Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

Given an isoparametric function $f$ on the $n$-dimensional round sphere, we consider functions of the form $u=w\circ f$ to reduce the semilinear elliptic problem \[ -\Delta_{g_0}u+\lambda u=\lambda\ | u\ | ^{p-1}u\qquad\text{ on }\mathbb{S}^n \] with $\lambda>0$ and $1<p$, into a singular ODE in $[0,\pi]$ of the form $w'' + \frac{h(r)}{\sin r} w' + \frac{\lambda}{\ell^2}\ (| w|^{p-1}w - w\ )=0$, where $h$ is an strictly decreasing function having exactly one zero in this interval and $\ell$ is a geometric constant. Using a double shooting method, together with a result for oscillating solutions to this kind of ODE, we obtain a sequence of sign-changing solutions to the first problem which are constant on the isoparametric hypersurfaces associated to $f$ and blowing-up at one or two of the focal submanifolds generating the isoparametric family. Our methods apply also when $p>\frac{n+2}{n-2}$, i.e., in the supercritical case. Moreover, using a reduction via harmonic morphisms, we prove existence and multiplicity of sign-changing solutions to the Yamabe problem on the complex and quaternionic space, having a finite disjoint union of isoparametric hipersurfaces as regular level sets.


Introduction
Let (M, g) be a closed (compact without boundary) Riemannian manifold of dimension n ≥ 3. We will consider the Yamabe type equations where λ ∈ C ∞ (M ), µ ∈ R and p > 1. In case λ = R g is the scalar curvature and p = p n := n+2 n−2 is the critical Sobolev exponent, equation (1.1) is the well known Yamabe equation, widely studied in the last 50 years (see, for example, [2,7,15] and the references therein). When p < n+2 n−2 , we will say that the equation (1.1) is subcritical and we will call it supercritical if p > n+2 n−2 . In the subcritical case, as the Sobolev embedding H 1 (M, g) ֒→ L p (M, g) is compact, the existence of positive and sign-changing solutions can be obtained using standard variational methods [43,44]. When M = Ω is a bounded domain of R n+1 with smooth boundary, there has been recent progress in handling supercritical exponent problems like (1.1). A fruitful  approach consists in reducing the supercritical problem to a more general elliptic critical or subcritical problem, either by considering rotational symmetries or by means of maps preserving the Laplace operator or by a combination of both, see [17] and the references therein. In case of closed Riemannian manifolds, these reduction methods also apply and have been combined with the Lyapunov-Schmidt reduction method in order to obtain sequences of positive and sign-changing solutions to similar supercritical problems, such that they blow-up or concentrate at minimal submanifolds of M [16,21,25,32,39].
The main interest of this paper is to seek for sign-changing solutions (also called nodal solutions) to the problem (1.2) − ∆ g0 u + λu = λ |u| p−1 u on (S n , g 0 ).
when p is either subcritical or supercritical. Here g 0 denotes the round metric and we will assume from now on that λ > 0 is constant. When p = p n , (1.2) is a renormalization of the Yamabe problem (1.1) and this kind of solutions have been studied in [13,15,19,20,37] and more recently in [22,31,40]. The slightly subcritical case has been studied in [42], where the authors obtained multiplicity of nodal solutions blowing-up at points, while the general subcritical case has been studied in [28] and in [8]. The method introduced in [28] allowed the authors to obtain more information about the qualitative behavior of the solutions, for they showed the existence of an infinite number of non constant positive solutions having prescribed level sets in terms of isoparametric hypersurfaces. This method has been further generalized to supercritical exponents in [6] to prove a similar result on general closed Riemannian manifolds, including the round sphere. Other results concerning the existence and concentration of positive solutions along minimal submanifolds for the supercritical and slightly supercritical can be found in [21,32]. However, very little is known about the existence, multiplicity and blow-up of nodal solutions for the supercritical problem on the sphere (1.2). One of the few results known by the authors is given in [27], where the existence of at least one signchanging solution to the supercritical problem was settled. In this direction we will follow and generalize the ideas introduced in [22] to obtain an infinite number of nodal solutions to the supercritical and subcritical problem (1.2), having as level and critical sets isoparametric hypersurfaces and its focal submanifolds.
To state our main result and to describe the method, we briefly recall some aspects of the theory of isoparametric functions and hypersurfaces. For the details, we refer the reader to [4,12]. A smooth function f : The regular level sets of f are called isoparametric hypersurfaces.
The theory of isoparametric hypersurfaces in the round sphere (S n , g 0 ) is very rich and it is a vast research topic. In this case, isoparametric hypersurfaces coincide with the hypersurfaces of constant principal curvatures. Its classification began with E. Cartan [9] and it is still an open problem, see [18,33,34] and the references therein. Some major progresses in the theory were made by Cartan himself, H. F. Münzner [35,36] and D. Ferus, H. Karcher and Münzner [23]. Given an isoparametric hypersurface, there exist a huge number of isoparametric functions having it as level hypersurface, for if f : S n → R is isoparametric, ν : Im(f ) → R is monotone and α ∈ R {0}, then α(ν • f ) is again isoparametric. However, there are "canonical" isoparametric functions, which are obtained by restricting Cartan-Münzner polynomials to the sphere [12,Section 3.5]. They are well understood and have some nice properties. For instance, if f : S n → R is obtained in this way, then Imf = [−1, 1], the inverse image of a regular value is a connected isoparametric hypersurface, its only critical values are t = ±1 and the functions a and b defined in (1.3) can be written explicitly. To give these explicit expressions, let ℓ be the number of distinct principal curvatures of the level sets of f . Münzner showed that ℓ ∈ {1, 2, 3, 4, 6} and if ℓ is odd, then all the multiplicities of the principal curvatures are the same, while if ℓ is even, there exist, at most, two different multiplicities m − and m + with 1 ≤ m − , m + ≤ n − 1. With this notation, if ∆ g0 f = a(f ) and The sets M − := f −1 (−1) and M + := f −1 (1) are smooth submanifolds of S n of dimension n − = (n−1)−m − and n + := (n−1)−m + , called focal submanifolds, see [12,Section 2.4]. The main feature of these submanifolds is that every isoparametric hypersurface is a tube around M − and M + . If u denotes a sign-changing smooth function defined on a Riemannian manifold, we define the nodal and the critical sets of u to be the sets {u = 0} and {∇u = 0}, respectively. We state the main result of this paper, which generalizes Theorem 1.3 in [22]. Theorem 1.1. Let S ⊂ S n be an isoparametric hypersurface and let n − ≤ n + be the dimensions of its corresponding focal submanifolds. Then, for any λ > 0, any k ∈ N and any p ∈ (1, n−n++2 n−n+−2 ), equation (1.2) admits a nodal solution u k such that its nodal set has at least k connected components, each of them being an isoparametric hypersurface diffeomorphic to S. The critical set of u k consists in the focal submanifolds M − and M + and, at least, k − 1 isoparametric hypersurfaces diffeomorphic to S. Moreover, the solutions u k satisfy for every x ∈ M − or for every x ∈ M + .
Here the numbers n−n±+2 n−n±−2 ≥ p n are just the critical Sobolev exponents in dimensions n − n ± . Our Theorem improves the existence result stated by Henry in [27], giving an infinite number of distinct solutions instead of one. It also extends the multiplicity result in [22] to the subcritical and supercritical exponents. However, this last result gives a better description of the nodal set of the solutions. We strongly believe that a refinement of our methods may give a prescribed number of connected components for the nodal sets of the sign-changing solutions to problem (1.1), as in Theorem 1.2 in [22].
The last assumption of Theorem 1.1 says that the sequence (u k ) is not compact with the C 0 topology and that the blow-up occurs on one of the focal submanifolds, which are minimal submanifolds of the sphere [12]. Other noncompactness phenomenon of the same nature appears in the solutions to the critical Yamabe problem obtained in [19], where the blow-up occurs at a single point. However, it was recently proved by Premoselli and Vétois that this sequence of solutions is uniformly bounded from below, but not from above [40]. This does not holds true in general, as we state next.
As another consequence of Theorem 1.1, we obtain a multiplicity result for the Yamabe problem on projective spaces.
admits a sequence of sign-changing solutions (u k ) such that the regular level sets of u k consist of isoparametric hypersurfaces in (M, g) and We describe briefly the method we shall use in order to prove Theorem 1.1. The details will be given in Section 2 and Section 3.
Let f : S n → R be an isoparametric function obtained as the restriction of a Cartan-Münzner polynomial, and let ℓ, m − and m + be the number of principal curvatures and the multiplicities associated to the isoparametric hypersurfaces that f defines, as it was explained before. Then, it is easy to see that z : [−1, 1] → R is a solution to the problem with a(t) := −ℓ(n + ℓ − 1)t + ℓ 2 (m+−m−) 2 and b(t) := −ℓ 2 t 2 + ℓ 2 , if and only if u = z • f is a solution to the problem (1.2) (Cf. [22]). Therefore, if u = z • f is a solution to (1.2), its regular level sets and the set of its critical points are conformed by isoparametric hypersurfaces and focal submanifolds. We can simplify equation (1.8) even more by considering the new variable w(r) = z(cos r), and, in this way, solving (1.8) is equivalent to solving the singular ODE Observe that the natural boundary conditions associated to this problem are given by w ′ (0) = w ′ (π) = 0. Theorem 1.1 will be a consequence of the following one.
We will prove this theorem in Section 3. The function h appearing in Equation (1.9) has a unique zero a 0 ∈ (0, π). To prove Theorem 1.4 we will use the double shooting method developed in [22], which consists in considering the solutions w d , w c of Equation (1.9) with initial conditions w ′ d (0) = w ′ c (π) = 0, w d (0) = d, w c (π) = c and consider the maps = 0, as one can readily see. To understand the intersections of the curves I, J one needs information of the functions w d , w c . In the next section we will prove that, for large d and c, these functions have many zeroes close to 0 and π (respectively) and then, generalizing an argument based on a Pohozaev-type identity and presented in [10], we will prove that |I(d)|, |J(c)| → ∞ as c, d → ∞. These two results will allow us to conclude that the curves I and J behave as spirals rotating in opposite directions and from this we will obtain the intersections needed to solve the double shooting problem.

Double shooting and the proof of Theorem 1.4.
We now develop the double shooting method used to prove Theorem 1.4. First, observe that the function h defined in (1. the same properties with m − and m + interchanged and a unique zero at π − a 0 . To handle both singularities in (1.9) at the same time, the strategy is to shoot solutions from each of them and expect that, for some suitable initial and final conditions, the solutions coincide. That is, we consider the initial value problem and the "final" value problem looking for initial and final conditions d and c such that Hence, by uniqueness of the solution, we would have a well defined solution to problem (1.9) given by To construct the solutions with an arbitrarily large number of zeroes, we will need to use that the number of zeroes before and after a 0 goes to infinity as |d|, |c| → ∞.
Actually, problem (2.2) can be written as an initial condition problem having the form of (2.1). Indeed, if we consider the function h(r) defined above, then w f solves (2.2) if and only if ω(r) = w f (π − r) solves the initial value problem So, in order to understand problem (2.1) it is enough to consider problem (2.2).
In what follows, we will consider the more general equation where A > 0, µ > 0 and H is a non negative C 1 function defined in the interval [0, A]. Notice that equations (2.1) and (2.3) are special cases of the former by Observe that now we are just dealing with a single singularity at r = 0.
A standard contraction map argument (Cf. [22,29]) yields the existence and uniqueness of the solutions to equation (2.4) with initial conditions w(0) = d ∈ R and w ′ (0) = 0, depending continuously on d. For d > 0, let w d := w(·, d) be the solution with initial values w d (0) = d and w ′ d (0) = 0. To assure the existence of an arbitrarily large number of zeroes, we use the following result, proven in [22,27].
Theorem 2.1. Suppose that H(0) > 0, p > 1 and that the following inequality holds true. Then, for any ε > 0 and any positive integer k there exists D k > 0 so that the solution w d of (2.4) has at least k zeroes in (0, ε) for any d ≥ D k .

In case of equations (2.1) and (2.3), we have that
implies the validity of inequality (2.5) for both equations (2.1) and (2.3).
Since p n ≤ m++3 m+−1 is true for every 1 ≤ m + ≤ n − 1, we may guarantee the existence of an arbitrary large number of zeroes for equations (2.1) and (2.3) when p < p n , corresponding to the subcritical Yamabe problem, or when p n < p < m++3 m+−1 , corresponding to the supercritical one. Also observe that inequality (2.6) is true when p = p n and 1 ≤ m − , m + < n − 1, a fact used in [22] to assure the existence of a prescribed number of zeroes to equation (1.9) in the critical case. For the rest of this section, we will suppose that p > 1 satisfies inequality (2.6).
Even if the number of zeroes is arbitrarily large, it can not be infinite as we next show.
Proof. First observe that if r 0 ∈ [0, A] is a zero of w d * , then uniqueness of the solution implies that w d * (r 0 ) = 0. Therefore w d * is monotone in a neighborhood of r 0 and it is an isolated zero. Now, to see that the critical points are isolated, suppose that w ′ d * (r 0 ) = 0 for some r 0 ∈ [0, A]. As d = 0, 1, by uniqueness of the solutions, w d * (r 0 ) = 0 and this together with equation (2.4) implies that w ′′ d * (r 0 ) = 0. Without loss of generality, suppose w ′′ d * > 0. By continuity, there exists ε > 0 such that {r 0 } and r 0 is an isolated critical point.
For ε < A, take D k > 0 as given in Theorem 2.1. Then, for d ≥ D k > 0, w d has at least k zeroes before A. Denote them by r 1 (d) < r 2 (d) < . . . < r k (d). The following statement holds true.
For any compact subset K of [0, A], the functions z d converge C 1 -uniformly on K to the unique solution of the limit Cauchy problem and as p > 1 satisfies inequality (2.6), v has an infinite number of isolated zeroes in (0, ∞) (see Lemma 3.2 and Theorem 3.3 in [22] and Proposition 3.6 in [26]). Now observe that and by a j the j-th zero of the solution v 0 to the limit problem (2.7). Then r * j (d) = λd Now we focus on equation (2.1) in order to do a phase plane analysis. Let a 0 be the unique zero of h in (0, π). Let w d be the solution of (2.1) with initial conditions It is then easy to see that we have a well defined continuous function θ : (0, ∞) → R such that θ(1) = 0 and θ(d) gives an angle between I(d) and the positive x-axis for any d > 0. Note that, in a similar way, there is a unique continuous function θ : (−∞, 0) → R such that θ(−1) = −π and θ(d) gives an angle between I(d) and the positive x-axis. Thus, we have that for any Note that θ and ϑ run in opposite directions. If n(d) denotes the number of zeroes of w d before a 0 and N (c) the number of zeroes of w c after a 0 , then the angles θ and ϑ are related with these numbers by the following formulas, proved in [22], where, as usual for x ∈ R, ⌊x⌋ denotes the maximum integer such that ⌊x⌋ ≤ x. As a consequence of these formulas and Theorem 2.1, we have the following limits We will show that the curves I and J behave like spirals turning in opposite directions. The above limits show that the spirals actually turn. Next we see that the spirals are not bounded. If w d is a solution to the initial value problem (2.1) and w c is a solution to (2.2), define ρ(r, d) = jπ}, These numbers are well defined by (2.9) and they form unbounded sequences by the same limit. Observe that d i and d i are first and last time that the curve R hits the line θ = −iπ, respectively, while c i and c j correspond to the first and last time that S hits the line ϑ = jπ. It was also shown in [22] that for any c, d > 0, θ(d) < π/2 and ϑ(c) > −π/2. So, it follows that the curve R is completely contained in (−∞, π/2)×R >0 , while S is contained in (−π/2, ∞)×R >0 , and that R restricted to [d 1 , ∞) and S restricted to [c 1 , ∞) do not intersect.
We can now prove Theorem 1.4.
Proof of Theorem 1.4. Let k ∈ N and for each i, j ∈ N, set x i := ρ(d i ), x i := ρ( d i ), y j := ̺(c j ) and y j := ̺( c j ). By Lemma 2.4, these sequences are unbounded. Therefore, we can find i, j > k and α > j such that Observe that the curves R and S−((α+j)π, 0) restricted to the intervals [d i , d α+i−j ] and [c j , c α ], respectively, are both contained in A k := [−(α + i − j)π, −iπ] × [y j , y α ]. As x α+i−j > y j and x i < y α and as R restricted to [d j , d α+j−i ] intersects A k only at the points (−(α+j −i)π, x α+j−i ) and (−jπ, x j ), the intermediate value Theorem implies that the curve R must intersect S − ((α + j)π, 0). Let d R > 1 and c S > 1 be the points such that R(d R ) = S(c S )−((α+j)π, 0). Using the formulas (2.8), we can argue as in the proof of Theorem 1.2 in [22] to conclude that w dR = w cS is a solution to the problem (1.9) with exactly α + j > k zeroes and, since w ′ dR (0) = w ′ dR (π) = 0, it has, at least, k + 1 critical points by Rolle's Theorem. The fact that the zeroes and that the critical points are isolated follows from Lemma 2.2.
This theorem implies 1.1 as follows.
Proof of Theorem 1.1. Associated to S, there is a Cartan-Münzner polynomial such that its restriction to the sphere is an isoparametric function f : S n → [−1, 1]. Then equation (1.2) can be reduced into equation (1.9). By Theorem 1.4, this equation admits a sign-changing solution w k having at least k isolated zeroes and at least k + 1 isolated critical points in [0, π]. Therefore, u k = w k (arccos f ) is a solution to the problem (1.2) having as regular level sets a disjoint union of connected isoparametric hypersurfaces diffeomorphic to S. As w k has at least k isolated zeroes, then the nodal set u −1 k (0) has at least k connected components, all of them being isoparametric hypersurfaces diffeomorphic to S. As w k has at least k − 1 isolated critical points in (0, π) and as w ′ k (0) = 0 = w ′ k (π), the critical set of u k consists in, at least, k − 1 connected isoparametric submanifolds diffeomorphic to S, together with the focal submanifolds M − = f −1 (−1) and M + = f −1 (1). Finally, by construction of w k , we have that w k (0) → ∞ or |w k (π)| → ∞ as k → ∞, for the number of zeroes increases as the initial or final conditions increase. Suppose, without loss of generality, that w k (0) → ∞. In this case, as arccos (f (x)) = 0 for all x ∈ M − we concluding the limit (1.4) for every x ∈ M − . Even if Theorem 1.1 holds for subcritical, critical and supercritical values of p > 1, the methods developed here do not allow us to prove the existence of solutions such that the final value w d (π) = c is negative. However, in case of the critical exponent p n , we can use the refined version of this theorem given in [22] to construct a sequence of solutions which is not uniformly bounded from below.
Proof of Corollary 1.2. As the focal manifolds that generate S have positive dimensional, the number of distinct principal curvatures ℓ must be bigger that 1. Let f be the Cartan-Münzner polynomial associated to S. Then, for any k ∈ N, Theorem 1.2 in [22] gives a solution to the Yamabe problem on the sphere (1.5) having the form u k = arctan(f (w k )), where w k is a solution to the problem (1.9), with p = p n , having exactly k zeroes in [0, π]. Lemma 2.4 together with Lemma 4.6 in [22] imply that the sequences (x i ) and (y j ) are both increasing and unbounded. In this way, the situation in Lemma 4.7 in [22] can not happen and the number of zeroes before and after a 0 must increase as the initial and final conditions, w k (0) and |w k (π)| respectively, increase. For k odd, by Lemma 2.2 and its proof, necessarily w k (0) > 0 and w k (π) < 0 and w k can not have an infinite number of zeroes before and after a 0 . Therefore there exists a subsequence (w kj ), with each k j odd, such that w kj (0) → ∞ and w kj (π) → −∞ as j → ∞. Since arctan(f (x)) = 0 for every x ∈ M + and arctan(f (x)) = π for every x ∈ M − , the sequence (u kj ) satisfies the desired limits as j → ∞.

Submersion with minimal fibers and the proof of Corollary 1.3.
We shall study Riemannian submersions of the form π : (S n , g 0 ) → (M m , g), where (M, g) is a closed Riemannian manifold with dimension m < n, and such that the fibers are minimal. It is well known that under these conditions, π is a harmonic morphism with dilation ϕ ≡ 1, see [3] or [24,Section 11.2]. In this case, a function v : M → R solves the equation with λ, µ > 0, if and only if u = µ λ 1 p−1 v •π is a solution to the Yamabe problem on the sphere (1.2) (Cf. [14]). Observe that if we are considering the critical exponent p n for the equation (1.2), then this same exponent is subcritical in equation (3.1) for, in this case, m < n implies p n < p m .
We next show that isoparametric functions are preserved by Riemannian submersions with minimal fibers.
Now suppose that f is an isoparametric function such that f (x) = f (y) if π(x) = π(y). Let a, b : R → R be such that |∇f | 2 h = a(f ) and ∆ h f = b(f ). As π : N → M is a submersion, it is a quotient map [30] and the function f passes to the quotient as an smooth function f : M → R such that f = f • π. Then, as before Since π is surjective, it has a right inverse and we conclude that |∇ f | 2 g = a( f ) and Let (M, g) denote (CP m ,g 0 ) or (HP m ,g 0 ), the complex and quaternionic spaces with their corresponding canonical metrics. Both of them are Einstein manifolds with constant positive scalar curvature (see [5,Theorem 14.39] and Sections 8.1 and 8.3 in [38]). We will denote it by Rg = Λ m > 0. Recall that CP m = S 2m+1 /S 1 and that H m = S 4m+3 /SU (2), where S 1 is the circle group and SU (2) is the group of unit quaternions. We need the following lemma. Proof. We begin with the proof of (1). In this case, we consider S 2m+1 ⊂ R 2m+2 . As m ≥ 3, we can write m = 2 + k with k ∈ N {0} and we can decompose Hence, we have an action of S 1 by isometries given as ζ(x, y) := (ζx, ζy), where ζ(x, y) := (ζx 1 , . . . , ζx α , ζy 1 , . . . , ζy β ), with ζ ∈ S 1 ⊂ C and x i , y j ∈ C. We can then consider the degree two Cartan-Münzner polynomial f : The restriction of f to S 2m+1 is an isoparametric function with focal submanifolds M − = S 2α−1 × {0} and M + = {0} × S 2β−1 , see [12].
Therefore, v is a solution to the Yamabe problem (1.6) if and only if u = µm λm 1 p 4m −1 v • π is a solution to the supercritical problem on the sphere By Lemma 3.2, there exists an SU (2)-invariant isoparametric function f : S 4m+3 → R, which is the restriction of a Cartan-Münzner polynomial and such that its corresponding focal submanifold have dimension at least 4. Hence, if we take and the supercritical problem (3.3) admits a sequence of nodal solutions (u k ) of the form u k = z k • f where z k is a solution to the problem (1.8) with at least k zeroes.
is a solution to the Yamabe problem (1.6). Moreover, by Lemma 3.1, f is an isoparametric function and, thus, v k has isoparametric hypersurfaces in HP m as level sets. Finally, the blow-up of the sequence (v k ) is a consequence of the limit (1.4) for the sequence (u k ) at M − or at M + .
Appendix A. Energy analysis and proof of Lemma 2.4 In this appendix we prove Lemma 2.4 for the solutions to the initial value problem (2.1). An entirely similar argument will hold for the problem (2.3) instead of (2.2). As in Section 2, we will suppose in what follows that the multiplicities of the principal curvatures of the isoparametric family satisfy 1 ≤ m − ≤ m + and that p satisfies inequality (2.6). To simplify the notation, we write equation (1.9) as where g(t) := λ ℓ 2 (|t| p−1 t − t). For the initial conditions w(0) = d and w ′ (0) = 0, let w d be the unique solution to problem (A.1) on [0, a 0 ]. Define the energy function Observe that E is nonincreasing on r ∈ [0, a 0 ], for The aim of this section is to prove that E(a 0 , d) → ∞ uniformly in [0, a 0 ] as d → ∞, for this will immediately imply Lemma 2.4. Since the proof is long and technical, we first sketch it in few lines, following the ideas given in [10] and [11]. First, in Lemma A.1 we prove the existence of the value r 0 (d) for which w d (r 0 ) = κd and d ≥ w d (r) ≥ κd for a suitable κ ∈ (0, 1) and for every r ∈ [0, r 0 ]. Since we will show that r 0 (d) → 0, we need to prove the existence of fixed T > 0, independent of d, such that E(r, d) → ∞ uniformly in [0, T ] as d → ∞. To see this, we establish a version of the Pohozaev identity [41] for equation (A.1), generalizing the ones given in [10,11]. This identity together with the properties of r 0 and inequality (2.6) will imply the existence of such T > 0. Finally, we prove that E(r, d) ≥ e −2T E(T, d) + C for every r ∈ [T, a 0 ], where C is a constant independent of d and r. The last inequality implies the desired uniform limit.
We begin with the existence of r 0 .
Proof. On the one hand, multiplying equation (A.1) by qw d , integrating from 0 to r ≤ a 0 and integrating by parts we obtain On the other hand, multiplying equation (A.1) by qζw ′ d , integrating from 0 to r, using integration by parts and (A.6) we have that But also, integration by parts yields Thus, adding (A.8) and (A.9), and using the above equality, identity (A.7) follows.
Observe that the derivative of w d does not appear in the right hand side of the identity, while the energy appears explicitly on the left hand side. Observe also that if r ∈ [0, a 0 ] is such that P (r, d) → ∞ as d → ∞, then also E(r, d) → ∞ as d → ∞. In this direction, we state and prove the following Lemma. The proof of this lemma is long and technical, and will take the following four pages. As will continue with the argument after its proof with Lemma A.4, the reader may skip it in a first reading.
Proof. The proof uses strongly that inequality (2.6) holds true. For the reader convenience, we divide it into three steps.
We begin with some estimates of r 0 (d) in terms of the initial condition d.

By
Step 1, r 0 (d) → 0 when d → ∞ and we can chooseD 2 ≥D 1 such that r 0 (d) < T for every d ≥D 2 . Since for every d ≥D 2 , we have that 1 ≤ κd ≤ w d (r) ≤ d for every r ∈ [0, r 0 ], and since the functions G(t) and tg(t) are nondecreasing when t ≥ 1, it follows that G(w d ) ≥ G(κd) ≥ 0 and that −g(w d )w d ≥ −g(d)d. Hence for every r ∈ [0, r 0 ] and every d ≥D 3 , whereD 3 ≥D 2 is such that G(κd)−g(d)d > 0 for every d ≥D 3 . First we prove the following limit lim d→∞ P (r 0 , d) = ∞ Indeed, since d ≥D 3 , the estimates (A.12) and the ones obtained in Step 1 yield that P (r 0 , d) = and since e (m+−1) d g(d) → 1 as d → ∞, (A.13) implies P (r 0 , d) → ∞ as d → ∞ as we wanted.