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Multiplicity results for the Yamabe problem on Sn

October 24, 2002
99 (24) 15252-15256

Abstract

We discuss some results related to the existence of multiple solutions for the Yamabe problem.
In this article we are concerned with a classical problem arising in differential geometry: the Yamabe problem. Let a compact n-dimensional Riemannian manifold (M, g) be given, and let Rg denote its scalar curvature. A metric g′ on M is conformally equivalent to g if there exists a positive function ρ such that g′ = ρg.
The Yamabe problem consists of seeking a metric g′ conformally equivalent to g such that the corresponding scalar curvature Rg′ is constant, say Rg′ ≡ 1. Hereafter, we will always deal with dimension n ≥ 3. Letting the conformality factor ρ be of the form ρ = u4/(n−2), u > 0, Rg′ and Rg are related by the formula
\[ \begin{equation*}R_{g^{\prime}}=u^{-\frac{n+2}{n-2}}\hspace{.167em} \left \left(R_{g}u-\frac{4(n-1)}{(n-2)}{\Delta}_{g}u\right) \right ,\end{equation*}\]
where Δg denotes the Laplace–Beltrami operator. Hence solving the Yamabe problem amounts to finding a positive solution u ∈ H1(M) of
\[ \begin{equation*}-\hspace{.167em}\frac{4(n-1)}{(n-2)}{\Delta}_{g}u+R_{g}u=u^{\frac{n+2}{n-2}}.\end{equation*}\]
We will discuss hereafter some new multiplicity results concerning Eq. 1 following recent papers (1, 2). First, we recall some existence results.

1. Existence Results

As far as the existence is concerned, the Yamabe problem has been completely solved. First, Aubin (3) has proved that a solution to Eq. 1 exists provided n ≥ 6 and (M, g) is not locally conformally flat. Next, Schoen (4) has handled the case of the dimensions n = 3, 4, 5 but still assuming that the metric g is not locally conformally flat. Finally, the flat case has been addressed by Schoen and Yau in ref. 5.
Let us give an idea of the proof in the case in which n ≥ 6 and g is not locally conformally flat. A positive solution of Eq. 1 is searched as a minimum u ∈ H1(M) of the Sobolev quotient
\[ \begin{equation*}Q(u)=\frac{\displaystyle\frac{1}{2}\hspace{.167em}{\int _{M}}\hspace{.167em}\displaystyle\frac{4(n-1)}{(n-2)}{\vert}{\nabla}_{g}u{\vert}^{2}+R_{g}u^{2}}{{\parallel}u{\parallel}^{2}_{2{^\ast}}}.\end{equation*}\]
The difficulty here is related to the fact that the embedding of H1(M) into L2*(M) is not compact and one does not know if a minimizing sequence converges. To overcome this problem, one argues as follows.

Step 1:

Let 𝒮 denote the best Sobolev constant, and suppose that um is a minimizing sequence such that
\[ \begin{equation*}Q(u_{m})\hspace{.167em}{\rightarrow}\hspace{.167em}c<\frac{1}{n}{\mathscr{S}}^{n/2}.\end{equation*}\]
Then um possesses a convergent subsequence. In other words, one proves that the Palais–Smale condition at level c, in short (PS)c, holds true provided c < 1/n𝒮n/2.

Step 2:

One looks for suitable test functions that satisfy Eq. 2. For example, in ref. 3 this is done by using the assumption that g is not locally conformally flat. Precisely, in such a case the Weyl tensor Wg is not identically zero and Eq. 2 is proved by taking appropriate functions that concentrate on points where Wg ≠ 0.
For a broader discussion on the existence results of the Yamabe problem, we refer to refs. 6 or 7. In the specific case that M = Sn and g is close to the standard metric g0, an indication of the proof of the existence of a solution of Eq. 1, for any n ≥ 3 and without the flatness requirement, is given later on; (see Remark 3.2).

2. Nonuniqueness

Unlike the existence, very few results are known about the nonuniqueness for the Yamabe problem. In ref. 8 it is taken M = Sn−1 × S\( \begin{equation*}_{{\mathit{T}}}^{{\mathit{1}}}\end{equation*}\), where S\( \begin{equation*}_{{\mathit{T}}}^{{\mathit{1}}}\end{equation*}\) denotes a circle with radius T. Letting T → ∞ it is proved that Eq. 1 has multiple solutions. Another known result deals with the case that (M,g) inherits a symmetry (see refs. 9 and 10), when one can find conditions ensuring that the minimum of Q on the symmetric functions of H1(M) is greater than the global minimum of Q on H1(M).
Below we outline, in a simplified version, a new multiplicity result discussed in ref. 1.
We take M = Sn and use stereographic coordinates. Then Eq. 1 becomes
\[ \begin{equation*}-\hspace{.167em}\frac{4(n-1)}{(n-2)}{\Delta}_{g}u+R_{g}u=u^{\frac{n+2}{n-2}},\hspace{1em}u\hspace{.167em}{\in}\hspace{.167em}D^{1,2}({\mathbb{R}}^{n}).\end{equation*}\]
Hereafter, we do not change notation and use the same symbol g to denote metric on ℝn corresponding to the original one on Sn.
Let us suppose that the metric h = (hij) is smooth and has compact support and consider a metric g = (gij) of the form
\[ \begin{equation*}g_{ij}(x)={\delta}_{ij}+{\varepsilon}h_{ij}(x),\hspace{.167em}x\hspace{.167em}{\in}\hspace{.167em}{\mathbb{R}}^{n},\hspace{1em}where\hspace{.167em}{\delta}_{jj}=1,\hspace{.167em}{\delta}_{ij}=0\hspace{.167em}for\hspace{.167em}i{\not=}j.\end{equation*}\]
Expanding Δg and Rg with respect to ɛ, Eq. 3 becomes of the form
\[ \begin{equation*}-\hspace{.167em}\frac{4(n-1)}{(n-2)}{\Delta}u=u^{\frac{n+2}{n-2}}+{\psi}({\varepsilon},\hspace{.167em}x,\hspace{.167em}u,\hspace{.167em}Du,\hspace{.167em}D^{2}u),\hspace{1em}u\hspace{.167em}{\in}\hspace{.167em}D^{1,2}({\mathbb{R}}^{n}).\end{equation*}\]
Above, the function ψ depends on x through h.
Solutions of Eq. 5 are the stationary points u ∈ D1,2(ℝn) of a functional such as
\[ \begin{equation*}f_{{\varepsilon}}(u)=f_{0}(u)+G({\varepsilon},\hspace{.167em}u),\end{equation*}\]
where
\[ \begin{equation*}f_{0}(u)=\frac{1}{2}\hspace{.167em}{\int _{{\mathbb{R}}^{n}}}\hspace{.167em}{\vert}{\nabla}u{\vert}^{2}dx-\frac{1}{2{^\ast}}\hspace{.167em}{\int _{{\mathbb{R}}^{n}}}\hspace{.167em}{\vert}u{\vert}^{2{^\ast}}dx,\end{equation*}\]
and G is related to ψ and is a perturbation term, in the sense that G(0, u) ≡ 0. More precisely, if we write
\[ \begin{equation*}R_{{\varepsilon}}={\varepsilon}R_{1}+{\varepsilon}^{2}R_{2}+o({\varepsilon}^{2}),\end{equation*}\]
there results
\[ \begin{equation*}f_{{\varepsilon}}(u)={\int _{{\mathbb{R}}^{n}}}\hspace{.167em}[{\mathscr{F}}_{1}({\varepsilon},\hspace{.167em}u){\cdot}{\mathscr{F}}_{2}({\varepsilon},\hspace{.167em}u)]dx+o({\varepsilon}^{2}),\end{equation*}\]
where
\[ \begin{equation*}{\mathscr{F}}_{1}=\frac{1}{2} \left \left({\vert}{\nabla}u{\vert}^{2}-{\varepsilon}\hspace{.167em}{{\sum_{i,j}}}\hspace{.167em}h_{ij}D_{i}uD_{j}u+{\varepsilon}^{2}\hspace{.167em}{{\sum_{i,j,l}}}\hspace{.167em}h_{il}h_{lj}D_{i}uD_{j}u+({\varepsilon}R_{1}+{\varepsilon}^{2}R_{2})u^{2}\right) \right -\frac{1}{2{^\ast}}{\vert}u{\vert}^{2{^\ast}}\end{equation*}\]
\[ \begin{equation*}{\mathscr{F}}_{2}=1+\frac{{\varepsilon}}{2}\hspace{.167em}trh+{\varepsilon}^{2} \left \left(\frac{1}{8}\hspace{.167em}(trh)^{2}-\frac{1}{4}\hspace{.167em}tr(h^{2})\right) \right .\end{equation*}\]
It follows that here fɛ has the form
\[ \begin{equation*}f_{{\varepsilon}}(u)=f_{0}(u)+{\varepsilon}G_{1}(u)+{\varepsilon}^{2}G_{2}(u)+o({\varepsilon}^{2}),\end{equation*}\]
where
\[ \begin{equation*}G_{1}(u)={\int } \left \left(-c_{n}\hspace{.167em}{{\sum_{i,j}}}\hspace{.167em}h_{ij}D_{i}uD_{j}u+\frac{1}{2}\hspace{.167em}R_{1}u^{2}+ \left \left(c_{n}{\vert}{\nabla}u{\vert}^{2}-\frac{1}{2{^\ast}}{\vert}u{\vert}^{2{^\ast}}\right) \right \frac{1}{2}\hspace{.167em}trh\right) \right dx,\end{equation*}\]
and
\[ \begin{equation*}G_{2}(u)={\int } \left \left[c_{n}\hspace{.167em}{{\sum_{i,j,l}}}\hspace{.167em}h_{il}h_{lj}D_{i}uD_{j}u+\frac{1}{2}\hspace{.167em}R_{2}u^{2}+ \left \left(c_{n}{\vert}{\nabla}u{\vert}^{2}-\frac{1}{2{^\ast}}{\vert}u{\vert}^{2{^\ast}}\right) \right \left \left(\frac{1}{8}\hspace{.167em}(tr\hspace{.167em}h)^{2}-\frac{1}{4}\hspace{.167em}tr(h^{2})\right) \right \right) \right \end{equation*}\]
\[ \begin{equation*} \left +\frac{1}{2}\hspace{.167em}trh \left \left(\frac{1}{2}\hspace{.167em}R_{1}u^{2}-c_{n}\hspace{.167em}{{\sum_{i,j}}}\hspace{.167em}h_{ij}D_{i}uD_{j}u\right) \right \right) \right dx.\hspace{.167em}\end{equation*}\]
The reason why we need to consider also the term ɛ2G2 will become clear later on, in the discussion before Eq. 11 below.
Functionals of the form Eq. 3 can be studied by means of a perturbation result in critical point theory (11–13), which has been used in refs. 14 and 15 to study the scalar curvature problem. Let
\[ \begin{equation*}z_{0}(x)={\kappa}_{n}{\cdot}\frac{1}{(1+{\vert}x{\vert}^{2})^{\displaystyle\frac{n-2}{2}}},\hspace{1em}{\kappa}_{n}=(4n(n-1))^{\frac{n-2}{4}}\end{equation*}\]
denote the unique (up to dilations and translations) positive solution to the problem
\[ \begin{equation*}-\hspace{.167em}\frac{4(n-1)}{(n-2)}{\Delta}u=u^{(n+2)/(n-2)},\hspace{1em}u\hspace{.167em}{\in}\hspace{.167em}D^{1,2}({\mathbb{R}}^{n})\end{equation*}\]
and consider the n + 1 dimensional manifold
\[ \begin{equation*}Z= \left \left\{z_{{\mu},{\xi}}={\mu}^{-\frac{n-2}{2}}z_{0} \left \left(\frac{x-{\xi}}{{\mu}}\right) \right {\vert}{\mu}>0,\hspace{.167em}{\xi}\hspace{.167em}{\in}\hspace{.167em}{\mathbb{R}}^{n}\right) \right .\end{equation*}\]
Clearly, every zμ,ξ ∈ Z is also a solution of Eq. 10 and hence is a critical point of f0. It turns out that the restriction of f"0(z) on (TzZ) is invertible for all z ∈ Z. Let Lz denote this inverse. The following lemma is a sort of Lyapunov–Schmidt reduction for fɛ.

Lemma 2.1.

For ɛ small there exists w = wɛ(μ, ξ) : Z → D1,2(ℝn)such that ∇fɛ(z + w) ∈ TzZ. Moreover, let φɛ : Z → ℝ,
\[ \begin{equation*}{\phi}_{{\varepsilon}}({\mu},\hspace{.167em}{\xi})\hspace{.167em}:=\hspace{.167em}f_{{\varepsilon}}(z_{{\mu},{\xi}}+w_{{\varepsilon}}({\mu},\hspace{.167em}{\xi})),\end{equation*}\]
and let (μ*,ξ*) be a critical point ofφɛ. Thenu*ɛ = zμ*,ξ* + wɛ(μ*,ξ*) is a critical point offɛ.
According to Lemma 2.1, it suffices to study the critical points of the finite dimensional functional φɛ. This is accomplished by taking the expansion of φɛ with respect to ɛ. Roughly, because G1(z) ≡ 0 on Z, the leading term of φɛ also depends on G2. More precisely, let b be the constant value of f0 on Z, and let Γ : Z → ℝ be defined by
\[ \begin{equation*}{\Gamma}(z)=G_{2}(z)-\frac{1}{2}\hspace{.167em}(L_{z}{\nabla}G_{1}(z),{\nabla}G_{1}(z)).\end{equation*}\]
Then, setting Γ(μ, ξ) := Γ(zμ,ξ) there results
\[ \begin{equation*}{\phi}_{{\varepsilon}}({\mu},\hspace{.167em}{\xi})=b+{\varepsilon}^{2}{\Gamma}({\mu},\hspace{.167em}{\xi})+o({\varepsilon}^{2}).\end{equation*}\]
It is clear that we have to control the behaviour of Γ at infinity and at the boundary of Z, ∂Z = {(0, ξ) : ξ ∈ ℝn}. This is done in the following two lemmas.

Lemma 2.2.

Γ can be extended smoothly to ∂Z by settingΓ(0, ξ) = 0. Moreover, there results
\[ \begin{equation*}{\Gamma}({\mu},{\xi})\hspace{.167em}{\rightarrow}\hspace{.167em}0,\hspace{1em}as\hspace{.167em}{\mu}+{\vert}{\xi}{\vert}\hspace{.167em}{\rightarrow}\hspace{.167em}+{\infty}.\end{equation*}\]

Lemma 2.3.

Let g be of the form of Eq. 4. Then there results
\[ \begin{equation*}\frac{{\partial}{\Gamma}}{{\partial}{\mu}}\hspace{.167em}(0,\hspace{.167em}{\xi})=0,\hspace{1em}\frac{{\partial}^{2}{\Gamma}}{{\partial}{\mu}^{2}}\hspace{.167em}(0,\hspace{.167em}{\xi})=0,\hspace{1em}\frac{{\partial}^{3}{\Gamma}}{{\partial}{\mu}^{3}}\hspace{.167em}(0,\hspace{.167em}{\xi})=0,\hspace{1em}{\forall}{\xi}\hspace{.167em}{\in}\hspace{.167em}{\mathbb{R}}^{n}.\end{equation*}\]
It turns out that the fourth derivative \( \begin{equation*}_{{\mathit{{\mu}}}}^{{\mathit{4}}}\end{equation*}\)Γ(0, ξ) depends on the Weyl tensor Wg. Precisely, let us expand the Weyl tensor Wg with respect to ɛ. One finds there exists a tensor W*h depending on h such that
\[ \begin{equation*}W_{g}={\varepsilon}W{^\ast}_{h}+o({\varepsilon}).\end{equation*}\]
Let us remark that if W*h is not identically zero, then g = δ + ɛh is not locally conformally flat.

Lemma 2.4.

If W*h ≢ 0, then there results
\[ \begin{equation*} \left \left\{\begin{matrix}\frac{{\partial}^{4}{\Gamma}}{{\partial}{\mu}^{4}}\hspace{.167em}(0,\hspace{.167em}{\xi})<0,\enskip\hfill &if\hspace{.167em}n>6\\ -{\infty}\enskip\hfill &if\hspace{.167em}n=6.\end{matrix} \right \end{equation*}\]
The preceding lemmas immediately imply that if n ≥ 6 and W*h ≢ 0, then Γ achieves the minimum at some (μ*, ξ*) with μ* > 0. From Eq. 11 it follows that (μ*, ξ*) is also a minimum of φɛ. Finally, according to Lemma 2.1, we infer that zμ*,ξ* is a solution of Eq. 3.
The advantage of our approach is that we can somewhat localize the minima, and we will use this fact to obtain multiple solutions. Precisely, let us take a function h such that
\[ \begin{equation*}h(x)={\alpha}(x)+{\beta}(x-x_{0}),\end{equation*}\]
where α, β have compact support. The functionals Γ corresponding to h, α, β will be distinguished by adding the corresponding index. One can show the following lemma.

Lemma 2.5.

Letting |x0| → ∞, there results
\[ \begin{equation*}{\Gamma}_{h}({\mu},\hspace{.167em}{\xi})={\Gamma}_{{\alpha}}({\mu},\hspace{.167em}{\xi})+{\Gamma}_{{\beta}}({\mu},\hspace{.167em}{\xi}-x_{0})+o(1).\end{equation*}\]
From Lemma 2.5 it readily follows that Γh has two distinct strict minima on Z provided |x0| is sufficiently large. In conclusion, another application of Lemma 3.1 yields Theorem 2.6.

Theorem 2.6.

Let g = δ + ɛ[α + β(⋅ − x0)], where α, β have compact support. Suppose that W*α, W*βare not identically =0 and let n ≥ 6. Then for|x0| sufficiently large problem(Eq. 3) has two distinct solutions provided |ɛ| is small enough.
Theorem 2.6 is a particular case of more general results (see ref. 1).

Remark 2.7:

The solutions found in Theorem 2.6 are minima of Γ as well as of φɛ and hence have a Morse index = 1, as critical points of fɛ.

3. Other Multiplicity Results

In this section we shortly discuss some nonuniqueness result from ref. 2. A common feature of these results is that the solutions, unlike the preceding ones, have Morse index greater than 1.
First of all, one can start from the two minima of φɛ found in Theorem 2.6 to prove that φɛ has a Mountain-Pass critical point on Z. Working directly with φɛ one needs to sharpen Lemma 2.2 by showing that not only Γ but also φɛ can be extended to μ = 0, and there results φɛ(0,ξ) ≡ b. This permits us to prove:

Theorem 3.1 (theorem 3.1 in ref. 2).

Suppose the same assumptions as in Theorem 2.6 hold true. Then, for |x0| large and|ɛ| small, Eq. 3 has a third solution with Morse index greater than or equal to 2.

Remark 3.2:

Because φɛ(0, ξ) ≡ b and φɛ(μ, ξ) → b as μ2 + |ξ|2 → ∞, one infers that φɛ has a critical point (μ*, ξ*) with μ* > 0. Such a point gives rise to a solution of Eq. 1, for any n ≥ 3 and any metric g close on Sn the standard metric g0. In other words, the preceding approach provides (in the perturbative case) a unified proof of the existence results by refs. 4, 5, and 7.
The next result deals with a new kind of solution, which is found near the sum of two elements of Z and hence is peaked near two points i, ξi), i = 1, 2. Such a solution will be called a two-bump solution. The main tool is an extension, developed in ref. 16, of the perturbation method sketched above. The problem handled in ref. 16 was the chaotic behaviour for a class of second-order Hamiltonian systems. The usual shadowing lemma is replaced by the existence of critical points of the Euler functional, say again fɛ, near a manifold of pseudocritical points obtained by gluing together two functions such as zμ,ξ ∈ Z. Dealing with the Yamabe problem, one needs more careful estimates, because the functions z ∈ Z have a polynomial decay at infinity, not exponential as in the case of Hamiltonian systems. However, one can show:

Theorem 3.3 (theorem 4.1 in ref. 2).

Suppose the same assumptions as in Theorem 2.6 hold true. Then, for |x0| large and |ɛ| small, Eq. 3 has a two-bump solutionuɛclose to z1 + z2, zi ∈ Z. Moreover, the Morse index of uɛis greater than or equal to2 and fɛ(uɛ) ∼ 2b.

Remark 3.4:

More in general, when h = Σ\( \begin{equation*}_{{\mathit{1}}}^{{\mathit{m}}}\end{equation*}\)αi(x−xi) and W*αi ≢ 0, one can find multibump solutions with a large Morse index.
The last result we discuss deals with the existence of infinitely many (one-bump) solutions of Eq. 3. Consider a metric g of the form
\[ \begin{equation*}g={\delta}+{\varepsilon}\hspace{.167em}{{\sum^{{\infty}}_{1}}}\hspace{.167em}{\sigma}_{i}{\alpha}_{i}(x-x_{i})\end{equation*}\]
where W*αi ≢ 0. Taking σi = i−b with b > 2/n, and letting |xi| ≫ 1, one can show that the functional φɛ possesses infinitely many critical points that give rise to infinitely many solutions of Eq. 3. To come back to solutions of the Yamabe problem on (Sn, g), some further restriction on the dimension n is in order. This is not surprising, because a compactness result by Schoen (ref. 4; see also refs. 17 and 18) implies that the possible solutions of Eq. 1 are bounded in C2 norm. One can prove:

Theorem 3.5 (theorem 1.3 in ref. 2).

Let k ≥ 2 and n ≥ 4k + 3. Then there exists a family of Ckmetrics gɛsuch that:
(i)  gɛconverges inCk(Sn) to the standard metricg0as ɛ → 0, and
(ii)  for ɛ small, the Yamabe problem for(Sn, gɛ) has a sequencevɛ,iof solutions such that∥vɛ,iL → ∞ asi → ∞.

Remark 3.6:

Theorem 3.5 shows that the Schoen compactness result cannot be extended to Ck manifolds of arbitrary dimension.

Note

This paper results from the National Academy of Sciences colloquium, “Nonlinear Partial Differential Equations and Applications,” held January 4–19, 1999, at the Arnold and Mabel Beckman Center of the National Academies of Science and Engineering in Irvine, CA.

Acknowledgments

This work was supported by Ministero dell'Universitá e della Ricerca Scientifica e Tecnologica under the national project Variational Methods and Nonlinear Differential Equations.

References

1
Ambrosetti A. & Malchiodi, A. (1999) J. Funct. Anal.168,529-561.
2
Berti M. & Malchiodi, A. (2001) J. Funct. Anal.180,210-241.
3
Aubin T. (1976) J. Math. Pures Appl.55,269-296.
4
Schoen R. (1984) J. Differ. Geom.20,479-495.
5
Schoen R. & Yau, S. T. (1988) Invent. Math.92,47-71.
6
Aubin T., (1998) Some Nonlinear Problems in Differential Geometry (Springer, New York).
7
Lee J. M. & Parker, T. H. (1987) Bull. Am. Math. Soc.17,37-91.
8
Schoen R., (1991) Differential Geometry: A Symposium in Honor of M. do Carmo (Wiley, New York), pp. 311–320.
9
Hebey E. & Vaugon, M. (1993) Bull. Sci. Math.117,241-286.
10
Hebey E. & Vaugon, M. (1992) Indiana Univ. J. Math.41,377-407.
11
Ambrosetti A. & Badiale, M. (1998) Ann. Inst. H. Poincaré Analyse Non Linéaire15,233-252.
12
Ambrosetti A. & Badiale, M. (1996) C. R. Acad. Sci. Ser. I323,753-758.
13
Ambrosetti A. & Badiale, M. (1998) Proc. R. Soc. Edinburgh Ser. A128,1131-1161.
14
Ambrosetti A., Garcia Azorero, J. & Peral, I. (1999) J. Funct. Anal.165,117-149.
15
Ambrosetti A. & Malchiodi, A. (2001) J. Differ. Equations170,228-245.
16
Berti M. & Bolle, P. (1999) Ann. Math. Pura Appl.176,323-378.
17
Li Y. (1995) J. Differ. Equations120,319-410.
18
Li Y. (1996) Comm. Pure Appl. Math.49,437-477.

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Proceedings of the National Academy of Sciences
Vol. 99 | No. 24
November 26, 2002
PubMed: 12399546

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    Published online: October 24, 2002
    Published in issue: November 26, 2002

    Acknowledgments

    This work was supported by Ministero dell'Universitá e della Ricerca Scientifica e Tecnologica under the national project Variational Methods and Nonlinear Differential Equations.

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    Antonio Ambrosetti
    Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Beirut 2-4, Trieste 34014, Italy

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      Multiplicity results for the Yamabe problem on Sn
      Proceedings of the National Academy of Sciences
      • Vol. 99
      • No. 24
      • pp. 15247-15831

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