independent of n . Lebesgue’s dominatedconvergence theorem then implies that this integral converges to as n → ∞ .The proof will be complete if we show: Z R N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U N +2 N − − ε ( y ) | x − y | N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) qq − d y C (1 + | x | ) ( N − qq − = CU qq − ( x ) . (3.21)This inequality follows from Lemma 3.4 because h := U ( N +2 N − − ε ) qq − ∈ L ( R N ) i.e., ( N − (cid:0) N +2 N − − ε (cid:1) qq − > N , and ( N − qq − < N are possible if ε is small enoughand q is large enough. (cid:3) The role of symmetries
The operator T is a compact perturbation of the identity and, as proved in Lemma 3.3,maps R × (cid:0) X ∩ ( X + k × X ± k ) (cid:1) into X + k × X ± k .We want to find solutions to our problem as zeroes of T and we will use the bifurcationtheory. As explained in the introduction, we want to find both radial and nonradialsolutions. In particular, to obtain the nonradial ones, we use some symmetry propertiesof the operator T that can be obtained by (3.5).We state the definition in a general way and we will then apply to some specific casesso to obtain different solutions. Let us introduce some notations. Let S be a subgroupof O ( N ) , where O ( N ) is the orthogonal group of R N , and let X S := (cid:8) v ∈ X + k (cid:12)(cid:12) ∀ s ∈ S , ∀ x ∈ R N , v ( s − ( x )) = v ( x ) (cid:9) (4.1)be the set of functions invariant by the action of S . Let σ : S → {− , } be a groupmorphism and define a second action of S on X by ( s ⋄ v )( x ) := σ ( s ) v (cid:0) s − ( x ) (cid:1) . (4.2) NTIRE RADIAL AND NONRADIAL SOLUTIONS FOR SYSTEMS WITH CRITICAL GROWTH 17
The invariant subspace of X + k × X ± k of interest is Z := (cid:8) z = ( z , z ) ∈ X + k × X ± k (cid:12)(cid:12) ∀ s ∈ S , z ( s − ( x )) = z ( x ) and σ ( s ) z ( s − ( x )) = z ( x ) (cid:9) . (4.3)Then we can prove the following result: Lemma 4.1.
The operator T defined in (3.14) maps R × ( X ∩ Z ) into Z .Proof. We will show that T = ( T , T ) is equivariant under the action of S , namely T (cid:0) α, z ( s − ( x )) , σ ( s ) z ( s − ( x )) (cid:1) = T (cid:0) α, z ( x ) , z ( x ) (cid:1) , and T (cid:0) α, z ( s − ( x )) , σ ( s ) z ( s − ( x )) (cid:1) = σ ( s ) T (cid:0) α, z ( x ) , z ( x ) (cid:1) . Let z = ( z , z ) ∈ X . First, notice that, thanks to (3.5), the functions f and f definedin (3.3)–(3.4) satisfy f (cid:0) | x | , z ( s − ( x )) , σ ( s ) z ( s − ( x )) (cid:1) = f ( | x | , z ( x )) , and f (cid:0) | x | , z ( s − ( x )) , σ ( s ) z ( s − ( x )) (cid:1) = σ ( s ) f ( | x | , z ( x )) . Second, because the Laplacian is equivariant under the action of the group O ( N ) , itreadily follows that ( − ∆) − (cid:0) σ ( s ) f ( s ( x )) (cid:1) = σ ( s ) (cid:0) ( − ∆) − f ( s ( x )) (cid:1) for any σ , s ∈ S and f ∈ L N/ ( N +2) .Putting these observations together concludes the proof. (cid:3) Lemma 4.2.
Assume β ( α ) = β n for all n ∈ N , with β n be as defined in (3.8) , and thatthe subspace of solutions in X S to the first equation of (3.7) has only the trivial solution.Still denote T the operator defined in (3.14) restricted to X ∩ Z . Then the linear map ∂ z T ( α, ,
0) :
Z → Z is invertible, where ∂ z T ( α, , is the Fréchet derivative of T withrespect to z at ( α, , .Proof. For any ( w , w ) ∈ X , one has, see (3.7), ∂ z T ( α, , (cid:18) w w (cid:19) = w − ( − ∆) − (cid:16) N +2 N − U N − w (cid:17) w − ( − ∆) − (cid:16) β ( α ) U N − w (cid:17) (4.4)with β ( α ) as defined in (F6). Since ∂ z T ( α, , is a compact perturbation of the identity(see Lemma 3.5 in [GGT]), it suffices to prove that ker (cid:0) ∂ z T ( α, , (cid:1) = { (0 , } in Z whenever β ( α ) = β n . Let ( w , w ) ∈ Z ⊆ X + k × X ± k . Notice that ∂ z T ( α, ,
0) ( w w ) = ( ) if and only if ( w , w ) is a solution to (3.7). By assumption we have that w ≡ andProposition 3.1 says that the only solutions to the second equation are given by (3.9) aswe assumed β ( α ) = β n . This gives the claim. (cid:3) Remark . From Lemma 4.2 we have that, when β ( α ) = β n for all n , deg (cid:0) T ( α, · ) , e B, (cid:1) = deg (cid:0) ∂ z T ( α, , , e B, (cid:1) = ( − m ( α ) (4.5) where e B is a suitable ball in Z centered at the origin and m ( α ) the sum of the algebraicmultiplicities of all eigenvalues λ belonging to (0 , of the problem − ∆ w = λ N +2 N − U N − w in R N , − ∆ w = λ β ( α ) U N − w in R N , ( w , w ) ∈ Z . (4.6) Proposition 4.4.
Assume the same hypotheses as in Lemma 4.2. Let n ∈ N and α ∗ n besuch that β ( α ∗ n ) = β n (recall that β n is defined in (3.8) ). For ε > small enough, thefollowing holds m ( α ∗ n + ε ) = m ( α ∗ n − ε ) + γ ( n ) (4.7) where γ ( n ) is the algebraic multiplicity of the solutions to − ∆ w = β n U N − w such that (0 , w ) ∈ Z .Proof. As the first equation of (4.6) does not depend on α , its contribution is the same tothe values m ( α ∗ n ± ε ) . Concerning the second one, since β ( α ) is a continuous increasingfunction we have get that β ( α ∗ n + ε ) ց β ( α ∗ n ) and then the contribution of the secondequation to m ( α ∗ n + ε ) is given by the algebraic multiplicity of the eigenvalues λ = n β ( α ∗ n + ε ) , . . . , β n β ( α ∗ n + ε ) o . In the same way, for ε small enough we have that m ( α ∗ n − ε ) isgiven by the algebraic multiplicity of the eigenvalues λ = n β ( α ∗ n + ε ) , . . . , β n − β ( α ∗ n + ε ) o . Thisgives the claim. (cid:3) Proposition 4.5.
Assume the same hypotheses as in Lemma 4.2 and let us suppose that γ ( n ) is an odd integer. Then the point ( α ∗ n , U, U ) is a bifurcation point from the curveof trivial solutions ( α, U, U ) to System (2.1) . Moreover the bifurcation is global, theRabinowitz alternative holds, and for any sequence ( α k , u ,k , u ,k ) of solutions convergingto ( α ∗ n , U, U ) , we have that ( u ,k = U + z ,k + z ,k u ,k = U + z ,k − z ,k and, up to a subsequence, ( u ,k = U + ε k Z n + o ( ε k ) ,u ,k = U − ε k Z n + o ( ε k ) , (4.8) as k → ∞ where Z n is a solution to the second equation in (3.7) such that (0 , Z n ) ∈ Z , k Z n k X = 1 and ε k = k z ,k k X → .Proof. From (4.5) and (4.7), it is standard to see that the curve of trivial solutions forthe operator T : R × ( X ∩ Z ) → Z bifurcates at the values α ∗ n with β ( α ∗ n ) = β n forany n such that γ ( n ) is odd, see [K, Theorem II.3.2] and the bifurcation is global. TheRabinowitz alternative finally follows from [K, Theorem II.3.3].Next let us show the expansion (2.11). Let ( z ,k , z ,k ) be solutions obtained by thebifurcation result to (2.1) as α k → α ∗ n (recall that ( z ,k , z ,k ) → (0 , in the space X ). NTIRE RADIAL AND NONRADIAL SOLUTIONS FOR SYSTEMS WITH CRITICAL GROWTH 19
First we show that k z ,k kk z ,k k C (4.9)where C is a constant independent of k and k·k = k·k X . First, notice that z ,k because, if it was, z ,k ∈ X S would satisfy − ∆ z = f ( | x | , z , but the assumption thatthe first equation of (3.7) has only the trivial solution in X S implies that this equationonly has trivial solutions for α ≈ α ∗ n . This contradicts the fact that ( z ,k , z ,k ) lies onthe branch of nontrivial solutions.To show (4.9), let us argue by contradiction: let us suppose that, up to subsequence, k z ,k kk z ,k k → + ∞ . Set w ,k = z ,k k z ,k k , w ,k = z ,k k z ,k k . The system satisfied by w ,k and w ,k is − ∆ w ,k = 1 k z ,k k (cid:20) F (cid:16) α k , U + k z ,k k w ,k + k z ,k kk z ,k k w ,k , U + k z ,k k w ,k − k z ,k kk z ,k k w ,k (cid:17) + F (cid:16) α k , U + k z ,k k w ,k + k z ,k kk z ,k k w ,k , U + k z ,k k w ,k − || z ,k |||| z ,k || w ,k (cid:17) − U ∗ − (cid:17)(cid:21) (4.10a) − ∆ w ,k = 1 k z ,k k (cid:20) F (cid:16) α k , U + k z ,k k k z ,k kk z ,k k w ,k + w ,k , U + k z ,k k k z ,k kk z ,k k w ,k − w ,k (cid:17) − F (cid:16) α k , U + k z ,k k k z ,k kk z ,k k w ,k + w ,k , U + k z ,k k k z ,k kk z ,k k w ,k − w ,k (cid:17)(cid:21) (4.10b) k w ,k k = k w ,k k = 1 (4.10c)Going if necessary to a subsequence, we can assume w ,k ⇀ w and w ,k ⇀ w in D , for some ( w , w ) ∈ Z . Arguing as in the first part of the proof of Lemma 3.5, we deducethat w ,k → w and w ,k → w in L ∗ ( R N ) and in D , . Using that F i ( α k , U, U ) = U ∗ − for i = 1 , , we can pass to the limit on Eq. (4.10a) and show that w ∈ X S satisfies − ∆ w = (cid:20) ∂F ∂u (cid:0) α ∗ n , U, U (cid:1) + ∂F ∂u (cid:0) α ∗ n , U, U (cid:1) + ∂F ∂u (cid:0) α ∗ n , U, U (cid:1) + ∂F ∂u (cid:0) α ∗ n , U, U (cid:1)(cid:21) w Moreover, arguing as in the second part of the proof of Lemma 3.5 on (4.10a), we canshow that k w ,k − w k D → . Thus w ,k → w in X and k w k = 1 . As in Section 3,using the properties of F we have that w ∈ X S satisfies − ∆ w = N + 2 N − U N − w in R N , This is a contradiction since in X S the previous equation admits only the trivial solution.So (4.9) holds. Hence, up to a subsequence, we have that k z ,k kk z ,k k → δ > . Passing to the limit in(4.10b), we get that − ∆ w = ∂F ∂u (cid:0) α ∗ n , U, U (cid:1) δw + w ∂F ∂u (cid:0) α ∗ n , U, U (cid:1) δw − w − ∂F ∂u (cid:0) α ∗ n , U, U (cid:1) δw + w − ∂F ∂u (cid:0) α ∗ n , U, U (cid:1) δw − w . and, arguing again as in the second part of the proof of Lemma 3.5, w ,k → w in X with k w k = 1 . As before, using the properties of F , we have that w solves − ∆ w = β ( α ) U N − w in R N , and hence w = Z n where Z n is a solution to the second equation in (3.7) such that (0 , Z n ) ∈ Z and k Z n k = 1 . Then z ,k = k z ,k k ( Z n + o (1)) . Next we show that z ,k = o (1) k z ,k k . (4.11)This is clear if lim k → + ∞ k z ,k kk z ,k k = 0 since in this case k z ,k k = k z ,k kk z ,k k k z ,k k = o (1) k z ,k k . (4.12)On the other hand, it is not possible that k z ,k kk z ,k k > D > because in this case we can passto the limit in (4.10a) and as before we get a contradiction. This shows (4.11). Comingback to the definition of ( u ,k , u ,k ) we have that (4.8) holds with ε k = k z ,k k . (cid:3) Now we specify some subgroups S that satisfy the assumptions of Lemma 4.2. Observethat when β ( α ) = β n the second equation in (3.7) does not possess solutions. The firstequation instead admits in X + k the solutions P Ni =1 a i x i (1+ | x | ) N/ . Then, the assumptionsof Lemma 4.2 are satisfied if the functions x i (1+ | x | ) N/ do not belong to X S . The firstexample is the radial case which allows to prove Theorem 2.1. The other examples,which are provided for every N > , prove the existence of different nonradial solutions.4.1. The radial case.
Following the previous notation we let S = O ( N ) and σ : S →{− , } be the group morphism such that σ ( s ) := 1 for all s ∈ O ( N ) . Thus X S = (cid:8) v ∈ X (cid:12)(cid:12) ∀ x ∈ R N , v ( x ) = v ( | x | ) (cid:9) , Z ≡ Z ± rad = (cid:8) z ∈ X + k × X ± k (cid:12)(cid:12) ∀ x ∈ R N , z ( x ) = z ( | x | ) (cid:9) . Proof of Theorem 2.1.
To prove the bifurcation result we define the operator T in (3.14)in the space Z + rad ⊆ X + k × X + k when n is even and in the space Z − rad ⊆ X + k × X − k when n is odd. Recalling the discussion at the beginning of Section 3, we have that the linearizedoperator ∂ z T ( α, , is invertible if and only if system (3.7) does not admit solutions in Z + rad when n is even ( Z − rad in case of n odd). From Proposition 3.1 we know that the firstequation in (3.7) does not depend on α and admits the unique radial solution W ( | x | ) which does not belong to X + k . The second equation in (3.7) instead admits solutions ifand only if β ( α ) = β n and the corresponding radial solution is W n ( | x | ) := W n, ( r ) . Hencethe assumption of Lemma 4.2 are satisfied. Moreover from (3.12) and the definition of NTIRE RADIAL AND NONRADIAL SOLUTIONS FOR SYSTEMS WITH CRITICAL GROWTH 21 the Jacobi polynomials we have that W n ∈ X + k if n is even and W n ∈ X − k if n isodd showing that γ ( n ) = 1 for any n . Further, using the monotonicity of β ( α ) , theglobal bifurcation result and the Rabinowitz alternative follows from Theorem II.3.2 andTheorem II.3.3 of [K]. Finally the fact that the curve is continuously differentiable nearthe bifurcation point follows from the bifurcation result of Crandall-Rabinowitz for one-dimensional kernel since the operator T is differentiable and the transversality conditionholds in Z because ∂ αz T ( α, , (cid:18) W n (cid:19) = − ∂ α β ( α ) − ∆) − (cid:16) U N − W n (cid:17)! , and so (cid:18)(cid:18) W n (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ∂ αz T ( α ∗ n , , (cid:18) W n (cid:19)(cid:19) ( D , ) = − ∂ α β ( α ∗ n ) Z R N U N − W n d x = 0 . (cid:3) Proof of Corollary 2.2 .
It is easy to check that (F1)–(F5) are satisfied. One readilycomputes that β ( α ) = ∗ − − p ) α − (2 ∗ − − p ) in (2.2) , ∗ α − in (1.1) , N − α − − NN − in (1.2) . (4.13)and so (F6) is also satisfied. Moreover (2.5) holds if and only if α ∗ = α ∗ n where α ∗ n isdefined by (2.8). Corollary 2.2 immediately follows. (cid:3) The first nonradial case.
Let h be the reflection through the hyperplane x N = 0 , S := h O ( N − , h i be the subgroup generated by O ( N − and h , and σ : S → {− , } be the group morphism such that σ ( s ) := 1 if s ∈ O ( N − and σ ( h ) := − ( σ iseasily seen to be well defined because h commutes with any element of O ( N − ). Thus X S = (cid:8) v ∈ X + k (cid:12)(cid:12) ∀ x = ( x ′ , x N ) ∈ R N , v ( x ′ , x N ) = v ( | x ′ | , − x N ) (cid:9) , Z ≡ Z ± = (cid:8) z ∈ X + k × X ± k (cid:12)(cid:12) ∀ x = ( x ′ , x N ) ∈ R N , z ( x ′ , x N ) = z ( | x ′ | , − x N ) and z ( x ′ , x N ) = − z ( | x ′ | , − x N ) (cid:9) . Observe that the odd symmetry helps to kill the radial solution in the kernel of thelinearized system while the even symmetries help to avoid the solutions given by thetranslation invariance of the problem. Indeed since functions in X S are even withrespect to each x i , i = 1 , . . . , N and belong to X + k from Proposition 3.1, it is easilydeduced that the solutions in X S of the first equation of (3.7) (see (3.9)) are the trivialones. Thus Lemma 4.2 applies and by Proposition 4.5 the bifurcation result can beproved when γ ( n ) is odd. Proposition 4.6.
With this choice of S = S and σ = σ , we have that γ ( n ) is odd ifand only if n = 4 ℓ + 1 or n = 4 ℓ + 2 for ℓ = 0 , , . . . Proof. In R N , we consider the spherical coordinates ( r, ϕ, θ , . . . , θ N − ) with r = | x | ∈ [0 , + ∞ ) , ϕ ∈ [0 , π ] , and θ i ∈ [0 , π ] as i = 1 , , . . . , N − with x = r cos ϕ sin θ · · · sin θ N − x = r sin ϕ sin θ · · · sin θ N − ... x N − = r sin θ N − cos θ N − x N = r cos θ N − . (4.14)Proposition 3.1 says that the solutions to − ∆ w = β n U N − w are, in radial coordinates,linear combinations of the n + 1 functions [0 , + ∞ ) × S N − → R : ( r, ϕ, θ , . . . , θ N − ) W n,k ( r ) Y k ( ϕ, θ , . . . , θ N − ) (4.15)for k = 0 , . . . , n , where Y k ( ϕ, θ , . . . , θ N − ) are spherical harmonics with eigenvalue k ( k + N − . For any k , there is only a single (up to a scalar multiple) spherical harmonicwhich is O ( N − -invariant and it is given by the function: Y k ( ϕ, θ , . . . , θ N − ) = Y k ( θ N − ) = P ( N − , N − ) k (cos θ N − ) where r cos θ N − = x N with θ N − ∈ [0 , π ] , (4.16)and P ( N − , N − ) k are the Jacobi Polynomials, see [G] for example. Then, the algebraicmultiplicity of the solutions to − ∆ w = β n U N − w that are O ( N − -invariant is n + 1 .By definition of the space Z , the solution (cid:0) , W n,k ( r ) Y k ( θ N − ) (cid:1) belongs to Z if andonly if Y k is odd with respect to x N , that is iff Y h ( θ N − ) = − Y h ( π − θ N − ) . Since theJacobi Polynomials are even if k is even and odd if k is odd, Y k ( θ N − ) is odd with respectto x N if and only if k is odd. This implies that to compute γ ( n ) we only have to considerthe odd indices k .The radial part corresponding to the index k is given by W n,k ( r ) = r k (1 + r ) k + N − P ( k + N − ,k + N − ) n − k (cid:18) − r r (cid:19) . If n = 2 j , we consider the operator T defined in X + k × X − k . In this way, | x | N − · W n,k (cid:0) x | x | (cid:1) = − W n,k ( x ) since n − k is odd for any k odd. Then γ ( n ) = P nk =0 , k odd j and it is odd if and only if j = 2 ℓ + 1 , or equivalently n = 4 ℓ + 2 .If, instead, n is odd, then n − k is even for any k odd and so we consider the operator T defined in X + k × X + k . Indeed, in this case, W n,k ( r ) ∈ X + k for every k odd and so γ ( n ) = j + 1 and it is odd if and only if j = 2 ℓ , equivalently n = 4 ℓ + 1 and thisconcludes the proof. (cid:3) Proof of Theorem 2.4.
As explained before we are in position to apply Proposition 4.5using Proposition 4.6. The expansion in (2.11) follows again from Proposition 4.5. Fi-nally let us show that our continuum of solutions contains nonradial functions. If bycontradiction we have that u and u are both radial we get that z = u − u is alsoradial. But z is odd in the last variable and so we get that z ≡ . Then u = u and NTIRE RADIAL AND NONRADIAL SOLUTIONS FOR SYSTEMS WITH CRITICAL GROWTH 23 by (F3)–(F4) we deduce that F i ( α, u , u ) = u ∗ − . This implies that u = u = U , acontradiction. (cid:3) The general case: proof of Theorem 2.6.
Since the general case involves hardnotations, for reader’s convenience we consider first the case m = 2 and prove Corollary2.8. The general case does not involve additional difficulties and we just will sketch it.Let h (resp. h ) be the reflection through the hyperplane x N = 0 (resp. x N − = 0 ), S = h O ( N − , h , h i and σ : S → {− , } be the group morphism that satisfies σ ( s ) = 1 whenever s ∈ O ( N − and σ ( h ) = σ ( h ) = − . Thus X S = (cid:8) v ∈ X (cid:12)(cid:12) ∀ x = ( x ′ , x N − , x N ) ∈ R N , v ( x ′ , x N − , x N ) = v ( | x ′ | , − x N − , x N ) ,v ( x ′ , x N − , x N ) = v ( | x ′ | , x N − , − x N ) (cid:9) , Z ≡ Z = (cid:8) z ∈ X + k × X ± k (cid:12)(cid:12) ∀ x = ( x ′ , x N − , x N ) ∈ R N ,z ( x ′ , x N − , x N ) = z ( | x ′ | , − x N − , x N ) ,z ( x ′ , x N − , x N ) = z ( | x ′ | , x N − , − x N ) ,z ( x ′ , x N − , x N ) = − z ( | x ′ | , − x N − , x N ) , and z ( x ′ , x N − , x N ) = − z ( | x ′ | , x N − , − x N ) (cid:9) , With this choice, arguing as in the previous case we have that the only solution in X S to the first equation of (3.7) is the trivial one. As a consequence, Proposition 4.5 appliesand a bifurcation occurs when γ ( n ) is odd.It remains to compute γ ( n ) . To do this we will compute the dimension of Y S k ( R N ) ,the space of spherical harmonics on R N related to the eigenvalue k ( k + N − whichare invariant by the action of S induced by σ (thus, for S = S , we select the sphericalharmonics which are invariant under the action of O ( N − and odd with respect to x N and x N − ).First, let use prove the following decomposition lemma: Lemma 4.7.
Let P S ( R N ) be the space of the polynomials in N variables which areinvariant by the action of O ( N − and such that ∀ x ∈ R N , v ( h i ( x )) = − v ( x ) , for i = 1 , . Then P S ( R N ) = x N x N − R [ r , x N − , x N ] where r = x + · · · + x N − (4.17) and R [ a , . . . , a k ] denotes the space of polynomials in the variables a , . . . , a k .Proof. The proof is similar as in Lemma 6.4 in [SW]. If p ( x ) is a polynomial in x N x N − R [ r , x N − , x N ] then it has an odd degree in x N and x N − and so it satisfies p ( h i ( x )) = − p ( x ) for i = 1 , . Moreover it depends on even powers of x + · · · + x N − and so it is invariant with respect to any s ∈ O ( N − . Thus x N x N − R [ r , x N − , x N ] ⊆P S ( R N ) .Conversely, let p ∈ P S ( R N ) . Since p ( h i ( x )) = − p ( x ) for i = 1 , then each termin p has to contain an odd power of x N − and x N . We can then define the polyno-mial q ( x ) := p ( x ) x N − x N which is even in x N − and x N . Now let s ∈ O ( N − suchthat s ( x , . . . , x N − ) = ( r, , . . . , with r = x + · · · + x N − . Then q is invari-ant so that q ( x , . . . , x N ) = q (cid:0) s ( x , . . . , x N − ) , x N − , x N (cid:1) = q ( r, . . . , , x N − , x N ) = q ( − r, . . . , , x N − , x N ) where the last equality comes from the fact that the map ( x , x , . . . , x N − ) ( − x , x , . . . , x N − ) belongs to O ( N − . Then q has to be evenin r and this implies that q ∈ R [ r , x N − , x N ] . (cid:3) Proposition 4.8.
With this choice of S = S and σ = σ , γ ( n ) is odd if and only if n = 8 ℓ + 2 , n = 8 ℓ + 3 , n = 8 ℓ + 4 or n = 8 ℓ + 5 for ℓ = 0 , , . . . Proof.
Recall that Y k ( R N ) , the space of spherical harmonics of eigenvalue k ( k + N − for − ∆ S N − consists of harmonic homogeneous polynomials of degree k . As stated inProposition 5.5 of [ABR], the space P k of homogeneous polynomials of degree k canbe decomposed as a direct sum of Y k ( R N ) with a subspace isomorphic to P k − . Thisdecomposition still holds when restricted to polynomials that are O ( N − -invariant andodd with respect to x N and x N − . This follows easily using the formula (5.6) of [ABR].As a consequence, dim Y S k ( R N ) = dim P S k ( R N ) − dim P S k − ( R N ) (4.18)where P S k ( R N ) is the space of homogeneous polynomials on R N of degree k which are O ( N − -invariant and odd with respect to x N and x N − .In view of (4.18), we have to compute the dimension of P S k ( R N ) using the decompo-sition in Lemma 4.7.It is not difficult to show that for any h ∈ N we have P S h +1 ( R N ) = { } since anypolynomial in it must contain x N − x N and powers of x + · · · + x N − and this is notpossible if the degree of the polynomial is odd. So we have proved that dim Y S h +1 ( R N ) =0 for any h and N .Then let us compute dim Y S h ( R N ) . Again from Lemma 4.7, we have that P S k ( R N ) =span (cid:8) x h +1 N x k − ℓ − h − N − r ℓ (cid:12)(cid:12) h = 0 , . . . , k − and ℓ = 0 , . . . , k − h − (cid:9) so that dim P S k ( R N ) = k − X h =0 k − h − X ℓ =0 k (cid:16) k (cid:17) and using (4.18) we get for k even dim Y S k ( R N ) = k (cid:16) k (cid:17) − k − (cid:16) k −
22 + 1 (cid:17) = k . (4.19)In this case the unique spherical harmonics which contribute to the computation of γ ( n ) are those of index k even. The corresponding radial part is W n,k ( r ) which belongs to X + if n is even and to X − if n is odd. Then, when n is even we define the operator T in the space X + × X + and we have that γ ( n ) = n X k =0 dim Y S k ( R N ) = ⌊ n ⌋ X j =0 dim Y S j ( R N ) = ⌊ n ⌋ X j =0 j = 12 j n k (cid:16)j n k + 1 (cid:17) NTIRE RADIAL AND NONRADIAL SOLUTIONS FOR SYSTEMS WITH CRITICAL GROWTH 25
Then γ ( n ) is odd when n = 8 j + 2 and n = 8 j + 4 . When n is odd instead, we definethe operator T in the space X + × X − and we have again that γ ( n ) = n X k =0 dim Y S k ( R N ) = ⌊ n ⌋ X j =0 dim Y S j ( R N ) = ⌊ n ⌋ X j =0 j = 12 j n k (cid:16)j n k + 1 (cid:17) Then γ ( n ) is odd when n = 8 j + 3 and n = 8 j + 5 concluding the proof. (cid:3) Proof of Corollary 2.8.
It is the same as the one of Theorem 2.4 (using Proposition 4.8). (cid:3)
Now we sketch the general case of Theorem 2.6. Let N > and m N − .For i = 1 , . . . , m let h i be the reflection through the hyperplane x N +1 − i = 0 , S m = h O ( N − m ) , h , . . . , h m i , and σ m : S m → {− , } be the group morphism defined by σ m ( s ) = 1 for s ∈ O ( N − m ) and σ m ( h i ) = − . Thus X S m = (cid:8) v ∈ X + k (cid:12)(cid:12) ∀ x = ( x ′ , x N − m +1 , . . . , x N ) ∈ R N − m × R m , ∀ i , . . . , i m ∈ N ,v ( x ) = v ( | x ′ | , ( − i x N − m +1 , . . . , ( − i m x N ) } , Z m = (cid:8) z ∈ X + k × X ± k (cid:12)(cid:12) ∀ x = ( x ′ , x N − m +1 , . . . , x N ) ∈ R N − m × R m , ∀ i , . . . , i m ∈ N ,z ( x ) = z ( | x ′ | , ( − i x N − m +1 , . . . , ( − i m x N ) , and z ( x ) = ( − i + ··· + i m z ( | x ′ | , ( − i x N − m +1 , . . . , ( − i m x N ) (cid:9) . As before we have that there is no nontrivial solution in X S m to the first equation of (3.7).Hence by Proposition 4.5 we only have to compute γ ( n ) . Analogously to the case m = 2 we use the following decomposition lemma: Lemma 4.9.
Let P S m ( R N ) be the space of the polynomials in N variables which areinvariant under the action of O ( N − m ) and such that ∀ x ∈ R N , v ( h i ( x )) = − v ( x ) forall i = 1 , . . . , m . Then P S m ( R N ) = x N − m +1 · · · x N R [ r , x N − m +1 , . . . , x N ] where r = x + · · · + x N − m (4.20) and R [ a , . . . , a k ] denotes the space of polynomials in the variables a , . . . , a k . Proposition 4.10.
With this choice of S = S m and σ = σ m , γ ( n ) is odd if and only if (cid:18) m + (cid:4) n − m (cid:5) m (cid:19) is an odd integer . Proof.
As in the proof of Proposition 4.8 we have that dim Y S m k ( R N ) = dim P S m k ( R N ) − dim P S m k − ( R N ) (4.21)where P S m k ( R N ) is the space of homogeneous polynomials on R N of degree k which are O ( N − m ) -invariant and odd with respect to x N − m +1 , . . . , x N . Because of Lemma 4.9,all non-zero polynomials invariant under the action induced by σ on S m must havedegree at least m and so P S m k ( R N ) = { } for k = 0 , . . . , m − , and dim P S m m ( R N ) = 1 .Moreover, as in the case m = 2 , P S m m +2 h +1 ( R N ) = { } for any h ∈ N . For P S m m +2 h ( R N ) , thedecomposition in Lemma 4.9 implies that is is isomorphic to P h ( a , . . . , a m +1 ) , the space of homogeneous polynomials of degree h in m + 1 variables. Thus dim P S m m +2 h ( R N ) =dim P h ( a , . . . , a m +1 ) = (cid:0) h + mm (cid:1) . Then, using (4.21), we get dim Y S m m +2 h ( R N ) = (cid:18) h + mm (cid:19) − (cid:18) h − mm (cid:19) = (cid:18) h + m − m − (cid:19) , h ∈ N . This implies that γ ( n ) = 0 for n m − . As when m = 2 we get γ ( n ) = ⌊ n − m ⌋ X h =0 dim Y S m m +2 h ( R N ) = ⌊ n − m ⌋ X h =0 (cid:18) h + m − m − (cid:19) . Now we use the so called hockey-stick identity ℓ X i = r (cid:18) ir (cid:19) = (cid:18) ℓ + 1 r + 1 (cid:19) which implies γ ( n ) = ⌊ n − m ⌋ X h =0 (cid:18) h + m − m − (cid:19) = (cid:18) m + (cid:4) n − m (cid:5) m (cid:19) . Finally the proof of Theorem 2.6 follows as in Theorem 2.4. (cid:3)
Proof of Theorem 2.6.
From Proposition 4.10 we have that γ ( n ) is odd when (cid:0) m + ⌊ n − m ⌋ m (cid:1) is odd. Then the proof follows from Proposition 4.5. (cid:3) Other solutions
The use of other symmetry subgroups of O ( N ) makes it possible to find differentsolutions. As an example we give another choice that generates nonradial solutions nonequivalent to the previous ones.For m > , let R m be the rotation of angle πm in ϕ , h i the reflection with respect to x i = 0 , i = 2 , . . . , N . Set S m = h R m , h , h , . . . , h N i , and σ m : S m → {− , } be thegroup morphism defined by σ m ( R m ) = 1 , σ m ( h ) = − , and σ m ( h i ) = 1 for i = 3 , . . . , N .(One easily checks that σ m is well defined using R m h R m = h .) Thus, using sphericalcoordinates, see (4.14), X S m = n v ∈ X + k (cid:12)(cid:12) ∀ x = ( r, ϕ, θ , . . . , θ N − ) ∈ R N ,v ( r, ϕ, θ , . . . , θ N − ) = v ( r, π − ϕ, π − θ , . . . , π − θ N − ) ,v ( r, ϕ, θ , . . . , θ N − ) = v (cid:16) r, ϕ + 2 πm , π − θ , . . . , π − θ N − (cid:17)o NTIRE RADIAL AND NONRADIAL SOLUTIONS FOR SYSTEMS WITH CRITICAL GROWTH 27
Z ≡ Z m = n z ∈ X + k × X + k (cid:12)(cid:12) ∀ x = ( r, ϕ, θ , . . . , θ N − ) ∈ R N ,z ( x ) = z ( r, π − ϕ, π − θ , . . . , π − θ N − ) ,z ( x ) = z (cid:16) r, ϕ + 2 πm , π − θ , . . . , π − θ N − (cid:17) ,z ( x ) = − z ( r, π − ϕ, π − θ , . . . , π − θ N − ) ,z ( x ) = z (cid:16) r, ϕ + 2 πm , π − θ , . . . , π − θ N − (cid:17)o . Let us show that, for any m > , the first equation in (3.7) admits only the trivialsolution. By Proposition 3.1, we have that w = P Ni =1 a i x i (1+ | x | ) N/ + bW . By (4.14) andthe definition of X S m , we get that a = a = 0 (using the invariance with respect to R m ) and a = · · · = a N − = 0 (using that cos θ i = cos( π − θ i ) , for any i = 1 , . . . , θ N − ).Finally b = 0 since W X + k . Thus the assumptions of Proposition 4.2 are satisfied. Toapply Proposition 4.4, we also need: Proposition 5.1.
Let m > , n = m , S := S n and σ := σ n . Then γ ( n ) = 1 . (5.1) Proof.
By Proposition 3.1 all solutions to the second equation of (3.7) corresponding to α ∗ m are given by P mk =0 A k W m,k ( r ) Y k ( ϕ, θ , . . . , θ N − ) . We know from [W] (see also [AG]for another use of this expansion in bifurcation theory) that Y k ( ϕ, θ , . . . , θ N − )= X j =0 ,...,ki i ··· i N − i = j, i N − = k N − Y ℓ =1 G i ℓ − i ℓ (cos θ ℓ , ℓ − (cid:16) B i ...i N − j cos jϕ + C i ...i N − j sin jϕ (cid:17) , (5.2)where G i ( · , ℓ ) are the Gegenbauer polynomials namely, ∞ X i =0 G i ( ω, ℓ ) x i = (1 − xω + x ) − (1+ ℓ ) / , while G ki ( ω, ℓ ) = (1 − ω ) k/ d k d ω k G i ( ω, ℓ ) . By definition of the space Z m , the solution (cid:0) , P mk =0 A j W m,k ( r ) Y k ( ϕ, θ , . . . , θ N − ) (cid:1) be-longs to Z m if and only if Y k ( ϕ, θ , . . . , θ N − ) is π/m periodic in ϕ , changes signunder the transformation ϕ π − ϕ , and is invariant under the transformations θ i π − θ i . The first two imply that Y k ( ϕ, θ , . . . , θ N − ) must not be constant in ϕ and k > j > m . Thus solutions to the second equation in (3.7) with Z m -invariance aremultiple of W m,m ( r ) Y m ( θ ) . Moreover, the unique nonzero coefficient in (5.2) is C m...mm .Because G i ( · , ℓ ) is a polynomial of degree ℓ , G mm ( ω, ℓ ) is a constant multiple of (1 − ω ) m/ . A straightforward computation shows that Y m ( ϕ, θ , . . . , θ N − ) = (sin θ N − ) m · · · (sin θ ) m sin( mϕ ) = ℑ m( x + i x ) m . Observe that Y m ( ϕ, θ , . . . , θ N − ) is invariant with respect to the reflection h i for i =3 , . . . , N , with respect to the rotation R m and it is odd in ϕ so that (0 , W m,m ( r ) Y m ( θ )) belongs to Z m . Recalling that W m,m ( r ) = r m (1+ r ) m + N − ∈ X + k , we have that γ ( m ) = 1 . (cid:3) Proof of Theorem 2.9.
From the previous discussion we have that the assumption ofLemma 4.2 are satisfied. Then the proof follows as in the case of Theorem 2.1 since wehave a one dimensional kernel. (cid:3)
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Francesca Gladiali, Dipartimento Polcoming, Università di Sassari - Via Piandanna 4,07100 Sassari, Italy.
E-mail address : [email protected] Massimo Grossi, Dipartimento di Matematica, Università di Roma La Sapienza, P.le A.Moro 2 - 00185 Roma, Italy.
E-mail address : [email protected] Christophe Troestler, Département de mathématique, Université de Mons, place duparc 20, B-7000 Mons, Belgium.
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