Doubling nodal solutions to the Yamabe equation in R n with maximal rank
aa r X i v : . [ m a t h . A P ] O c t DOUBLING NODAL SOLUTIONS TO THE YAMABE EQUATION IN R n WITH MAXIMAL RANK
MARIA MEDINA AND MONICA MUSSO
Abstract.
We construct a new family of entire solutions to the Yamabe equation − ∆ u = n ( n − | u | n − u in D , ( R n ) . If n = 3 our solutions have maximal rank, being the first example in odd dimension. Ourconstruction has analogies with the doubling of the equatorial spheres in the construction ofminimal surfaces in S (1). Introduction
Consider the problem − ∆ u = γ | u | p − u in R n , γ := n ( n − , u ∈ D , ( R n ) , (1.1)where n > p := n +2 n − and D , ( R n ) is the completion of C ∞ ( R n ) with the norm k∇ u k L ( R n ) .Problem (1.1) corresponds to the steady state of the energy-critical focusing nonlinear waveequation ∂ t u − ∆ u − | u | n − u = 0 , ( t, x ) ∈ R × R n , whose study (see for instance [4, 6, 7, 14, 15]) naturally relies on the complete classification ofthe set of non-zero finite energy solutions to (1.1), which is defined byΣ := (cid:26) Q ∈ D , ( R n ) \{ } : − ∆ Q = n ( n − | Q | n − Q (cid:27) , in particular in connection with the soliton resolution conjecture for which only a few exampleshave become known [13, 14, 5, 6, 7]. Observe that (1.1) is the Euler-Lagrange equation of thefunctional defined by e ( u ) := 12 Z R n |∇ u | dx − γ ( n − n Z R n | u | nn − dx. (1.2)Positive solutions to (1.1) solve the Yamabe problem on the sphere (after a stereographic pro-jection) and are the extremal functions for the Sobolev embedding. Thanks to the classical workof Caffarelli-Gidas-Spruck [2], it is known that all positive solutions to (1.1) are given by the socalled bubble and all its possible translations and dilations, that is, U ( y ) := (cid:18)
21 + | y | (cid:19) n − and U α,y ( y ) := α − n − U (cid:18) y − y α (cid:19) , α > , y ∈ R n , (1.3) The first author was supported by the European Union’s Horizon 2020 research and innovation programmeunder the Marie Sklodowska-Curie grant agreement N 754446 and UGR Research and Knowledge Transfer Found- Athenea3i. The second author is supported by EPSRC Research Grant EP/T008458/1. previously discovered independently by Aubin [1] and Talenti [19]. In fact, all radial solutionsin Σ have the form (1.3). Sign-changing solutions belonging to Σ have thus to be non radial.Using the classical theory of Ljusternik-Schnirelman category, Ding proved in [3] the existenceof infinitely many elements in Σ that are non radial, sign-changing, and with arbitrary largeenergy. The key idea in [3] is to look for solutions to (1.1) that are invariant under the action of O (2) × O ( n − ⊂ O ( n ) to recover compactness for the functional e ( u ). No further informationthough is known on the solutions found by Ding. Recently, more explicit constructions for sign-changing (non radial) solutions to (1.1) have been obtained by del Pino-Musso-Pacard-Pistoiaand Medina-Musso-Wei (see [8, 9, 16]). The solutions obtained in [8] are invariant under theaction of D k × O ( n − D k is the dihedral group of rotations and reflections leavinga regular polygon with k sides invariant. More precisely, for any k large enough, the authorsconstruct a solution to (1.1) looking like the bubble U in (1.3) surrounded by k negative scaledcopies of U arranged along the vertices of a k -regular polygon in R . At main order the solutionlooks like U ( y ) − k X j =1 λ − n − U (cid:0) λ − ( y − ξ j ) (cid:1) , (1.4)where ξ j := ( e π ( j − ik , , . . . ), λ = O ( k − ) if n > λ = O (( k ln k ) − ) if n = 3, as k → ∞ .Observe that k X j =1 λ − n − U (cid:0) λ − ( y − ξ j ) (cid:1) ⇀ c n δ Γ , as k → ∞ , for a positive constant c n , where δ Γ is the Dirac-delta at the equatorial on the ( y , y )-planeΓ = { y ∈ R n : y + y = 1 } in S n (1). We thus can think of the solutions obtained in [8]and described at main order in (1.4) as the sum of a positive fixed central bubble surroundedby a negative smooth function that desingularizes a Dirac-delta along the equatorial Γ, in thelimit as k → ∞ . We call this construction a desingularization of the equatorial , in analogy withsimilar desingularization constructions for minimal surfaces in Riemannian three-manifolds [10].We remark that these solutions are not invariant under the action of O (2) × O ( n − Q ∈ Σ, consider the linearoperator L Q := − ∆ − γp | Q | p − and define the null space Z Q := { f ∈ D , ( R n ) : f = 0 , L Q ( f ) = 0 } . Duyckaerts-Kenig-Merle [5] introduced the following definition of non-degeneracy for a solutionof problem (1.1): Q ∈ Σ is said to be non degenerate if Z Q coincides with the vector spacegenerated by the elements in Z Q related to the group of isometries in D , ( R n ) under whichproblem (1.1) is invariant, given by translations, scalings, rotations and Kelvin transformation.More precisely, Q is non degenerate if Z Q = ˜ Z Q , where ˜ Z Q := span (2 − n ) x j Q + | x | ∂ x j Q − x j x · ∇ Q, ∂ x j Q, j n, ( x j ∂ x k − x k ∂ x j ) Q, j < k n, n − Q + x · Q . OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 3
The rank of a solution Q ∈ Σ is the dimension of the vector space ˜ Z Q , and it cannot exceed thenumber N := 2 n + 1 + n ( n − , being this the largest possible dimension for ˜ Z Q . It is well known for instance that U in (1.3)is non degenerate and its rank is n + 1 (see [18]). In [17] it is proven that the solutions built in[8], looking at main order as in (1.4), are non degenerate and their rank is 3 n .A question we address in [16] is about the existence of solutions Q ∈ Σ to (1.1) whose rankis maximal. In fact, observe that not the bubbles in (1.3) nor the solutions built in [8] havemaximal rank. We partially answer this question building a new family of solutions to (1.1),where another polygon with a large number of sides is replicated for n > U ( y ) − k X j =1 λ − n − U ( λ − ( y − ξ j )) − h X j =1 µ − n − U ( µ − ( y − ξ j )) , (1.5)where ξ j := ( e π ( j − ik , , . . . ), ξ j := (0 , , e π ( j − ih , , . . . ), λ = O ( k − ) and µ = O ( h − ), for k and h sufficiently large, see [16]. Using the terminology we introduced before, this construction is a desingularization of the two equators , one in the ( y , y )-plane and the other in the ( y , y )-plane.These solutions are non degenerate. Furthermore, their rank is 5 ( n − maximal when n = 4. A generalization of this result is to consider non degenerate solutions obtained by gluinga central positive bubble with negative scaled copies of U centered at the vertices of ℓ regularpolygons with a large number of sides and lying in consecutive planes, which have maximal rankprovided the dimension n is 2 ℓ . In other words, a desingularization of ℓ equators in consecutiveplanes would provide an example of non degenerate solutions with maximal rank in any evendimension 2 ℓ . If the dimension n is odd, the existence of non degenerate solutions for (1.1) withmaximal rank remains an open problem and a different construction is required.Roughly speaking, the solution built in [16] breaks the radial behavior of the bubble in thefirst four coordinates, loosing the related invariances. This fact adds extra terms in the kernelof the linearized operator, being all the possible precisely when n = 4. Thus, to prove theanalogue in odd dimension one needs to find a solution breaking the radiality in an odd numberof coordinates.The aim of this work is to address this question and to provide a new family of sign-changingsolutions for problem (1.1) that we claim to have maximal rank in dimension n = 3. We provethe following result. Theorem 1.1.
Let n > and let k be a positive integer. Then for any sufficiently large k thereis a finite energy solution to (1.1) of the form u ( y ) = U ( y ) − k X j =1 λ − n − U ( λ − ( y − ξ j )) − k X j =1 λ − n − U ( λ − ( y − ξ j )) + o k (1)(1 + λ − n − ) , where ξ j := R ( p − τ e π ( j − k i , τ, , . . . ) , ξ j := R ( p − τ e π ( j − k i , − τ, , . . . ) , j = 1 , . . . , k, M. MEDINA AND M. MUSSO and λ := ℓ n − k , τ := tk − n − , if n > , λ := ℓ k (ln k ) , τ := t √ ln k , if n = 3 . Here λ + R = 1 , and η < ℓ, t < η − , for some positive fixed number η independent of k . The term o k (1) → uniformly on compactsets of R n as k → ∞ .These solutions have maximal rank in dimension . Some remarks are in order.
Remark . Let us briefly discuss our construction. The solution predicted by Theorem 1.1looks at main order as u ( y ) := U ( y ) − k X j =1 λ − n − U ( λ − ( y − ξ j )) − k X j =1 λ − n − U ( λ − ( y − ξ j )) . The polygonal distribution of the points ξ , . . . , ξ k and ξ , . . . ξ k makes u a function with severalimportant symmetries: it is invariant under rotation of angle πk in the ( y , y )-plane, and evenin the other variables y j , j = 3 , . . . , n . The assumption that λ + R = 1 gives that u is alsoinvariant under Kelvin transformation. We will take great advantage of these symmetries inmany different ways in our proof. For instance they allow us to choose the same scaling factor λ for each one of the negative bubbles centred at the different points ξ j and ξ j , j = 1 , . . . , k , whichin principle may not be the same, reducing substantially the number of scaling parameters toadjust. Taking λ and τ small positive parameters as k → ∞ , a formal computation shows thatthe energy functional defined in (1.2) and evaluated at u = u has the following expansion, as k → ∞ , e ( u ) ∼ (2 k + 1) a n + k (cid:18) λ n − ( b n − c n τ ) − d n λ n − k n − − e n λ n − τ n − (cid:19) , when dimension n >
4, for some explicit positive constants a n , b n , c n , d n and e n . Our choicefor λ and τ in terms of k is to get at main order the balance ∇ λ,τ e ( u ) ∼ . We will justify this heuristic argument in Section 2.
Remark . The construction obtained in Theorem 1.1 differs from the desingularization ofthe equatorial in (1.4) obtained in [8] or of two equators in (1.5) in [16]. In fact the solutionsin Theorem 1.1 can be thought as the sum of a positive fixed central bubble surrounded by anegative smooth function that desingularizes Dirac-deltas located in points on two circles thatare collapsing into a Dirac-delta supported along the equatorial Γ, in the limit as k → ∞ .We call this construction a doubling of the equatorial in the ( y , y )-plane, in analogy withsimilar doubling constructions for minimal surfaces in Riemannian three-manifolds obtained in[10, 11, 12]. OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 5
Remark . Let u be the solution predicted in Theorem 1.1 and define the following 4 n − z ( y ) := n − u ( y ) + ∇ u ( y ) · y, z α ( y ) := ∂∂y α u ( y ) , α = 1 , . . . , n,z n +1 ( y ) := − y z ( y ) + | y | z ( y ) , z n +2 ( y ) := − y z ( y ) + | y | z ( y ) ,z n +3 ( y ) := − y z ( y ) + | y | z ( y ) ,z n + α +2 ( y ) := − y α z ( y ) + y z α ( y ) , α = 2 , . . . , n,z n + α ( y ) := − y α z ( y ) + y z α ( y ) , α = 3 , . . . , n,z n + α − ( y ) := − y α z ( y ) + y z α ( y ) , α = 4 , . . . , n. These functions belong to D , ( R n ) and also to the kernel of L u := − ∆ − γp | u | p − . Furthermore,they are linearly independent (see Appendix B). Consider now the case n = 3: we have that4 n − N = 10. Thus the solutions constructed in Theorem 1.1 have maximal rank in dimension3. The non-degeneracy of the solution remains an open problem. Remark . The doubling of the equatorial in R presumably closes the question about solutionswith maximal rank. Indeed, for any odd dimension we could combine a doubling of the equatorialin three coordinates with a desingularization of the equatorial as in [8] in two coordinates or a desingularization of two equators as in [16] in four coordinates, as many times as needed. Weconjecture that combining these three structures we can build a maximal solution in any odddimension. Remark . This doubling construction also provides an alternative example of solution withmaximal rank for even dimensions of the form n = 6 ℓ , ℓ ∈ N . For these dimensions, in [16] theauthors propose to replicate 3 ℓ times the desingularization of [8]. We conjecture that combining2 ℓ structures like the doubling of the equatorial of Theorem 1.1 would provide a different solutionwith maximal rank in these dimensions. Remark . The construction described in Theorem 1.1 is not the only possible way to doublethe equatorial
Γ. For any even integer 2 m , we can construct another sequence of solutions that doubles the equatorial in the form of the sum of a positive fixed central bubble surrounded by anegative smooth function that desingularizes Dirac-deltas located in points on 2 m circles thatare collapsing into a Dirac-delta supported along the equatorial Γ, in the limit as k → ∞ , where m of these circles collapse onto Γ from above and m from below.We can also combine desingularization of the equatorial and doubling of the equatorial . Forany odd integer 2 m + 1, we can construct a sequence of solutions with the form of the sum ofa positive fixed central bubble surrounded by a negative smooth function that consists of twoparts. One part desingularizes Dirac-deltas located in points on 2 m circles that are collapsinginto a Dirac-delta supported along the equatorial Γ, in the limit as k → ∞ . The other partdesingularizes Dirac-deltas located at points along the equatorial, desingularizing a Dirac-deltaalong the equatorial Γ, in the limit as k → ∞ .Since the proofs of these constructions are in the same spirit as the one of Theorem 1.1, wewill briefly describe them in section 6, explaining the principal differences.The proof of Theorem 1.1 is based on a Lyapunov-Schmidt reduction in the spirit of [8]: wedefine a first approximation and we look for a solution in a nearby neighborhood, by linearizingaround the approximation and, after developing an appropriate linear theory, solving by a fixed M. MEDINA AND M. MUSSO point argument. This allows to reduce the original problem to the solvability of a finite dimen-sional one. However, for the construction in Theorem 1.1 this last step is rather delicate. Infact, the finite dimensional reduction leads to two equations (in the two parameters to adjust, λ and τ ) where the sizes of the error and the non linear term play a fundamental role. If onefollows the strategy of [8], the nonlinear term cannot be controlled and the reduced problemcannot be solved. For this reason, we need to carry on a much more refined argument.The key point is the following: if one pays attention to the error term near the bubbles, thiscan be decomposed in a (relatively) large but symmetric part, and a smaller but non symmetricpart (see Proposition 2.2). In the final argument (the reduction procedure) the symmetric partis orthogonal to the element of the kernel, and therefore not seen. Thus, the part playing a rolein the reduction is the non symmetric one, that is significatively smaller. Roughly speaking, thisallows to solve the linearized problem in two parts, a symmetric and large but irrelevant part,and a non symmetric but small one. Indeed, if our solution has the form u ∗ + φ , being u ∗ theapproximation, we will split φ near each bubble as φ = φ s + φ ∗ , where φ s is symmetric withrespect to the hyperplane y = τ (or analogously y = − τ ). This behavior is also inherited by thenonlinear part of the equation and we will be able to perform the fixed point argument settingthe size of φ ∗ very small (see Proposition 4.6), what will allow us to conclude the reduction. Thisnew strategy requires delicate decompositions and estimates of every term of the equations, aswell as the development of a sharp linear invertibility theory.The structure of the article is the following. In section 2 we detail the approximation and theerror associated, estimating it near and far from the bubbles in different norms, and identifyingthe symmetric and non symmetric parts and their sizes. Section 3 is devoted to the linear theory,where a refinement of the theory in [8] is developed. Section 4 is the core of the strategy, wherethe gluing scheme is performed, together with the precise decomposition of the function in theirsymmetric and non symmetric part. In section 5 we carry on the dimensional reduction, con-cluding the proof of Theorem 1.1. The appendix contains fundamental computations concerningthe shape of the approximation.2. Doubling construction: a first approximation
Let n > τ ∈ (0 , R n we fix the following points P := ( p − τ , , τ, , . . . , , P := ( p − τ , , − τ, , . . . , . Let λ ∈ (0 ,
1) be a positive number, and define R as λ + R = 1 . (2.1)Let k be an integer number and u [ λ, τ ]( y ) := U ( y ) − k X j =1 λ − n − U y − ξ j λ !| {z } U j ( y ) − k X j =1 λ − n − U y − ξ j λ !| {z } U j ( y ) (2.2)where ξ j := R ( p − τ cos θ j , p − τ sin θ j , τ, , . . . , ,ξ j := R ( p − τ cos θ j , p − τ sin θ j , − τ, , . . . , , with θ j := 2 π j − k . OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 7
Observe that ξ = RP and ξ = RP , while ξ j and ξ j are obtained respectively from P and P after a rotation in the ( y , y )-plane of angle 2 π j − k . Thanks to (2.1), the functions U j , U j areinvariant under Kelvin transform, so that u is also invariant under this transformation, that is u ( y ) = | y | − n u (cid:18) y | y | (cid:19) . (2.3)A direct observation reflects that u also shares the following symmetries u ( y , . . . , − y j , . . . , y n ) = u ( y , . . . , y j , . . . , y n ) , j = 2 , . . . , n, (2.4) u ( e π ( j − k i ¯ y, y , . . . , y n ) = u (¯ y, y , . . . , y n ) , ¯ y = ( y , y ) , j = 2 , . . . , k. (2.5)In our construction, we assume that the integer k is large and the parameters λ and τ are givenby λ := ℓ n − k , τ := tk − n − , if n > ,λ := ℓ k (ln k ) , τ := t √ ln k , if n = 3 , where η < ℓ, t < η − , (2.6)for some η small and fixed, independent of k , for any k large enough. The error function, definedas E [ ℓ, t ]( y ) := ∆ u + γ | u | p − u , y ∈ R n , (2.7)inherits the symmetries (2.4), (2.5). As a consequence, fixing a small δ > k , itis enough to describe the error function in the sets B ( ξ , δk ) and R n \ S kj =1 (cid:16) B ( ξ j , δk ) ∪ B ( ξ j , δk ) (cid:17) in order to know it in the whole space R n .For our purpose it is convenient to measure the error using the following weighted L q norm k h k ∗∗ := k (1 + | y | ) n +2 − nq h k L q ( R n ) . (2.8)For the moment we request that q is a fixed number with n < q < n . Later on, we will needa more restrictive assumption on q , n < q < n − n − . We will evaluate the k · k ∗∗ -norm of E in the interior regions B ( ξ j , δk ) and B ( ξ j , δk ), for any j = 1 , . . . , k , and in the exterior region R n \ S kj =1 (cid:16) B ( ξ j , δk ) ∪ B ( ξ j , δk ) (cid:17) . The error in the interior regions B ( ξ j , δk ) and B ( ξ j , δk ) , j = 1 , . . . , k . To describe the error E in each one of the balls B ( ξ j , δk ), B ( ξ j , δk ), it is enough to do it in B ( ξ , δk ), as already observed.In this region the dominant term of the function u in (2.2) is U . Thus, for some s ∈ (0 , γ − E ( x ) = p U ( x ) + s [ X j =1 U j ( x ) + k X j =1 U j ( x ) − U ( x )] p − − X j =1 U j ( x ) − k X j =1 U j ( x ) + U ( x ) + X j =1 U pj ( x ) + k X j =1 U pj ( x ) − U p ( x ) , M. MEDINA AND M. MUSSO for x ∈ B ( ξ , δk ). Let us introduce the change of variable λy = x − ξ , so that U ( λy + ξ ) = λ − n − U ( y ). In these expanded variables, the error takes the form λ n +22 γ − E ( ξ + λy ) = p U ( y ) + sλ n − [ X j =1 U j ( λy + ξ ) + k X j =1 U j ( λy + ξ ) − U ( λy + ξ )] p − ×× λ n − X j =1 U j ( λy + ξ ) + k X j =1 U j ( λy + ξ ) − U ( λy + ξ ) + λ n +22 X j =1 U pj ( λy + ξ ) + k X j =1 U pj ( λy + ξ ) − U p ( λy + ξ ) , (2.9)for some s ∈ (0 , | y | < δλk . A direct Taylor expansion gives that U j ( λy + ξ ) = 2 n − λ n − ( λ + | λy + ξ − ξ j | ) n − for j = 1= 2 n − λ n − | ξ − ξ j | n − " − ( n −
2) ( y, ξ − ξ j ) | ξ − ξ j | λ + O λ (1 + | y | ) | ξ − ξ j | ! ,U j ( λy + ξ ) = 2 n − λ n − ( λ + | λy + ξ − ξ j | ) n − for j = 1 , . . . , k = 2 n − λ n − | ξ − ξ j | n − " − ( n −
2) ( y, ξ − ξ j ) | ξ − ξ j | λ + O λ (1 + | y | ) | ξ − ξ j | ! ,U ( λy + ξ ) = U ( ξ ) (cid:20) − ( n −
2) ( y, ξ )1 + | ξ | λ + O (cid:18) λ | y | | ξ | (cid:19)(cid:21) , (2.10)uniformly for | y | δλk . In the Appendix we will show that k X j =2 | ξ − ξ j | n − = k X j =2 | ξ − ξ j | n − = ( A n k n − (cid:0) O ( τ ) (cid:1) if n > ,A k ln k (cid:0) O ( τ ) (cid:1) if n = 3 , (2.11)and k X j =1 | ξ − ξ j | n − = B n kτ n − (cid:0) O (( τ k ) − ) (cid:1) if n > ,B n kτ n − (1 + O ( τ )) if n = 4 ,A k ln (cid:0) πτ (cid:1) (cid:0) O ( | ln τ | − ) (cid:1) if n = 3 , (2.12)where A := π − and, if n > A n := 2(2 π ) n − ∞ X j =1 j − n , B n := 22 n − π Z ∞ ds (1 + s ) n − . OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 9
From (2.9), (2.10), (2.11) and (2.12) we get that | λ n +22 γ − E ( ξ + λy ) | C " λ n − | y | + λ n +22 for | y | δλk . Notice that, to obtain this estimate, we do not need to use the precise informationgathered in (2.11) and (2.12), but only the order in k . Indeed, it is enough to get the upperbound | U j ( λy + ξ ) | C for | y | δλk , and the estimate follows since, as (2.12) reflects, | U j ( λy + ξ ) | is smaller.Direct computations (see [8]) give k (1 + | y | ) n +2 − nq λ n +22 γ − E ( ξ + λy ) k L q ( | y | < δλk ) ( Ck − nq if n > , Ck ln k if n = 3 , for some fixed constant C >
0. Therefore, by symmetry we conclude that, for any j = 1 , . . . , k , k (1 + | y | ) n +2 − nq λ n +22 γ − E ( ξ j + λy ) k L q ( | y | < δλk ) (cid:26) Ck − nq if n > , Ck ln k if n = 3 , (2.13)and k (1 + | y | ) n +2 − nq λ n +22 γ − E ( ξ j + λy ) k L q ( | y | < δλk ) (cid:26) Ck − nq if n > , Ck ln k if n = 3 . (2.14) The error in R n \ S kj =1 (cid:16) B ( ξ j , δk ) ∪ B ( ¯ ξ j , δk ) (cid:17) . For y in this region we have | E ( y ) | C | y | ) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X j =1 " λ n − | y − ξ j | n − + λ n − | y − ¯ ξ j | n − n − × k X j =1 λ n − | y − ξ j | n − + λ n − | y − ξ j | n − ! C λ n − (1 + | y | ) k X j =1 | y − ξ j | n − + 1 | y − ξ j | n − ! , (2.15)where we have used that in this exterior region k X j =1 λ n − | y − ξ j | n − C, k X j =1 λ n − | y − ξ j | n − C. (2.16) For n >
4, using (2.6), k (1 + | y | ) n +2 − nq E k L q ( R n \ S kj =1 (cid:16) B ( ξ j , δk ) ∪ B (¯ ξ j , δk ) (cid:17) ) Cλ n − k X j =1 Z | y − ξ j | > δk (1 + | y | ) ( n +2) q − n − q | y − ξ j | ( n − q dy ! q + Z | y − ¯ ξ j | > δk (1 + | y | ) ( n +2) q − n − q | y − ξ j | ( n − q dy ! q Cλ n − k Z δk t n − t ( n − q dt ! q Ck − nq , (2.17)for some constant C . Similarly, if n = 3, we get k (1 + | y | ) n +2 − nq E k L q ( R n \ S kj =1 ( B ( ξ j , δk ) ∪ B (¯ ξ j , δk ))) C ln k . (2.18)Estimates (2.13), (2.14), (2.17) and (2.18) merge in the following Proposition 2.1.
Assume that λ and τ satisfy (2.6) . There exist an integer k and a positiveconstant C such that for all k > k the following estimates hold true k E k ∗∗ Ck − nq if n > and k E k ∗∗ C | ln k | − if n = 3 . We refer to (2.8) for the definition of the k · k ∗∗ -norm. The following result is the key point of the argument carried out for the gluing procedure andthe reduction method in sections 4 and 5. It provides a decomposition of the interior error termin two parts: one is relatively large but symmetric, and the other is non symmetric but smallerin size. Thanks to this observation we can refine the fixed point argument to fix a smaller sizeof the functions involved in the reduction.
Proposition 2.2.
Assume that λ and τ satisfy (2.6) and | y | < δλk . Then, there exists a decom-position E ( ξ + λy ) = E s ( ξ + λy ) + E ∗ ( ξ + λy ) , such that E s ( ξ + λy ) is even with respect to y α for all α = 1 , . . . , n and there exists an integer k such that, for k > k , | λ n +22 E ∗ ( ξ + λy ) | C λ n − k | y | if n > ,C λ / k (ln k ) | y | if n = 3 , , | y | < δλk . (2.19) OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 11
Proof.
The idea is to identify in (2.9) the main order terms using (2.10). Indeed, we write, for | y | < δkλ , γ − λ n +22 E s ( ξ + λy ) := pU ( y ) p − λ n − X j =1 n − λ n − | ξ − ξ j | n − + k X j =1 n − λ n − | ξ − ξ j | n − − U ( ξ ) + λ n +22 X j =1 n − λ n − | ξ − ξ j | n − ! p + k X j =1 n − λ n − | ξ − ξ j | n − ! p − U p ( ξ ) , (2.20)that is even with respect to every y α , α = 1 , . . . , n . Let us define E ∗ ( ξ + λy ) := E ( ξ + λy ) − E s ( ξ + λy ) . Using (2.11) (renaming n − n − n >
4) we get that (cid:12)(cid:12)(cid:12)(cid:12) λ n − X j =1 ( y, ξ − ξ j ) | ξ − ξ j | n λ (cid:12)(cid:12)(cid:12)(cid:12) λ n | y | X j =1 | ξ − ξ j | n − Cλ n k n − | y | ( C | y | k if n > ,C | y | k (ln k ) if n = 3 , and (2.19) follows for | y | < δλk . (cid:3) The linear theory
Consider the linear problem L ( ϕ ) = h ( y ) in R n , L ( ϕ ) := ∆ ϕ + pγU p − ϕ. (3.1)It is known that ker { L } = span { Z , Z , . . . , Z n +1 } , with Z α := ∂ y α U, α = 1 , . . . , n, Z n +1 := y · ∇ U + n − U. (3.2)Defining the norm k ϕ k ∗ := k (1 + | y | n − ) ϕ k L ∞ ( R n ) , (3.3)in [8, Lemma 3.1] the following existence result is proved. Lemma 3.1.
Assume n < q < n in the definition of k · k ∗∗ . Let h be a function such that k h k ∗∗ < + ∞ and Z R n Z α h = 0 for all α = 1 , . . . , n + 1 . Then (3.1) has a unique solution ϕ with k ϕ k ∗ < + ∞ such that Z R n U p − Z α ϕ = 0 for all α = 1 , . . . , n + 1 , and k ϕ k ∗ C k h k ∗∗ , for some constant C depending only on q and n . We will also need a priori estimates of the gradient of such solution.
Lemma 3.2.
Let ϕ be the solution of (3.1) predicted by Lemma 3.1 and assume k (1+ | y | n +2 ) h k L ∞ ( R n ) < + ∞ . Then, there exists C > depending only on n such that k∇ ϕ k L ∞ ( R n ) C ( k ϕ k ∗ + k (1 + | y | n +2 ) h k L ∞ ( R n ) ) . Proof.
By standard elliptic estimates, k∇ ϕ k L ∞ ( B ) C (cid:0) k ϕ k L ∞ ( B ) + k h k L ∞ ( B ) (cid:1) C (cid:0) k ϕ k ∗ + k (1 + | y | n +2 ) h k L ∞ ( R n ) (cid:1) . (3.4)Defining ˜ ϕ ( y ) := | y | − n ϕ ( | y | − y ) it can be checked that∆ ˜ ϕ + pγU p − ˜ ϕ = ˜ h in R n \ { } , with ˜ h ( y ) := | y | − n − h ( y | y | − ), and thus k∇ ˜ ϕ k L ∞ ( B ) C (cid:16) k ˜ ϕ k L ∞ ( B ) + k ˜ h k L ∞ ( B ) (cid:17) . Noticing that k ˜ ϕ k L ∞ ( B ) = k| y | n − ϕ k L ∞ ( R n \ B / ) k ϕ k ∗ , k ˜ h k L ∞ ( B ) = k| y | n +2 h k L ∞ ( R n \ B / ) k (1 + | y | n +2 ) h k L ∞ ( R n ) , we obtain k∇ ˜ ϕ k L ∞ ( B ) C (cid:0) k ϕ k ∗ + k (1 + | y | n +2 ) h k L ∞ ( R n ) (cid:1) . (3.5)Writing ϕ ( y ) = | y | − ( n − ˜ ϕ ( | y | − y ) it can be seen that |∇ ϕ ( y ) | C | y | n − | ˜ ϕ ( | y | − y ) | + C | y | n |∇ ˜ ϕ ( | y | − y ) | , and thus k∇ ϕ k L ∞ ( R n \ B ) C (cid:0) k ˜ ϕ k L ∞ ( B ) + k∇ ˜ ϕ k L ∞ ( B ) (cid:1) (3.6)Putting together (3.4), (3.5) and (3.6) the estimate follows. (cid:3) The gluing scheme
Our goal will be to find a solution of the form u = u + φ, (4.1)with φ a small function (in a sense that will be precised later). Thus, u is a solution of (1.1) ifand only if ∆ φ + pγ | u | p − φ + E + γN ( φ ) = 0 , (4.2)where N ( φ ) := | u + φ | p − ( u + φ ) − | u | p − u − p | u | p − φ, and E was defined in (2.7).Let ζ ( s ) be a smooth function such that ζ ( s ) = 1 for s < ζ ( s ) = 0 for s >
2, and let δ > k . Let us define ζ j ( y ) := ( ζ ( kδ − | y | − | y − ¯ ξ j | y | | ) if | y | > ,ζ ( kδ − | y − ¯ ξ j | ) if | y | , ζ j ( y ) := ζ j ( y , y , − y , . . . , y n ) (4.3)for j = 1 , . . . , k . We notice that a function φ of the form φ = k X j =1 ( φ j + φ j ) + ψ OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 13 is a solution of (4.2) provided the functions φ j , φ j and ψ solve the following system of couplednon linear equations∆ φ j + pγ | u | p − ζ j φ j + ζ j (cid:2) pγ | u | p − ψ + E + γN ( φ ) (cid:3) = 0 , j = 1 , . . . , k, (4.4)∆ φ j + pγ | u | p − ζ j φ j + ζ j (cid:2) pγ | u | p − ψ + E + γN ( φ ) (cid:3) = 0 , j = 1 , . . . , k, (4.5)∆ ψ + pγU p − ψ + pγ ( | u | p − − U p − )(1 − k X j =1 ( ζ j + ζ j ))+ pγU p − k X j =1 ( ζ j + ζ j ) ψ + pγ | u | p − k X j =1 (1 − ζ j ) φ j + pγ | u | p − k X j =1 (1 − ζ j ) φ j + − k X j =1 ( ζ j + ζ j ) ( E + γN ( φ )) = 0 . (4.6)Given our setting it is natural to ask for some symmetry properties on φ j and φ j . In particular,denoting ˆ y := ( y , y ) and y ′ := ( y , . . . , y n ), we want them to satisfy φ j (ˆ y, y ′ ) = φ ( e π ( j − k i ˆ y, y ′ ) , j = 1 , . . . k, (4.7)where φ ( y , . . . , y α , . . . , y n ) = φ ( y , . . . , − y α , . . . , y n ) , α = 2 , , . . . , n,φ ( y ) = | y | − n φ ( | y | − y ) , (4.8)and φ ( y ) = φ ( y , y , − y , . . . , y n ) . (4.9) Remark . The functions φ j and φ j are not even in the third coordinate separately but φ j + φ j is so, that is, ( φ j + φ j )( y ) = ( φ j + φ j )( y , y , − y , . . . , y n ) . Likewise, the functions ( ζ j + ζ j ) and ( ζ j φ j + ζ j φ j ) are even in the third coordinate.For ρ > k φ k ∗ ρ, (4.10)where φ ( y ) := λ n − φ ( ξ + λy ) and k · k ∗ is defined in (3.3). Proposition 4.2.
There exist constants k , C and ρ such that, for all k > k and ρ < ρ , if φ j and φ j satisfy conditions (4.7) - (4.10) then there exists a unique solution ψ = Ψ( φ ) of (4.6) such that ψ ( y , . . . , y α , . . . ) = ψ ( y , . . . , − y α , . . . ) , α = 3 , . . . , n, (4.11) ψ (ˆ y, y ′ ) = ψ ( e π ( j − k i ˆ y, y ′ ) , j = 1 , . . . k, (4.12) ψ ( y ) = | y | − n ψ ( | y | − y ) , (4.13) and k ψ k ∗ C (cid:16) k φ k ∗ + k − nq (cid:17) if n > , k ψ k ∗ C (cid:16) k φ k ∗ + (ln k ) − (cid:17) if n = 3 . Furthermore, given two functions φ , φ the operator Ψ satisfies k Ψ( φ ) − Ψ( φ ) k ∗ C k φ − φ k ∗ . Proof.
We prove the result by combining a linear theory with a fixed point argument as in [8,Lemma 4.1]. Indeed, consider first the problem∆ ψ + pγU p − ψ = h, (4.14)with h satisfying (4.11), (4.12), k h k ∗∗ < + ∞ and h ( y ) = | y | − n − h ( | y | − y ) . (4.15)Proceeding as in [8, Lemma 4.1] one can apply Lemma 3.1 to conclude the existence of a uniquebounded solution ψ = T ( h ) of (4.14) satisfying symmetries (4.11)-(4.13) and k ψ k ∗ C k h k ∗∗ , where C is a positive constant depending only on n and q .Let us denote V ( y ) := pγ ( | u | p − − U p − ) − k X j =1 ( ζ j + ζ j ) | {z } V ( y ) + pγU p − k X j =1 ( ζ j + ζ j ) | {z } V ( y ) , (4.16)and M ( ψ ) := − k X j =1 ( ζ j + ζ j ) ( E + γN ( φ )) . (4.17)Thus, in order to solve (4.6) by a fixed point argument, we write ψ = − T V ψ + pγ | u | p − k X j =1 (1 − ζ j ) φ j + k X j =1 (1 − ζ j ) φ j + M ( ψ ) =: M ( ψ ) , where ψ ∈ X , the space of continuous functions with k · k ∗ < + ∞ and satisfying (4.11)-(4.13).Pointing out Remark 4.1 and the special symmetries of u it can be checked that V ψ + pγ | u | p − k X j =1 (1 − ζ j ) φ j + k X j =1 (1 − ζ j ) φ j + M ( ψ )satisfies (4.11), (4.12) and (4.15) for every ψ ∈ X and thus M ( ψ ) is well defined. Let us see that M is actually a contraction mapping in the k · k ∗ norm in a small ball around the origin in X .Proceeding as in [8, Lemma 4.1] we see that k V ψ k ∗∗ ( Ck − nq k ψ k ∗ if n > , C ln k k ψ k ∗ if n = 3 , k U p − k X j =1 ( ζ j + ζ j ) k ∗∗ ( Ck − nq k φ k ∗ if n > , C ln k k φ k ∗ if n = 3 , (4.18)whenever | y − ξ j | > δk , | y − ξ j | > δk . Using Proposition 2.1 we get k M ( ψ ) k ∗∗ ( Ck − nq (1 + k φ k ∗ ) + C k ψ k ∗ if n > ,C k (1 + k φ k ∗ ) + C k ψ k ∗ if n = 3 . (4.19) OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 15
Similarly, for ψ , ψ such that k ψ k ∗ < ρ , k ψ k ∗ < ρ it can be seen that k M ( ψ ) − M ( ψ ) k ∗∗ Cρ k ψ − ψ k ∗ . This estimate, together with (4.18)-(4.19), allows us to conclude that for ρ small enough (inde-pendent of k ) the operator M is a contraction map in the set of functions ψ ∈ X with k ψ k ∗ ( C ( k φ k ∗ + k − nq ) if n > ,C ( k φ k ∗ + (ln k ) − ) if n = 3 . Then the lemma follows by a fixed point argument. (cid:3)
We can also establish a uniform bound on the gradient of the function.
Proposition 4.3.
Let ψ be the solution of (4.6) provided by Proposition 4.2. Then there exists C > , depending only on n , such that k∇ ψ k L ∞ ( R n ) C. (4.20) Proof.
The goal is to estimate the terms in (4.6) to apply the a priori estimate in Lemma 3.2.Consider V and V defined in (4.16). Thus,(1 + | y | n +2 ) | V ψ | C k ψ k ∗ (1 + | y | ) U p − k X j =1 λ n − | y − ξ j | n − + λ n − | y − ξ j | n − ! C k ψ k ∗ , (1 + | y | n +2 ) | V ψ | k ψ k ∗ (1 + | y | ) U p − C k ψ k ∗ . (4.21)Analogously, noticing that (1 + | y | n +2 ) U p ≈
1, we have(1 + | y | n +2 ) | u | p − k X j =1 (1 − ζ j ) | φ j | + | u | p − k X j =1 (1 − ζ j ) | φ j | C k φ k ∗ , and(1 + | y | n +2 ) − k X j =1 ( ζ j + ζ j ) | N ( φ ) | (1 + | y | n +2 ) U p − (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 ( φ j + φ j ) (cid:12)(cid:12)(cid:12)(cid:12) + | ψ | C ( k φ k ∗ + k ψ k ∗ ) . (4.22)On the other hand (1 + | y | n +2 )(1 − k X j =1 ( ζ j + ζ j )) | E | C, as a consequence of (2.15) and (2.16). Using the estimates in Proposition 4.2 and Lemma 3.2we conclude (4.20). (cid:3) Let ψ = Ψ( φ ) given by Proposition 4.2. Notice that, thanks to the imposed conditions (4.7)and (4.9), solving the systems (4.4) and (4.5) can be reduced to solving the equation for φ , thatis, ∆ φ + pγ | u | p − ζ φ + ζ h pγ | u | p − Ψ( φ ) + E + γN ( φ ) i = 0 in R n , or equivalently ∆ φ + pγ | U | p − φ + ζ E + γ N ( φ , φ ) = 0 , (4.23) where N ( φ , φ ) := p (cid:0) | u | p − ζ − | U | p − (cid:1) φ + ζ h p | u | p − Ψ( φ ) + N ( φ ) i . Denote Z α ( y ) := λ − n − Z α (cid:18) y − ξ λ (cid:19) , α = 1 , . . . , n + 1where Z α was defined in (3.2). In order to solve (4.23) we will deal first with a projected linearversion. Given a general function h we consider∆ φ + pγ | U | p − φ + h = c U p − Z + c n +1 U p − Z n +1 , (4.24)where c := R R n hZ R R n U p − Z , c n +1 := R R n hZ n +1 R R n U p − Z n +1 . Lemma 4.4.
Suppose that h is even with respect to y , y , . . . , y n and satisfies (4.15) , andassume that h ( y ) := λ n +22 h ( ξ + λy ) satisfies k h k ∗∗ < + ∞ .Then problem (4.24) has a unique solution φ = T ( h ) that is even with respect to y , y , . . . , y n and satisfies φ ( y ) = | y | − n φ ( | y | − y ) , Z R n φU p − Z n +1 = 0 , Z R n φU p − Z = 0 , k φ k ∗ C k h k ∗∗ , with φ ( y ) := λ n − φ ( ξ + λy ) .Proof. Notice that, up to redefining h as h − c U p − Z − c n +1 U p − Z n +1 , we can assume Z R n hZ = Z R n hZ n +1 = 0 , i.e., c = c n +1 = 0 and thus equation (4.24) is equivalent to∆ φ + pγ | U | p − φ = − h in R n . We want to apply [8, Lemma 3.1] to solve this problem, and therefore we need to prove that Z R n hZ α = 0 for all α = 1 , , , , . . . , n. This follows straightforward for α = 2 , , , . . . , n due to the evenness of h . The case α = 1holds as a consequence of (4.15) (see the proof of [8, Lemma 4.2]). Then the result follows by[8, Lemma 3.1]. (cid:3) If instead of satisfying condition (4.15) the function h is even in all its coordinates we canprove a similar result (notice that in such case c = 0). Lemma 4.5.
Suppose that h ( y ) := λ n +22 h ( ξ + λy ) is even with respect to y α for every α =1 , . . . , n and k h k ∗∗ < + ∞ . Then problem (4.24) has a unique solution φ = T ( h ) that is evenwith respect to y α for every α = 1 , . . . , n and satisfies Z R n φU p − Z n +1 = 0 , k φ k ∗ C k h k ∗∗ , with φ ( y ) := λ n − φ ( ξ + λy ) . OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 17
Proof.
The result follows as in the previous Lemma just by noticing that Z R n hZ α = 0 for all α = 1 , . . . , n, due to the evenness of h . (cid:3) As a consequence of these lemmas we are able to solve the projected version of (4.23). Indeed,consider ∆ φ + pγ | U | p − φ + ζ E + γ N ( φ , φ ) = c U p − Z + c n +1 U p − Z n +1 , (4.25)with c := R R n ( ζ E + γ N ( φ , φ )) Z R R n U p − Z , c n +1 := R R n ( ζ E + γ N ( φ , φ )) Z n +1 R R n U p − Z n +1 . (4.26) Proposition 4.6.
There exists a unique solution φ = φ ( ℓ, t ) of (4.25) , that satisfies k φ k ∗ Ck − nq if n > , k φ k ∗ Ck ln k if n = 3 , and k N ( φ , φ ) k ∗∗ Ck − nq if n > , k N ( φ , φ ) k ∗∗ C ( k ln k ) if n = 3 , where φ ( y ) := λ n − φ ( ξ + λy ) and N ( φ , φ )( y ) := λ n +22 N ( φ , φ )( ξ + λy ) .Proof. We will solve (4.25) by means of a fixed point argument, writting φ = T ( ζ E + γ N ( φ , φ )) =: M ( φ ) , where T is the linear operator specified in Lemma 4.4. To do so, we begin analyzing the nonlinearterm, that can be decomposed as N ( φ , φ ) = f + f + f + f , where f := pζ ( | u | p − − U p − ) φ , f := ( ζ − U p − φ ,f := ζ p | u | p − Ψ( φ ) , f := ζ N ( φ ) . To estimate these terms we proceed in the same way as [8, Proposition 4.1], so we just highlightthe differences. Given a general function f , let us denote ˜ f ( y ) := λ n +22 f ( ξ + λy ). Assume first n >
4. Thus, noticing that k X j =1 U ( y + λ − ( ξ j − ξ )) Cλ n − k n − k X j =1 j n − + 1(2 τ ) n − Cλ n − , and using Proposition 4.2 one gets | ˜ f ( y ) | Cλ n − U ( y ) p − | φ ( y ) | for | y | < δλk , k ˜ f k ∗∗ Cλ n q k φ k ∗ , (4.27) | ˜ f ( y ) | CU ( y ) p − | φ ( y ) | for | y | > cλ − / , k ˜ f k ∗∗ Cλ n q k φ k ∗ , | ˜ f ( y ) | CU p − ( y ) λ n − | ψ ( ξ + λy ) | for | y | > cλ − / , k ˜ f k ∗∗ Cλ n q ( k φ k ∗ + k − nq ) . Notice that ˜ N ( φ ) = | V ∗ + ˆ φ | p − ( V ∗ + ˆ φ ) − | V ∗ | p − V ∗ − p | V ∗ | p − ˆ φ, where ˆ φ ( y ) := λ n − φ ( ξ + λy ) and V ∗ ( y ) := − U ( y ) − X j =1 U ( y − λ − ( ξ j − ξ )) − k X j =1 U ( y − λ − ( ξ j − ξ )) + λ n − U ( ξ + λy ) . Hence | ˜ f ( y ) | CU ( y ) p − ( | φ ( y ) | + λ n − | ψ ( ξ + λy ) | ) for | y | < δλk , k ˜ f k ∗∗ Cλ n q ( k φ k ∗ + k − nq ) . Finally, defining f := ζ E and using estimate (2.13) we also have k ˜ f k ∗∗ Cλ n q . In the case n = 3 one has k ˜ f k ∗∗ Ck ln k k φ k ∗ , k ˜ f k ∗∗ Ck ln k k φ k ∗ , k ˜ f k ∗∗ Ck ln k (cid:18) k φ k ∗ + 1 k ln k (cid:19) , (4.28) k ˜ f k ∗∗ Ck ln k (cid:18) k φ k ∗ + 1 k ln k (cid:19) , k ˜ f k ∗∗ Ck ln k . (4.29)Applying Proposition 2.1, estimates (4.27)-(4.29), Proposition 4.2 and Lemma 4.4, we concludethat M is a contraction that maps functions φ with k φ k ∗ Ck − nq if n > , k φ k ∗ Ck ln k if n = 3 , into the same class of functions whenever n < q < n − n − . Analogously, it can be proved theLipschitz character of the operators and thus, applying a fixed point argument, we conclude theproof. (cid:3) Proposition 4.7.
Let φ be the solution of (4.25) provided by Proposition 4.6. Then thereexists C > , depending only on n , such that | φ ( y ) | C λ n − (1 + | y | ) α where α = 2 if n > ,α = 1 if n = 4 , < α < if n = 3 , where φ ( y ) := λ n − φ ( ξ + λy ) .Proof. Denote L ( φ ) := ∆ φ + pγ | U | p − φ . Thus, (4.25) can be written in the form L ( φ ) + a ( y ) φ = g ( y ) + c U p − Z + c n +1 U p − Z n +1 , with a ( y ) := λ pγ ( | u | ζ − | U | p − )( ξ + λy ). Hence | a ( y ) | CU p − , | g ( y ) | C λ n − (1 + | y | ) . Applying [8, Lemma 3.2] with ν = 4 for n > ν = 3 for n = 4 and 2 < ν < n = 3 we getthe desired estimates on φ . (cid:3) To perform the reduction procedure in section 5 we will need more precise estimates on thepointwise behavior of φ , in particular on the part that will not be orthogonal to the kernel,whose size is smaller. OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 19
Proposition 4.8.
Let φ be the solution of (4.25) provided by Proposition 4.6 and denote φ ( y ) := λ n − φ ( ξ + λy ) . Then there exists a decomposition φ = φ s + φ ∗ such that φ s is evenwith respect to y α for every α = 1 , . . . , n and | φ ∗ ( y ) | C λ n − k | y | if n > ,C λ / k (ln k ) | y | α < α < , if n = 3 . . (4.30) Proof.
The idea of the proof is to identify in the equation (4.25) the largest terms, that happento be symmetric, and will produce a large but symmetric solution. The remaining terms willgive a function φ ∗ non symmetric but smaller. Denote V s := λ n − X j =1 n − λ n − | ξ − ξ j | n − + k X j =1 n − λ n − | ξ − ξ j | n − − U ( ξ ) . Consider E s given by (2.20) and let us define λ n − N s ( ξ + λy ) := ζ ( ξ + λy ) (cid:16) ˜ f s + ˜ f s + ˜ f s (cid:17) + ( ζ ( ξ + λy ) −
1) ˜ f s , where ˜ f s := p (cid:2) ( U ( y ) + V s ) p − − U ( y ) p − (cid:3) φ s , ˜ f s := U ( y ) p − φ s , ˜ f s := pλ n − ( U ( y ) + V s ) p − ψ ( ξ ) , ˜ f s := ( U ( y ) + V s ) p − ( ˆ φ s ) , being ˆ φ s := φ s + X j =1 λ n +22 φ sj ( ξ + λy ) + k X j =1 λ n +22 φ sj ( ξ + λy ) . Notice that, if φ s is even in all its coordinates, then N s ( ξ + λy ) is also even and groups thelargest terms of N ( ξ + λy ). Thus, proceeding as in the previous Proposition we can find asolution to ∆ φ s + pγ | U | p − φ s + ζ E s + γ N s ( φ , φ ) = c sn +1 U p − Z n +1 , with c sn +1 := R R n ( ζ E s + γ N s ( φ , φ )) Z n +1 R R n U p − Z n +1 , φ s ( y ) = λ n − φ s ( ξ + λy ) , by applying Lemma 4.5 to perform a fixed point argument in the set of functions φ s which areeven in all their coordinates and have size k φ s k ∗ Ck − nq if n > , k φ s k ∗ Ck ln k if n = 4 . Furthermore, proceeding as in Proposition 4.7 we can conclude that | φ s ( y ) | C λ n − (1 + | y | ) α where α = 2 if n > ,α = 1 if n = 4 , < α < n = 3 . Let us define φ ∗ := φ − φ s . Hence it solves∆ φ ∗ + pγ | U | p − φ ∗ + ζ E ∗ + γ N ∗ ( φ , φ ) = c U p − Z + c ∗ n +1 U p − Z n +1 , (4.31) where E ∗ is defined in Proposition 2.2, N ∗ ( φ , φ ) := N ( φ , φ ) − N s ( φ , φ ) , and c := R R n ( ζ E ∗ + γ N ∗ ( φ , φ )) Z R R n U p − Z , c ∗ n +1 := R R n ( ζ E ∗ + γ N ∗ ( φ , φ )) Z n +1 R R n U p − Z n +1 = c n +1 − c sn +1 . The key point is that, without the previous symmetric part, the terms left are smaller and moreprecise estimates con be done, Indeed, denote V ( y ) := λ n − X j =1 U j ( ξ + λy ) + k X j =1 U j ( ξ + λy ) − U ( ξ + λy ) . Thus we can write λ n − N ∗ ( ξ + λy ) := ζ ( ξ + λy ) (cid:16) ˜ f ∗ + ˜ f ∗ + ˜ f ∗ (cid:17) + ( ζ ( ξ + λy ) −
1) ˜ f ∗ , where ˜ f ∗ ( y ) := p (cid:2) ( U ( y ) + V s ( y )) p − − U p − ( y ) (cid:3) φ ∗ ( y )+ p (cid:2) ( U ( y ) + V ( y )) p − − ( U ( y ) + V s ( y )) p − (cid:3) φ ( y ) , ˜ f ∗ ( y ) := U ( y ) p − φ ∗ ( y ) , ˜ f ∗ ( y ) := pλ n − ( U ( y ) + V ( y )) p − λ ∇ ψ ( η ) y + pλ n − h ( U ( y ) + V ( y )) p − − ( U ( y ) + V s ( y )) p − i ψ ( ξ ) , ˜ f ∗ ( y ) := ˜ f ( y ) − ˜ f s ( y ) , with ˜ f defined in Proposition 4.6. Noticing that (cid:12)(cid:12)(cid:12)(cid:12) (cid:2) ( U ( y ) + V ( y )) p − − ( U ( y ) + V s ( y )) p − (cid:3) φ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) CU p − ( y ) | V ( y ) − V s |k φ k ∗ and | V ( y ) − V s | C λ n − | y | k if n > , | V ( y ) − V s | C λ / | y | k (ln k ) if n = 3 , it can be proved that (cid:12)(cid:12)(cid:12)(cid:12) (cid:2) ( U ( y ) + V ( y )) p − − ( U ( y ) + V s ( y )) p − (cid:3) φ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) C λ n − k | y | if n > ,C λ / k (ln k ) | y | if n = 3 , and the same bound can be proved for ˜ f ∗ and ˜ f ∗ by using the estimates in Proposition 4.2 andProposition 4.3. Using this together with Proposition 2.2 we can proceed as in the proof ofProposition 4.7 to estimate the size of | φ ∗ | . That is, we can rewrite problem (4.31) as L ( φ ∗ ) + a ∗ ( y ) φ ∗ = g ∗ ( y ) + c U p − Z + c ∗ n +1 U p − Z n +1 , where L ( φ ) := ∆ φ + pγ | U | p − φ, | a ∗ ( y ) | CU ( y ) p − , OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 21 and | g ∗ ( y ) | C λ n − k | y | if n > ,C λ / k (ln k ) | y | if n = 3 . . Applying again [8, Lemma 3.2] with ν = 3 for n > < ν < n = 3 we obtain(4.30). (cid:3) Likewise, we will need accurate estimates on the non symmetric part of ψ . Proposition 4.9.
Let ψ be the solution of (4.6) provided by Proposition 4.2. Then ψ ( ξ + λy ) = ψ s ( y ) + ψ ∗ ( y ) , y ∈ B (0 , δλk ) , where ψ s is even with respect to y and | ψ ∗ ( y ) | C (cid:16) k φ k ∗ + k ψ k ∗ + τ o k (1) (cid:17) λ | y | (1 + | y | ) , where o k (1) is a function that goes to 0 when k → ∞ .Proof. Since ψ is a solution of (4.6) we can write, making the convolution with the fundamentalsolution of the Laplace equation, ψ ( ξ + λy ) = c n Z R n | ξ + λy − x | n − W ( ψ )( x ) dx, y ∈ B (0 , δλk ) ,W ( ψ ) := V ψ + M ( ψ ) + pγ | u | p − k X j =1 ((1 − ζ j ) φ j + (1 − ζ j ) φ j ) − pγU p − ψ, where c n is a constant depending only on the dimension and V , M were defined in (4.16) and(4.17) respectively. Furthermore,1 | ξ + λy − x | n − =: A ( x, y ) + B ( x, y ) + B ( x, y ) , where A ( x, y ) := 1 | x − ξ | n − " − (cid:18) n − (cid:19) λ y + 2 λ P ni =1 ,i =3 y i ( x − ξ ) i | x − ξ | ,B ( x, y ) := − ( n − λy x − τ | x − ξ | n , B ( x, y ) := O (cid:18)(cid:18) ( λ y + 2 λ ( y, x − ξ )) | x − ξ | n +2 (cid:19)(cid:19) . Notice that A ( x, y ) is even with respect to y and thus we can define ψ s ( y ) := c n Z R n A ( x, y ) W ( ψ )( x ) dx, that inherits this symmetry. Therefore we have to estimate ψ ∗ ( y ) := c n Z R n ( B ( x, y ) + B ( x, y )) W ( ψ )( x ) dx. Since | W ( ψ )( x ) | C (1+ | x | ) and | y | < cλ − / it easily follows that (cid:12)(cid:12)(cid:12)(cid:12) Z R n B ( x, y ) W ( ψ )( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) λτ | y | o k (1) . (4.32) To estimate the term with B ( x, y ) notice first that (cid:12)(cid:12)(cid:12)(cid:12) Z R n B ( x, y ) U p − ψ dx (cid:12)(cid:12)(cid:12)(cid:12) Cλ | y | Z R n | x − ξ | n − | x | ) | ψ ( x ) | dx Cλ | y |k ψ k ∗ (4.33)and likewise (cid:12)(cid:12)(cid:12)(cid:12) Z R n B ( x, y ) V ψ dx (cid:12)(cid:12)(cid:12)(cid:12) Cλ | y |k ψ k ∗ . (4.34)Observing that | φ j ( y ) | C k φ k ∗ λ n − | y − ξ j | n − , | φ j ( y ) | C k φ k ∗ λ n − | y − ξ j | n − , we get (cid:12)(cid:12)(cid:12)(cid:12) Z R n B ( x, y ) | u | p − k X j =1 ((1 − ζ j ) φ j + (1 − ζ j ) φ j ) (cid:12)(cid:12)(cid:12)(cid:12) Cλ | y |k φ k ∗ , (4.35) (cid:12)(cid:12)(cid:12)(cid:12) Z R n B ( x, y ) − k X j =1 ( ζ j + ζ j ) N ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) Cλ | y | ( k φ k ∗ + k ψ k ∗ ) , (4.36)and thus the only term left is the one concerning the error, namely Z R n B ( x, y ) − k X j =1 ( ζ j + ζ j ) E dx = Z R n B ( x, y ) − k X j =1 ζ j E dx | {z } B − k X j =1 Z R n B ( x, y ) ζ j E dx | {z } B . Let
R > B = Z B ( ξ ,R ) B ( x, y )(1 − k X j =1 ζ j ) E dx + O ( | y | k − ( n − ) . The desired estimate will follow by noticing that the largest terms of the error are orthogonalto B ( x, y ). To see this, we write the error as γ − E ≃ pU p − k X j =1 U j + k X j =1 U j − k X j =1 U pj − k X j =1 U pj = pU p − ( ξ ) k X j =1 U j − k X j =1 U pj + pU p − ( ξ ) k X j =1 U j − k X j =1 U pj + p ( U p − − U p − ( ξ )) k X j =1 U j + p ( U p − − U p − ( ξ )) k X j =1 U j . Notice first that, since B ( x, y ) is odd with respect to the hyperplane x = τ , there holds Z B ( ξ ,R ) B ( x, y )(1 − k X j =1 ζ j ) pU p − ( ξ ) k X j =1 U j − k X j =1 U pj dx = 0 . OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 23
On the other hand, writting U j ( x ) = λ n − ( λ + | ξ − ξ j | + | x − ξ | ) n − " − ( n −
2) ( x − ξ , ξ − ξ j ) λ + | ξ − ξ j | + | x − ξ | + O ( x − ξ , ξ − ξ j ) λ + | ξ − ξ j | + | x − ξ | ! , and applying again the evenness of its main terms with respect to x = τ we have that (cid:12)(cid:12)(cid:12)(cid:12) Z B ( ξ ,R ) B ( x, y ) − k X j =1 ζ j pU p − ( ξ ) k X j =1 U j dx (cid:12)(cid:12)(cid:12)(cid:12) Cλ n − | y | Z B ( ξ ,R ) | x − τ || x − ξ | n k X j =1 τ | x − τ | ( λ + | ξ − ξ j | + | x − ξ | ) n dx + Cλ n − | y | Z B ( ξ ,R ) | x − τ || x − ξ | n k X j =1 | x − ξ | | ξ − ξ j | ( λ + | ξ − ξ j | + | x − ξ | ) n +22 dx Cλ n − | y | Z B ( ξ ,R ) | x − ξ | n − k X j =1 τ | ξ − ξ j | n − + 1 | ξ − ξ j | n − ! dx = λτ | y | o k (1) , where in the last inequality we have used (2.12). Proceeding analogously with the other termsit can be concluded that Z B ( ξ ,R ) B ( x, y ) − k X j =1 ζ j E dx = λτ o k (1) | y | , and therefore B = λτ o k (1) | y | . (4.37)Likewise, | B | Cλ | y | k X j =1 Z B ( ξ j , δk ) | x − τ || ξ − ξ j | n k X i =1 λ n − | ξ − ξ j | n − dx = λτ o k (1) | y | . (4.38)Putting together (4.32)-(4.36) with (4.37) and (4.38) the result follows. (cid:3) Proof of Theorem 1.1
The goal of this section is to find positive parameters ℓ and t that enter in the definition of λ and τ in (2.6) and are independent of k , in such a way that c and c n +1 (defined in (4.26))vanish. In fact, if such a choice is possible, then the solution φ found in Proposition 4.6 solves(4.23) and thus, applying Proposition 4.2 and (4.7)-(4.9), we can conclude that φ solves (4.2)and therefore u = u + φ is a solution of (1.1). Thus, we want to prove the existence of ℓ and t so that c ( ℓ, t ) := Z R n ( ζ E + γ N ( φ , φ )) Z = 0 , c n +1 ( ℓ, t ) := Z R n ( ζ E + γ N ( φ , φ )) Z n +1 = 0 . It is worth pointing out that, due to some symmetry, the main order term of c vanishes, whatmakes necessary an expansion of c at lower order. This is usually a delicate issue and it requiressharper estimates on the non linear term, that is, a finer control on the size of the terms ψ and φ in the spirit of Proposition 4.7. However in this case this type of estimates are not enough,since they do not produce a non linear term sufficiently small. We need to identify a precisedecomposition of φ and ψ in one symmetric part whose contribution to the computation of c is zero, and a smaller non-symmetric part (see Claim 6). This decompositions were developedin Proposition 4.8 and Proposition 4.9.What we obtain at the end is that, for n > c ( ℓ, t ) = D n tℓ nn − k n +1 − n − (cid:20) d n ℓt n − − (cid:21) + 1 k α Θ k ( ℓ, t ) ,c n +1 ( ℓ, t ) = E n ℓk n − (cid:2) e n ℓ − (cid:3) + 1 k β Θ k ( ℓ, t ) , (5.1)where α > n + 1 − n − , β > n − , and c ( ℓ, t ) = F ℓ t √ ln k ( k ln k ) (cid:20) f ℓt − (cid:21) + 1 k (ln k ) Θ k ( ℓ, t ) ,c ( ℓ, t ) = G ℓk ln k (cid:2) gℓ − (cid:3) + 1 k ln k ln (cid:16) π √ ln k (cid:17) ln k Θ k ( ℓ, t ) , (5.2)for n = 3, where D n , d n , E n , e n , F, f, G, g are fixed positive numbers (depending only on n ) andΘ k ( ℓ, t ) is a generic function, smooth on its variables, and uniformly bounded as k → ∞ . Hence,by a fixed point argument we can conclude the existence of ℓ and t such that c ( ℓ, t ) = c n +1 ( ℓ, t ) = 0 . (5.3)By simplicity we detail the argument in the case of (5.2). With abuse of notation on the functionΘ, that always stands for a generic function smooth on its variables and uniformly bounded as k → ∞ , (5.3) is equivalent to f ℓ − t + o k (1)Θ k ( ℓ, t ) = 0 ,gℓ − o k (1)Θ k ( ℓ, t ) = 0 . Defining ρ := t and η := ℓ we can rewrite the system as ρ = f η / + o k (1)Θ k ( η, ρ ) ,η = 1 g + o k (1)Θ k ( η, ρ ) . Suppose 0 < ρ C fixed. Hence the second equation can be expressed as η = F ρ ( η ) , with F a ( s ) := 1 g + o k (1)Θ k ( s, a ) . OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 25
Consider the set X := { η ∈ R : 0 < η g } . Using the smoothness of Θ k it is easy to see that F ρ maps X into itself and that it is a contraction for k large enough. Thus, for any fixed ρ thereexists a fixed point η ρ ∈ X such that η ρ = F ρ ( η ρ ).Replacing on the first equation this translates into ρ = f ( F ρ ( η ρ )) / + o k (1)Θ k ( η ρ , ρ ) = fg / + o k (1)Θ k ( η ρ , ρ ) =: G ( ρ ) . Considering the set Y := { ρ ∈ R : 0 < ρ fg / } and using the smoothness of Θ it can bechecked that G is a contraction that maps the set Y into itself, and therefore we conclude theexistence of a fixed point ρ = G ( ρ ), what concludes the argument.The rest of the section is devoted to prove (5.1) and (5.2). Let us consider first the case of c n +1 , that follows analogously to [8]. We write c n +1 ( ℓ, t ) = Z R n EZ n +1 − Z R n (1 − ζ ) EZ n +1 + γ Z R n N ( φ , φ ) Z n +1 = 0 . Thus, for ℓ and t as in (2.6) we have: Claim 1: Z R n EZ n +1 = E n ℓk n − (cid:2) e n ℓ − (cid:3) + k n − n − n − Θ k ( ℓ, t ) if n > ,G ℓk ln k (cid:2) gℓ − (cid:3) + k ln k ln (cid:16) π √ ln k (cid:17) ln k Θ k ( ℓ, t ) if n = 3 . Claim 2: Z R n (1 − ζ ) EZ n +1 = ( k n − Θ k ( ℓ, t ) if n > , k ln k ) Θ k ( ℓ, t ) if n = 3 . Claim 3: Z R n N ( φ , φ ) Z n +1 = ( k n + nq − Θ k ( ℓ, t ) if n > , k ln k ) Θ k ( ℓ, t ) if n = 3 . Notice that these claims together give the second equation in (5.1) and (5.2).
Proof of Claim 1.
We decompose Z R n EZ n +1 = Z B ( ξ , δk ) EZ n +1 + Z Ext EZ n +1 + X j =1 Z B ( ξ j , δk ) EZ n +1 + k X j =1 Z B ( ξ j , δk ) EZ n +1 , (5.4)where δ is a positive constant independent of k and Ext := {∩ kj =1 {| y − ξ j | > δk }} ∩ {∩ kj =1 {| y − ξ j | > δk }} . Denoting V ( y ) := λ n − X j =1 U j ( λy + ξ ) − k X j =1 U j ( λy + ξ ) − U ( λy + ξ ) we have, for some s ∈ (0 , γ − Z B ( ξ , δk ) EZ n +1 = λ n +22 Z B (0 , δλk ) E ( λy + ξ ) Z n +1 ( y )= p X j =1 λ n − Z B (0 , δλk ) U p − U j ( λy + ξ ) Z n +1 + p k X j =1 λ n − Z B (0 , δλk ) U p − U j ( λy + ξ ) Z n +1 − pλ n − Z B (0 , δλk ) U p − U ( λy + ξ ) Z n +1 + p Z B (0 , δλk ) (cid:2) ( U + sV ) p − − U p − (cid:3) V Z n +1 + X j =1 λ n +22 Z B (0 , δλk ) U pj ( λy + ξ ) Z n +1 + k X j =1 λ n +22 Z B (0 , δλk ) U pj ( λy + ξ ) Z n +1 − λ n +22 Z B (0 , δλk ) U p ( λy + ξ ) Z n +1 . (5.5)Thus, defining I := R R n U p − Z n +1 , from (2.10) follows that λ n − Z B (0 , δλk ) U p − U j ( λy + ξ ) Z n +1 = 2 n − λ n − I | ξ − ξ j | n − λ | ξ − ξ j | Θ k ( ℓ, t ) ! ,λ n − Z B (0 , δλk ) U p − U j ( λy + ξ ) Z n +1 = 2 n − λ n − I | ξ − ξ j | n − λ | ξ − ξ j | Θ k ( ℓ, t ) ! , and λ n − Z B (0 , δλk ) U p − U ( λy + ξ ) Z n +1 = λ n − U ( ξ ) I (cid:18) λ | ξ | Θ k ( ℓ, t ) (cid:19) , which are the main order terms in (5.5). Indeed, (cid:12)(cid:12)(cid:12)(cid:12) X j =1 λ n +22 Z B (0 , δλk ) U pj ( λy + ξ ) Z n +1 (cid:12)(cid:12)(cid:12)(cid:12) C X j =1 λ n +2 | ξ j − ξ | n +2 Z B (0 , δλk ) | y | ) n − C ( λk ) − P j =1 λ n +2 | ξ j − ξ | n +2 if n > , | ln( λk ) | P j =1 λ n +2 | ξ j − ξ | n +2 if n = 3 , (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 λ n +22 Z B (0 , δλk ) U pj ( λy + ξ ) Z n +1 (cid:12)(cid:12)(cid:12)(cid:12) C X j =1 λ n +2 | ξ j − ξ | n +2 Z B (0 , δλk ) | y | ) n − C ( λk ) − P j =1 λ n +2 | ξ j − ξ | n +2 if n > , | ln( λk ) | P j =1 λ n +2 | ξ j − ξ | n +2 if n = 3 , OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 27 and (cid:12)(cid:12)(cid:12)(cid:12) λ n +22 Z B (0 , δλk ) U p ( λy + ξ ) Z n +1 dy (cid:12)(cid:12)(cid:12)(cid:12) Cλ n +22 Z B (0 , δλk ) | y | ) n − C ( λ n − k − if n > ,λ n +22 | ln( λk ) | if n = 3 . Finally these three estimates, together with the mean value theorem, also imply (cid:12)(cid:12)(cid:12)(cid:12) p Z B (0 , δλk ) (cid:2) ( U + sV ) p − − U p − (cid:3) V Z n +1 (cid:12)(cid:12)(cid:12)(cid:12) C ( λ n − k − + ( λk ) − ( λk ) n +2 if n > ,λ n +22 | ln( λk ) | + ( λk ) n +2 | ln( λk ) | if n = 3 . Proceeding like in [8, Proof of Claim 2] we obtain the estimates of the other terms in (5.4), thatis, (cid:12)(cid:12)(cid:12)(cid:12) Z
Ext EZ n +1 (cid:12)(cid:12)(cid:12)(cid:12) C ( k n − if n > , k ln k ) if n = 3 , (5.6) (cid:12)(cid:12)(cid:12)(cid:12) X j =1 Z B ( ξ j , δk ) EZ n +1 (cid:12)(cid:12)(cid:12)(cid:12) C ( k n − if n > , k ln k ) if n = 3 , (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 Z B ( ξ j , δk ) EZ n +1 (cid:12)(cid:12)(cid:12)(cid:12) C ( k n − if n > , k ln k ) if n = 3 . (5.7)Claim 1 follows using estimates (2.11) and (2.12). Proof of Claim 2.
Noticing that (cid:12)(cid:12)(cid:12)(cid:12) Z R n ( ζ − EZ n +1 (cid:12)(cid:12)(cid:12)(cid:12) C (cid:12)(cid:12)(cid:12)(cid:12) Z {| y − ξ | > δk } EZ n +1 (cid:12)(cid:12)(cid:12)(cid:12) , the result follows using (5.6) and (5.7). Proof of Claim 3.
Decomposing the non linear term as in the proof of Proposition 4.6 andusing Proposition 4.7 it can be seen that (cid:12)(cid:12)(cid:12)(cid:12) Z R n N ( φ , φ ) Z n +1 (cid:12)(cid:12)(cid:12)(cid:12) Ck − n − nq Z R n U p − | Z n +1 | , and the claim holds for n >
4. If n = 3 it follows from estimates (4.28), (4.29) and the fact that (cid:12)(cid:12)(cid:12)(cid:12) Z R n N ( φ , φ ) Z n +1 (cid:12)(cid:12)(cid:12)(cid:12) C k λ n +22 N ( φ , φ )( λy + ξ ) k ∗∗ (cid:18)Z R n dy (1 + | y | ) n (cid:19) q − q . Next we proceed to compute c . We write c ( ℓ, t ) = Z R n EZ − Z R n (1 − ζ ) EZ + γ Z R n N ( φ , φ ) Z = 0 , and we affirm that, for ℓ and t as in (2.6), Claim 4: Z R n EZ = D n tℓ nn − k n +1 − n − (cid:2) d n ℓt n − − (cid:3) + k n +1 Θ k ( ℓ, t ) if n > ,F ℓ t √ ln k ( k ln k ) (cid:2) f ℓt − (cid:3) + k (ln k ) Θ k ( ℓ, t ) if n = 3 . Claim 5: Z R n (1 − ζ ) EZ = ( k n +1 Θ k ( ℓ, t ) if n > , k (ln k ) Θ k ( ℓ, t ) if n = 3 . Claim 6: Z R n N ( φ , φ ) Z = ( k α Θ k ( ℓ, t ) if n > , k (ln k ) Θ k ( ℓ, t ) if n = 3 , where α > n + 1 − n − .These claims together imply the validity of the first equation in (5.1) and (5.2). Proof of Claim 4.
We decompose again as Z R n EZ = Z B ( ξ , δk ) EZ + Z Ext EZ + X j =1 Z B ( ξ j , δk ) EZ + k X j =1 Z B ( ξ j , δk ) EZ , (5.8)Proceeding as in (5.6) and (5.7) we get (cid:12)(cid:12)(cid:12)(cid:12) Z Ext EZ (cid:12)(cid:12)(cid:12)(cid:12) Cλ n − k n − λ − n − k − n q − q k (1 + | y | ) n +2 − nq E k L q ( Ext ) , (5.9) (cid:12)(cid:12)(cid:12)(cid:12) X j =1 Z B ( ξ j , δk ) EZ (cid:12)(cid:12)(cid:12)(cid:12) C X j =1 λ n − ( λk ) − nq | ξ j − ξ | n − k (1+ | y | ) n +2 − nq λ n +22 γ − E ( ξ j + λy ) k L q ( | y | < δλk ) , (5.10) (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 Z B ( ξ j , δk ) EZ (cid:12)(cid:12)(cid:12)(cid:12) C k X j =1 λ n − ( λk ) − nq | ξ j − ξ | n − k (1+ | y | ) n +2 − nq λ n +22 γ − E ( ξ j + λy ) k L q ( | y | < δλk ) . (5.11)For the first integral in (5.8) we separate as in (5.5). Noticing that U j ( λy + ξ ) is even withrespect to the third coordinate, it follows that λ n − Z B (0 , δλk ) U p − U j ( λy + ξ ) Z = 0 . (5.12)Furthermore, using (2.10), λ n − Z B (0 , δλk ) U p − U j ( λy + ξ ) Z = c n λ n − τ λI | ξ − ξ j | n (cid:0) λ Θ k ( ℓ, t ) (cid:1) , and λ n − Z B (0 , δλk ) U p − U ( λy + ξ ) Z = ˜ c n λ n − τ λI (cid:18) λ | ξ | Θ k ( ℓ, t ) (cid:19) , where I := R R n U p − y Z . One also can compute the lower order terms (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 λ n +22 Z B (0 , δλk ) U pj ( λy + ξ ) Z (cid:12)(cid:12)(cid:12)(cid:12) C ( λk ) − ( λk ) n +2 and (cid:12)(cid:12)(cid:12)(cid:12) λ n +22 Z B (0 , δλk ) U p ( λy + ξ ) Z dy (cid:12)(cid:12)(cid:12)(cid:12) Cλ n k − . (5.13)Thus, decomposing as in (5.5) Claim 4 is obtained from estimates (5.9)-(5.13) together with(2.13), (2.14) and (6.14). OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 29
Proof of Claim 5.
It follows straightforward from (5.9), (5.10) and (5.11).
Proof of Claim 6.
Assume first n > N ( φ , φ ) = f + f + f + f asin the proof of Proposition 4.6. Changing variables and using (4.27) it can be seen that (cid:12)(cid:12)(cid:12)(cid:12) Z R n f ( y ) Z ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) Cλ n − Z B (0 , δλk ) U ( y ) p − | φ ( y ) || Z | . Notice that in this region λ / c | y | and thus, by Proposition 4.7 we can write | φ ( y ) | C λ β λ n − − β (1 + | y | ) C λ β (1 + | y | ) n − β , where β := − n − + ε , ε > (cid:12)(cid:12)(cid:12)(cid:12) Z R n f ( y ) Z ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) Ck β + n − Z R n | y | ) − n +6 | y | (1 + | y | ) n | y | ) n − β dy Ck β + n − for ε small enough. Notice also that 2 β + n − > n + 1 − n − , that is the order of the mainterm. Likewise, using the estimate on φ , (cid:12)(cid:12)(cid:12)(cid:12) Z R n f ( y ) Z ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) Cλ n − Z {| y | >cλ − / } U p − | Z | (1 + | y | ) dy Cλ n − +2 Z R n | y | (1 + | y | ) n +2 dy Ck n +2 . To estimate the projection of f we first point out that f ≈ ζ p | U | p − ψ . Due to the cancellationin (5.12) the main order term in Claim 4 is rather small, and this makes necessary sharp estimateson the size of the projection of the nonlinear term. Indeed, to prove this claim we will have tomake use of the decomposition of ψ in a large but symmetric part (that happens to be orthogonalto Z ) and a non symmetric but small part specified in Proposition 4.9. Thus, Z R n f ( y ) Z ( y ) dy ≈ λ n − Z B (0 , δλk ) U ( y ) p − ψ ( ξ + λy ) Z = λ n − Z B (0 , δλk ) U p − ψ s ( y ) Z dy + λ n − Z B (0 , δλk ) U p − ψ ∗ ( y ) Z dy. The first integral in the right hand side vanishes due to the oddness of Z and the evennes of ψ s in the third coordinate. The second can be estimated as (cid:12)(cid:12)(cid:12)(cid:12) λ n − Z B (0 , δλk ) U p − ψ ∗ ( y ) Z dy (cid:12)(cid:12)(cid:12)(cid:12) Cλ n − +1 ( k φ k ∗ + k ψ k ∗ + τ o k (1)) Z R n U p − | y | (1 + | y | ) | Z | dy Ck n − nq , and hence, choosing n < q < n − n − we conclude that (cid:12)(cid:12)(cid:12)(cid:12) Z R n f ( y ) Z ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) Ck α with α > n + 1 − n − . Analogously, noticing that f ≈ | U | p − (cid:16)P kj =1 ( φ j + φ j ) + ψ (cid:17) , estimate (cid:12)(cid:12)(cid:12)(cid:12) Z R n f ( y ) Z ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) Ck α with α > n + 1 − n − , follows by Proposition 4.3. This completes the proof of Claim 6 for n > n = 3 ,
4. Using the decomposition of φ found in Proposition 4.8 wenotice that Z R n ˜ f s ( y ) Z ( y ) dy = 0 , Z R n ˜ f s ( y ) Z ( y ) dy = 0 , and we obtain, for n = 3, (cid:12)(cid:12)(cid:12)(cid:12) Z R n f ( y ) Z ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z R n ˜ f ∗ ( y ) Z ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) C λ Z B (0 , δλk ) U ( y ) p − | φ ∗ ( y ) || Z ( y ) | dy + λ k (ln k ) Z B (0 , δλk ) | y | U ( y ) p − | φ ( y ) || Z ( y ) | dy ! Ck (ln k ) , and (cid:12)(cid:12)(cid:12)(cid:12) Z R n f ( y ) Z ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z {| y | >cλ − / } f ∗ ( y ) Z ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z {| y | >cλ − / } U ( y ) p − φ ∗ ( y ) Z ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) C λ λk (ln k ) Z R n | y | ) | y | (1 + | y | ) n Ck (ln k ) , where in the last inequalities we have applied Proposition 4.7 and Proposition 4.8.Likewise, for n = 4, (cid:12)(cid:12)(cid:12)(cid:12) Z R n f ( y ) Z ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) Ck , (cid:12)(cid:12)(cid:12)(cid:12) Z R n f ( y ) Z ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) Ck . The terms involving ˜ f and ˜ f are estimated in a similar way and Claim 6 follows.6. Remark 1.7: The general construction
This section is devoted to the constructions described in Remark 1.7. The first is the constructionof the doubling of the equatorial
Γ with an even number of circles, which is done in subsection6.1. The second is a combination of the doubling and the desingularization of the equatorial withan odd number of circles. This is done in subsection 6.2.6.1.
Even number of circles.
Let m be a fixed integer. Let τ i ∈ (0 , i = 1 , . . . , m, and fixthe points P i := ( q − τ i , , τ i , , . . . , , P i := ( q − τ i , , − τ i , , . . . , . Let λ i ∈ (0 , i = 1 , . . . , m, be positive numbers, and define R i as λ i + R i = 1 . We use thenotation λ = ( λ , . . . , λ m ) , τ = ( τ , . . . , τ m ) . Let k be an integer number and u m [ λ, τ ]( y ) := U ( y ) − k X j =1 (cid:20) m X i =1 λ − n − U y − ξ ij λ !| {z } U ij ( y ) + m X i =1 λ − n − U y − ξ ij λ !| {z } U ij ( y ) (cid:21) (6.1) OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 31 for y ∈ R n , where, for i = 1 , . . . , m , and j = 1 , . . . , k , ξ ij := R i ( q − τ i cos θ j , q − τ i sin θ j , τ i , , . . . , ,ξ ij := R i ( q − τ i cos θ j , q − τ i sin θ j , − τ i , , . . . , , with θ j := 2 π j − k . Observe that the function (6.1) satisfies the symmetries (2.3), (2.4) and (2.5). We assume thatthe integer k is large, and that the parameters λ and τ are given by λ i := ℓ n − i k , τ i := t i k − n − , if n > ,λ i := ℓ i k (ln k ) , τ i := t i √ ln k , if n = 3 , where η < ℓ i , t i < η − (6.2)for some η small and fixed, independent of k , for any k large enough. The doubling of theequatorial Γ with an even number of circles is the content of next
Theorem 6.1.
Let n > and let k be a positive integer. Then for any sufficiently large k thereis a finite energy solution to (1.1) of the form u ( y ) = u m [ λ, τ ]( y ) + o k (1)(1 + | λ | − n − ) , where the term o k (1) → uniformly on compact sets of R n as k → ∞ . The solution in Theorem 6.1 has the form u ( y ) = u m [ λ, τ ]( y ) + φ ( y ) , φ = ψ + k X j =1 m X i =1 ( ¯ φ ij + φ ij )where φ ij , φ ij , i = 1 , . . . , m , j = 1 , . . . , k , and ψ solve the following system of coupled non linearequations ∆ φ ij + pγ | u m | p − ζ ij φ ij + ζ ij (cid:2) pγ | u m | p − ψ + E m + γN ( φ ) (cid:3) = 0 , (6.3)∆ φ ij + pγ | u m | p − ζ ij φ ij + ζ ij (cid:2) pγ | u m | p − ψ + E m + γN ( φ ) (cid:3) = 0 , (6.4)∆ ψ + pγU p − ψ + pγ ( | u m | p − − U p − )(1 − k X j =1 m X i =1 ( ζ ij + ζ ij ))+ pγU p − [ k X j =1 m X i =1 ( ζ ij + ζ ij ) ψ + pγ | u m | p − k X j =1 m X i =1 (1 − ζ ij ) φ ij + pγ | u m | p − k X j =1 m X i =1 (1 − ζ ij ) φ ij + − k X j =1 m X i =1 ( ζ ij + ζ ij ) ( E m + γN ( φ )) = 0 . (6.5)The functions ζ ij are defined as ζ j in (4.3) with ¯ ξ j replaced by ¯ ξ ij , and ζ ij ( y ) := ζ ij ( y , y , − y , . . . , y n ) , E m ( y ) := ∆ u m + γ | u m | p − u m , y ∈ R n , and N ( φ ) := | u m + φ | p − ( u m + φ ) − | u m | p − u m − p | u m | p − φ . One can prove that k E m k ∗∗ Ck − nq if n > , k E m k ∗∗ C (ln k ) − if n = 3 . Denoting ˆ y := ( y , y ) and y ′ := ( y , . . . , y n ), we assume that the functions φ ij and φ ij satisfy φ ij (ˆ y, y ′ ) = φ i ( e π ( j − k i ˆ y, y ′ ) , φ i ( y ) = | y | − n φ i ( | y | − y ) ,φ i ( y , . . . , y α , . . . , y n ) = φ i ( y , . . . , − y α , . . . , y n ) , α = 2 , , . . . , n, and φ ij ( y ) = φ ij ( y , y , − y , . . . , y n ) . Moreover ( φ ij + φ ij )( y ) = ( φ ij + φ ij )( y , y , − y , . . . , y n ) , as well as ( ζ ij + ζ ij ) and ( ζ ij φ ij + ζ ij φ ij ).For ρ > m X i =1 k φ i k ∗ ρ, where φ i ( y ) := λ n − φ i ( ξ i + λy ) and k · k ∗ is defined in (3.3).Arguing as in Proposition 4.2, one proves that there exists a unique solution ψ = Ψ( φ , . . . , φ m )of (6.5), satisfying (4.11), (4.12) and (4.13). Besides k ψ k ∗ C m X i =1 k φ i k ∗ + k − nq ! if n > , k ψ k ∗ C m X i =1 k φ i k ∗ + (ln k ) − ! if n = 3 . We replace the solution ψ = Ψ( φ , . . . , φ m ) of (6.5) in (6.3) and (6.4). Using the symmetrieswe described before, it is enough to solve (6.3) for j = 1. We are thus left with a system of m equations in φ = ( φ , . . . , φ m ) unknowns∆ φ i + pγ | u m | p − ζ i φ i + ζ i (cid:2) pγ | u m | p − ψ + E m + γN ( φ ) (cid:3) = 0 , i = 1 , . . . , m. Instead of solving it directly, we first solve the auxiliary problem∆ φ i + pγ | U i | p − φ i + ζ i E m + γ N i ( φ , φ ) = c i U p − i Z i + c i,n +1 U p − i Z i,n +1 , (6.6)where N i ( φ , φ ) := p ( | u m | p − ζ − | U i | p − ) φ i + ζ h p | u m | p − Ψ( φ ) + N ( φ ) i ,Z iα ( y ) := λ − n − Z α (cid:18) y − ξ i λ (cid:19) , α = 3 , n + 1and c i := R R n ( ζ i E m + γ N i ( φ , φ )) Z i R R n U p − i Z i , c i,n +1 := R R n ( ζ i E m + γ N i ( φ , φ )) Z i,n +1 R R n U p − i Z i,n +1 . Arguing as in Proposition 4.6, one proves that there exists a unique solution φ i = φ i ( l, t ) of(6.6), that satisfies k φ i k ∗ Ck − nq if n > , k φ i k ∗ Ck ln k if n = 3 , and k N ( φ i , φ ) k ∗∗ Ck − nq if n > , k N ( φ i , φ ) k ∗∗ C ( k ln k ) if n = 3 , OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 33 where φ i ( y ) := λ n − φ i ( ξ i + λy ) and N ( φ i , φ )( y ) := λ n +22 N ( φ i , φ )( ξ i + λy ). Furthermore,proceeding as in Proposition 4.8 and Proposition 4.9, there exist decompositions φ i = φ si + φ ∗ i , ψ ( ξ i + λy ) = ψ s ( y ) + ψ ∗ ( y ) , where φ si and ψ s are even with respect to y and | φ ∗ i ( y ) | C λ n − k | y | if n > ,C λ / k (ln k ) | y | α , < α < , if n > , | ψ ∗ ( y ) | Cλk − nq | y | (1 + | y | ) . In order to complete the proof of Theorem 6.1, we need to find positive parameters ℓ , . . . ℓ m and t , . . . , t m entering in the definition of λ and τ in (6.2) so that for all i = 1 , . . . , mc i ( ℓ, t ) = c i,n +1 ( ℓ, t ) = 0 , ℓ = ( ℓ , . . . ℓ m ) , t = ( t , . . . , t m ) . (6.7)In dimension n >
4, this system decouples and becomes a n ℓ i − k n − n − Θ i,n +1 ,k ( ℓ , . . . ℓ m , t , . . . , t m ) = 0 , i = 1 , . . . , m,b n ℓ i t n − i − k n − n − Θ i, ,k ( ℓ , . . . ℓ m , t , . . . , t m ) = 0 , i = 1 , . . . , m, where a n , b n are positive constants that are independent of k , and Θ i,n +1 ,k , Θ i, ,k are smoothfunctions of their argument, which are uniformly bounded, together with their first derivatives,as k → ∞ .In dimension n = 3, the system becomes a ℓ i − k ln k Θ i, ,k ( ℓ , . . . ℓ m , t , . . . , t m ) = 0 , i = 1 , . . . , m,b ℓ i t i − k ln k Θ i, ,k ( ℓ , . . . ℓ m , t , . . . , t m ) = 0 , i = 1 , . . . , m, where a , b are positive constants, and Θ i, ,k , Θ i, ,k are smooth functions of their argument,which are uniformly bounded, together with their first derivatives, as k → ∞ . A fixed pointargument gives the existence of ℓ and t solutions to (6.7). This concludes the proof of Theorem6.1.6.2. Odd number of circles.
Let µ ∈ (0 ,
1) and define R so that µ + R = 1 . Let k , m beinteger numbers and u m +1 [ µ, λ, τ ]( y ) := U ( y ) − k X j =1 " U j ( y ) + m X i =1 (cid:0) U ij ( y ) + U ij ( y ) (cid:1) (6.8)where λ , τ , U ij and U ij are defined at the beginning of subsection 6.1 and (6.1), while U j ( y ) := µ − n − U ( y − ξ j µ ) , ξ j := R (cos θ j , sin θ j , , , . . . , θ j := 2 π j − k . The function (6.8) satisfies the symmetries (2.3), (2.4) and (2.5). We assume that the integer k is large, and that the parameters µ , λ and τ are given by µ := ℓ n − k if n > , µ := ℓ k (ln k ) if n = 3 ,λ i := ℓ n − i k , τ i := t i k − n − , if n > , λ i := ℓ i k (ln k ) , τ i := t i √ ln k , if n = 3 , (6.9)where η < ℓ, ℓ i , t i < η − for some η small and fixed, independent of k , for any k large enough.We have Theorem 6.2.
Let n > and let k be a positive integer. Then for any sufficiently large k thereis a finite energy solution to (1.1) of the form u ( y ) = u m +1 [ µ, λ, τ ]( y ) + o k (1)(1 + ( µ + | λ | ) − n − ) , where the term o k (1) → uniformly on compact sets of R n as k → ∞ . The solution in Theorem 6.2 has the form u ( y ) = u m +1 [ µ, λ, τ ]( y ) + φ ( y ) , φ = ψ + k X j =1 [ φ j + m X i =1 ( ¯ φ ij + φ ij )] , with φ j (ˆ y, y ′ ) = φ ( e π ( j − k i ˆ y, y ′ ) , φ ij (ˆ y, y ′ ) = φ i ( e π ( j − k i ˆ y, y ′ ) , (6.10)where ˆ y := ( y , y ) and y ′ := ( y , . . . , y n ). The functions φ j , φ ij and φ ij also satisfy φ ( y ) = | y | − n φ ( | y | − y ) , φ i ( y ) = | y | − n φ i ( | y | − y ) ,φ i ( y , . . . , y α , . . . , y n ) = φ i ( y , . . . , − y α , . . . , y n ) , α = 2 , , . . . , n, and φ ij ( y ) = φ ij ( y , y , − y , . . . , y n ) . Moreover φ j ( y ) = φ j ( y , y , − y , . . . , y n ) , ( φ ij + φ ij )( y ) = ( φ ij + φ ij )( y , y , − y , . . . , y n ) . OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 35
Thanks to (6.10), it is enough to describe φ , φ i , i = 1 , . . . , m . The functions φ , φ i and ψ solve the following system of coupled non linear equations∆ ψ + pγU p − ψ + pγ ( | u m +1 | p − − U p − ) − k X j =1 " ζ j + m X i =1 ( ζ ij + ζ ij ) + pγU p − k X j =1 [ ζ j + m X i =1 ( ζ ij + ζ ij ) ψ + pγ | u m +1 | p − k X j =1 m X i =1 (1 − ζ ij ) φ ij + pγ | u m +1 | p − k X j =1 m X i =1 (1 − ζ ij ) φ ij + pγ | u m +1 | p − k X j =1 (1 − ζ j ) φ j + − k X j =1 " ζ j + m X i =1 ( ζ ij + ζ ij ) ( E m +1 + γN ( φ )) = 0 , ∆ φ + pγ | U | p − φ + ζ E m +1 + γ N ( φ ) = c n +1 U p − Z n +1 , and, for i = 1 , . . . , m ,∆ φ i + pγ | U i | p − φ i + ζ i E m +1 + γ N i ( φ ) = c i U p − i Z i + c i,n +1 U p − i Z i,n +1 . Here E m +1 ( y ) := ∆ u m +1 + γ | u m +1 | p − u m +1 , y ∈ R n , and N ( φ ) := | u m +1 + φ | p − ( u m +1 + φ ) − | u m +1 | p − u m +1 − p | u m +1 | p − φ . For any j , ζ j , ζ ij are defined as ζ j in (4.3) with ¯ ξ j replaced respectively by ξ j and ¯ ξ ij , and ζ ij ( y ) := ζ ij ( y , y , − y , . . . , y n ). Moreover, N ( φ ) := p ( | u m +1 | p − ζ − | U | p − ) φ + ζ (cid:2) p | u m +1 | p − ψ + N ( φ ) (cid:3) , N i ( φ ) := p ( | u m +1 | p − ζ i − | U i | p − ) φ i + ζ i (cid:2) p | u m +1 | p − ψ + N ( φ ) (cid:3) ,c n +1 := R R n ( ζ E m +1 + γ N ( φ )) Z n +1 R R n U p − Z n +1 , and c i := R R n ( ζ i E m +1 + γ N i ( φ )) Z i R R n U p − i Z i , c i,n +1 := R R n ( ζ i E m +1 + γ N i ( φ )) Z i,n +1 R R n U p − i Z i,n +1 . where Z α ( y ) := µ − n − Z α (cid:18) y − ξ µ (cid:19) , Z iα ( y ) := λ − n − Z α (cid:18) y − ξ i λ (cid:19) , α = 3 , n + 1 . It can be proved that k ψ k ∗ C m X i =1 k φ i k ∗ + k − nq ! if n > , k ψ k ∗ C m X i =1 k φ i k ∗ + (ln k ) − ! if n = 3 , k φ k ∗ Ck − nq if n > , k φ k ∗ Ck ln k if n = 3 , k φ i k ∗ Ck − nq if n > , k φ i k ∗ Ck ln k if n = 3 , where φ ( y ) := µ n − φ ( ξ + µy ) and φ i ( y ) := λ n − φ i ( ξ i + λy ), and the corresponding estimateson their non symmetric part.In order to complete the proof of Theorem 6.2, we need to find positive parameters ℓ, ℓ , . . . ℓ m and t , . . . , t m entering in the definition of µ , λ and τ in (6.9) so that for all i = 1 , . . . , mc n +1 (¯ ℓ, t ) = c i (¯ ℓ, t ) = c i,n +1 (¯ ℓ, t ) = 0 , ¯ ℓ = ( ℓ, ℓ , . . . ℓ m ) , t = ( t , . . . , t m ) . (6.11)In dimension n >
4, this system decouples at main order and becomes a n ℓ − k n − n − Θ n +1 ,k ( ℓ, ℓ , . . . ℓ m , t , . . . , t m ) = 0 ,a n ℓ i − k n − n − Θ i,n +1 ,k ( ℓ, ℓ , . . . ℓ m , t , . . . , t m ) = 0 , i = 1 , . . . , m,b n ℓ i t n − i − k n − n − Θ i, ,k ( ℓ, ℓ , . . . ℓ m , t , . . . , t m ) = 0 , i = 1 , . . . , m, where a n , b n are positive constants, and Θ n +1 ,k , Θ i,n +1 ,k , Θ i, ,k are smooth functions of theirargument, which are uniformly bounded, together with their first derivatives, as k → ∞ .In dimension n = 3, system (6.11) decouples and becomes a ℓ − k ln k Θ ,k ( ℓ, ℓ , . . . ℓ m , t , . . . , t m ) = 0 ,a ℓ i − k ln k Θ i, ,k ( ℓ, ℓ , . . . ℓ m , t , . . . , t m ) = 0 , i = 1 , . . . , m,b ℓ i t i − k ln k Θ i, ,k ( ℓ, ℓ , . . . ℓ m , t , . . . , t m ) = 0 , i = 1 , . . . , m, where a , b are positive constants, and Θ ,k , Θ i, ,k Θ ,k are smooth functions of their argument,which are uniformly bounded, together with their first derivatives, as k → ∞ . A fixed pointargument gives the existence of ℓ and t solutions to (6.11). Appendix A: Some useful computations
Proof of (2.11) and (2.12) . By definition k X j =2 | ξ − ξ j | n − = 1 R n − (1 − τ ) n − k X j =2 − cos θ j )] n − , θ j = 2 π j − k . Using the symmetry of the construction we have k X j =2 − cos θ j )] n − = k X j =2 − cos θ j )] n − + 14 n − if k even , k − X j =2 − cos θ j )] n − if k odd , OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 37 where 0 < θ j < π . Hence,2(1 − cos θ j ) = θ j " − cos( η j ) θ j for some 0 η j θ j , and 1 − π − cos( η j ) θ j π . Then, for k large and odd, we get k X j =2 − cos θ j )] n − = 2 k − X j =2 θ n − j + 2 k − X j =2 θ n − j − (cid:20) − cos( η j ) θ j (cid:21) n − (cid:20) − cos( η j ) θ j (cid:21) n − = ( k n − A n (1 + O ( σ k )) if n > ,k ln kA (1 + O ( σ k )) if n = 3 , where A n := 2(2 π ) n − ∞ X j =1 j − n , A := π − and σ k := k − if n > ,k − ln k if n = 5 ,k − if n = 4 , (ln k ) − if n = 3 . The case of k even can be analogously treated, and therefore (2.11) follows.In the same spirit we also observe that, for n > k odd, we have k X j =1 | ξ − ξ j | n − = 2 R n − k − X j =1 τ + 2(1 − τ )(1 − cos θ j )] n − = 2(2 R ) n − τ n − k − X j =1 π − τ τ ( j − k ) ] n − (1 + O ( τ )) , and for n = 3, k X j =1 | ξ − ξ j | = 1 Rτ k − X j =1 π − τ τ ( j − k ) ] n − (1 + O ( | ln τ | − )) . If n >
4, using (2.6) we get k − X j =1 π − τ τ ( j − k ) ] n − = Z k − dx (1 + π − τ τ ( xk ) ) n − ! (1 + O (( kτ ) − ))= kτπ √ − τ Z π √ − τ τ k − k ds (1 + s ) n − (1 + O (( kτ ) − ))= kτπ √ − τ Z ∞ ds (1 + s ) n − ! (1 + O (( kτ ) − ) + O ( τ n − )) . (6.12)If n = 3, k − X j =1 π − τ τ ( j − k ) ] = kτπ √ − τ Z π √ − τ τ k − k ds (1 + s ) (1 + O (( kτ ) − ))= kτπ √ − τ ln( π √ − τ τ k − k + s π − τ τ (cid:18) k − k (cid:19) ) (1 + O (( kτ ) − ))= kτπ √ − τ ln (cid:16) πτ (cid:17) (1 + O ( τ | ln τ | − )) . (6.13)Combining (6.12) and (6.13) we obtain the validity of (2.12) for k odd. The even case followsin the same way.Analogous computations provide k X j =1 | ξ − ξ j | n = ( C n kτ n − (cid:0) O (( τ k ) − ) (cid:1) if n > ,C n kτ n − (cid:0) O ( τ n − ) (cid:1) if n = 3 , , C n := 22 n π Z ∞ ds (1 + s ) n . (6.14) Appendix B: linear independence of the functions z j ( y ) . We give here the proof of the linear independence of the functions z j ( y ), j = 0 , . . . , n − n − c j with n − X j =0 c j z j ( y ) = 0 , ∀ y ∈ R n , (6.15) OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 39 then c j = 0 for all j . The result will follow from evaluating this expression in different points,properly chosen. Notice that, by definition, (6.15) can be written as[ c − y c n +1 − y c n +2 − y c n +3 ] z ( y ) + " c + | y | c n +1 − n X α =2 y α c n + α +2 z ( y )+ " c + | y | c n +2 + c n +4 y − n X α =3 c n + α y α z ( y )+ " c + | y | c n +3 + c n +5 y + c n +3 y − n X α =4 c n + α − y α z ( y )+ n X l =4 [ c l + c n + l +2 y + c n + l y + c n + l − y ] z l ( y ) = 0 . (6.16)Since our solution u satisfies u ( y , y , . . . , y j , . . . , y n ) = u ( y , y , . . . , − y j , . . . , y n ) , ∀ j = 3 , . . . n, we have that necessarily, for every j = 3 , . . . n , z j ( q ) = 0 ∀ q = ( q , . . . , q n ) with q j = 0 . Taking q = 0 in (6.16) we get c z (0) + c z (0) + c z (0) = 0 , (6.17)and taking q = ( q , q , , , .., = 0, with ( q , q ) a generic point in the plane ( y , y ), andevaluating in (6.16), we get c z ( q ) + c z ( q ) + c z ( q ) + c n +1 ( − q z ( q ) + | q | z ( q ))+ c n +2 ( − q z ( q ) + | q | z ( q )) + c n +4 ( − q z ( q ) + q z ( q )) = 0 . (6.18)Choose now 5 points of the form q = ( q , q , , , .., = 0 in (6.18) to obtain 5 independentequations. This, combined with (6.17) gives c = c = c = c n +1 = c n +2 = c n +4 = 0 . Fix q = (0 , , q , , , .., = 0. Thus c z ( q ) + c n +3 ( − q z ( q ) + | q | z ( q ))+ c n +5 ( − q z ( q )) + c n +3 ( − q z ( q )) = 0 . Hence we can choose 4 points of the form q = (0 , , q , , , .., = 0 to obtain c = c n +3 = c n +5 = c n +3 = 0 . Similarly, taking q = (0 , , , q j , , , .., = 0, for j = 4 , . . . , n , a generic point and evaluating in(6.16) we get c j z j ( q ) + c n + j +2 ( − q j z ( q )) + c n + j ( − q j z ( q )) = 0 . Choosing 3 appropriate points of the form q = (0 , , , q j , , , .., = 0, for j = 4 , . . . , n , wededuce c j = c n + j +2 = c n + j = 0 . So far we have proven that c j = 0 , ∀ j = 0 , . . . , n, and hence it remains to see that n X j =4 c n + j − z j ( y ) = 0 ∀ y, implies c n +4 = . . . = c n − = 0. Let us evaluate this linear combination at the points ¯ q j =(0 , , , ...q j , .. ), for j = 4 , . . . , n . We get c n + j − ( − q j z (¯ q j ) + z j (¯ q j )) = 0 . Choosing q j so that − q j z (¯ q j ) + z j (¯ q j ) = 0 we obtain c n + j − = 0 for all j = 4 , . . . , n and weconclude. References [1] T. Aubin, Equations diff´ e rentielles non lin´ e aires et probl` e me de Yamabe concarnant la courbure scaalaire, J.Math. Pures Appl.
55 (1976), 269-290.[2] L.A. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equationswith critical Sobolev growth,
Comm. Pure Appl. Math.
42 (1989), 271-297.[3] W. Ding, On a conformally invariant elliptic equation on R n , Communications in Mathematical Physics
Geom. Funct. Anal.
27 (2017), no. 4, 798–862.[5] T. Duyckaerts, C. Kenig, F. Merle, Solutions of the focusing nonradial critical wave equation with the com-pactness property,
Ann. Sc. Norm. Super. Pisa Cl. Sci.
Vol. XV (2016), 731-808.[6] T. Duyckaerts, C. Kenig, F. Merle, Profiles of bounded radial solutions of the focusing, energy-critical waveequation.
Geom. Funct. Anal.
22 (2012), no. 3, 639-698.[7] T. Duyckaerts, C. Kenig, F. Merle, Universality of the blow-up profile for small type II blow-up solutions ofthe energy-critical wave equation: the nonradial case.
J. Eur. Math. Soc. (JEMS)
14 (2012), no. 5, 1389-1454.[8] M. del Pino, M. Musso, F. Pacard, A. Pistoia, Large energy entire solutions for the Yamabe equation.
Journalof Differential Equations
251 (2011), 2568–2597.[9] M. del Pino, M. Musso, F. Pacard. A. Pistoia. Torus action on S n and sign changing solutions for conformallyinvariant equations. Annali della Scuola Normale Superiore di Pisa (5) 12 (2013), no. 1, 209–237.[10] N. Kapouleas. Doubling and desingularization constructions for minimal surfaces. Surveys in geometric anal-ysis and relativity,
Adv. Lect. Math. (ALM) , 20, Int. Press, Somerville, MA, (2011), 281-325.[11] N. Kapouleas. Minimal surfaces in the round three-sphere by doubling the equatorial two-sphere, I.
J. Dif-ferential Geom.
106 (2017), no. 3, 393–49.[12] N. Kapouleas, P. McGrath. Minimal surfaces in the round three-sphere by doubling the equatorial two-sphere,II.
Comm. Pure Appl. Math.
72 (2019), no. 10, 2121–2195.[13] C. Kenig, F.Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linearSchr¨odinger equation in the radial case,
Invent. Math.
166 (2006), 645-675.[14] C. Kenig, F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing nonlinearwave equation,
Acta Math.
201 (2008), 147-212.[15] J. Krieger, W. Schlag, D. Tataru, Slow blow-up solutions for the H ( R ) critical focusing semilinear waveequation. Duke Math. J.
147 (2009), 1-53.[16] M. Medina, M. Musso, J. Wei, Desingularization of Clifford torus and nonradial solutions to Yamabe problemwith maximal rank.
Journal of Functional Analysis
OUBLING NODAL SOLUTIONS WITH MAXIMAL RANK 41 [17] M. Musso, J. Wei, Nondegeneracy of Nonradial Nodal Solutions to Yamabe Problem.
Communications inMathematical Physics
J. Funct. Anal.
89 (1990), no. 1, 1-52.[19] G. Talenti, Best constants in Sobolev inequality,
Ann. Mat. Pura Appl.
110 (1976), 353–372.(Mar´ıa Medina)
Departamento de Matem´aticas, Universidad Aut´onoma de Madrid, Ciudad Uni-versitaria de Cantoblanco, 28049 Madrid, Spain
Email address : [email protected] (Monica Musso) Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UnitedKingdom
Email address ::