# Multiplicity results for the Yamabe problem on *S*^{n}

^{n}

## Abstract

We discuss some results related to the existence of

*multiple solutions*for the Yamabe problem.### Sign up for PNAS alerts.

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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*R*denote its scalar curvature. A metric_{g}*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 where We will discuss hereafter some new multiplicity results concerning Eq. 1 following recent papers (1, 2). First, we recall some existence results.

*g′*conformally equivalent to*g*such that the corresponding scalar curvature*R*is constant, say_{g′}*R*. Hereafter, we will always deal with dimension_{g′}≡ 1*n ≥ 3*. Letting the conformality factor ρ be of the form*ρ = u*and^{4/(n−2)}, u > 0, R_{g′}*R*are related by the formula_{g}\[ \begin{equation*}R_{g^{\prime}}=u^{-\frac{n+2}{n-2}}\hspace{.167em} \left \left(R_{g}u-\frac{4(n-1)}{(n-2)}{\Delta}_{g}u\right) \right ,\end{equation*}\]

*Δ*denotes the Laplace–Beltrami operator. Hence solving the Yamabe problem amounts to finding a positive solution_{g}*u ∈ H*of^{1}(M)\[ \begin{equation*}-\hspace{.167em}\frac{4(n-1)}{(n-2)}{\Delta}_{g}u+R_{g}u=u^{\frac{n+2}{n-2}}.\end{equation*}\]

## 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 The difficulty here is related to the fact that the embedding of

*n ≥ 6*and*g*is not locally conformally flat. A positive solution of Eq. 1 is searched as a minimum*u ∈ H*of the Sobolev quotient^{1}(M)\[ \begin{equation*}Q(u)=\frac{\displaystyle\frac{1}{2}\hspace{.167em}{\int _{M}}\hspace{.167em}\displaystyle\frac{4(n-1)}{(n-2)}{\vert}{\nabla}_{g}u{\vert}^{2}+R_{g}u^{2}}{{\parallel}u{\parallel}^{2}_{2{^\ast}}}.\end{equation*}\]

*H*into^{1}(M)*L*is not compact and one does not know if a minimizing sequence converges. To overcome this problem, one argues as follows.^{2*}(M)### Step 1:

Let 𝒮 denote the best Sobolev constant, and suppose that Then

*u*is a minimizing sequence such that_{m}\[ \begin{equation*}Q(u_{m})\hspace{.167em}{\rightarrow}\hspace{.167em}c<\frac{1}{n}{\mathscr{S}}^{n/2}.\end{equation*}\]

*u*possesses a convergent subsequence. In other words, one proves that the_{m}*Palais–Smale*condition at level*c*, in short*(PS)*, holds true provided_{c}*c < 1/n𝒮*.^{n/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*W*is not identically zero and Eq. 2 is proved by taking appropriate functions that concentrate on points where_{g}*W*._{g}≠ 0For a broader discussion on the existence results of the Yamabe problem, we refer to refs. 6 or 7. In the specific case that

*M = S*and^{n}*g*is close to the standard metric*g*, an indication of the proof of the existence of a solution of Eq. 1, for any_{0}*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 = S*\( \begin{equation*}_{{\mathit{T}}}^{{\mathit{1}}}\end{equation*}\), where^{n−1}× S*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*H*is greater than the global minimum of^{1}(M)*Q*on*H*.^{1}(M)Below we outline, in a simplified version, a new multiplicity result discussed in ref. 1.

We take Hereafter, we do not change notation and use the same symbol

*M = S*and use stereographic coordinates. Then Eq. 1 becomes^{n}\[ \begin{equation*}-\hspace{.167em}\frac{4(n-1)}{(n-2)}{\Delta}_{g}u+R_{g}u=u^{\frac{n+2}{n-2}},\hspace{1em}u\hspace{.167em}{\in}\hspace{.167em}D^{1,2}({\mathbb{R}}^{n}).\end{equation*}\]

*g*to denote metric on ℝ^{n}corresponding to the original one on*S*.^{n}Let us suppose that the metric Expanding Above, the function ψ depends on

*h = (h*is smooth and has compact support and consider a metric_{ij})*g = (g*of the form_{ij})\[ \begin{equation*}g_{ij}(x)={\delta}_{ij}+{\varepsilon}h_{ij}(x),\hspace{.167em}x\hspace{.167em}{\in}\hspace{.167em}{\mathbb{R}}^{n},\hspace{1em}where\hspace{.167em}{\delta}_{jj}=1,\hspace{.167em}{\delta}_{ij}=0\hspace{.167em}for\hspace{.167em}i{\not=}j.\end{equation*}\]

*Δ*and_{g}*R*with respect to ɛ, Eq. 3 becomes of the form_{g}\[ \begin{equation*}-\hspace{.167em}\frac{4(n-1)}{(n-2)}{\Delta}u=u^{\frac{n+2}{n-2}}+{\psi}({\varepsilon},\hspace{.167em}x,\hspace{.167em}u,\hspace{.167em}Du,\hspace{.167em}D^{2}u),\hspace{1em}u\hspace{.167em}{\in}\hspace{.167em}D^{1,2}({\mathbb{R}}^{n}).\end{equation*}\]

*x*through*h*.Solutions of Eq. 5 are the stationary points whereand there resultswhereIt follows that here whereandThe reason why we need to consider also the term

*u ∈ D*of a functional such as^{1,2}(ℝ^{n})\[ \begin{equation*}f_{{\varepsilon}}(u)=f_{0}(u)+G({\varepsilon},\hspace{.167em}u),\end{equation*}\]

\[ \begin{equation*}f_{0}(u)=\frac{1}{2}\hspace{.167em}{\int _{{\mathbb{R}}^{n}}}\hspace{.167em}{\vert}{\nabla}u{\vert}^{2}dx-\frac{1}{2{^\ast}}\hspace{.167em}{\int _{{\mathbb{R}}^{n}}}\hspace{.167em}{\vert}u{\vert}^{2{^\ast}}dx,\end{equation*}\]

*G*is related to ψ and is a perturbation term, in the sense that*G(0, u) ≡ 0*. More precisely, if we write\[ \begin{equation*}R_{{\varepsilon}}={\varepsilon}R_{1}+{\varepsilon}^{2}R_{2}+o({\varepsilon}^{2}),\end{equation*}\]

\[ \begin{equation*}f_{{\varepsilon}}(u)={\int _{{\mathbb{R}}^{n}}}\hspace{.167em}[{\mathscr{F}}_{1}({\varepsilon},\hspace{.167em}u){\cdot}{\mathscr{F}}_{2}({\varepsilon},\hspace{.167em}u)]dx+o({\varepsilon}^{2}),\end{equation*}\]

\[ \begin{equation*}{\mathscr{F}}_{1}=\frac{1}{2} \left \left({\vert}{\nabla}u{\vert}^{2}-{\varepsilon}\hspace{.167em}{{\sum_{i,j}}}\hspace{.167em}h_{ij}D_{i}uD_{j}u+{\varepsilon}^{2}\hspace{.167em}{{\sum_{i,j,l}}}\hspace{.167em}h_{il}h_{lj}D_{i}uD_{j}u+({\varepsilon}R_{1}+{\varepsilon}^{2}R_{2})u^{2}\right) \right -\frac{1}{2{^\ast}}{\vert}u{\vert}^{2{^\ast}}\end{equation*}\]

\[ \begin{equation*}{\mathscr{F}}_{2}=1+\frac{{\varepsilon}}{2}\hspace{.167em}trh+{\varepsilon}^{2} \left \left(\frac{1}{8}\hspace{.167em}(trh)^{2}-\frac{1}{4}\hspace{.167em}tr(h^{2})\right) \right .\end{equation*}\]

*f*has the form_{ɛ}\[ \begin{equation*}f_{{\varepsilon}}(u)=f_{0}(u)+{\varepsilon}G_{1}(u)+{\varepsilon}^{2}G_{2}(u)+o({\varepsilon}^{2}),\end{equation*}\]

\[ \begin{equation*}G_{1}(u)={\int } \left \left(-c_{n}\hspace{.167em}{{\sum_{i,j}}}\hspace{.167em}h_{ij}D_{i}uD_{j}u+\frac{1}{2}\hspace{.167em}R_{1}u^{2}+ \left \left(c_{n}{\vert}{\nabla}u{\vert}^{2}-\frac{1}{2{^\ast}}{\vert}u{\vert}^{2{^\ast}}\right) \right \frac{1}{2}\hspace{.167em}trh\right) \right dx,\end{equation*}\]

\[ \begin{equation*}G_{2}(u)={\int } \left \left[c_{n}\hspace{.167em}{{\sum_{i,j,l}}}\hspace{.167em}h_{il}h_{lj}D_{i}uD_{j}u+\frac{1}{2}\hspace{.167em}R_{2}u^{2}+ \left \left(c_{n}{\vert}{\nabla}u{\vert}^{2}-\frac{1}{2{^\ast}}{\vert}u{\vert}^{2{^\ast}}\right) \right \left \left(\frac{1}{8}\hspace{.167em}(tr\hspace{.167em}h)^{2}-\frac{1}{4}\hspace{.167em}tr(h^{2})\right) \right \right) \right \end{equation*}\]

\[ \begin{equation*} \left +\frac{1}{2}\hspace{.167em}trh \left \left(\frac{1}{2}\hspace{.167em}R_{1}u^{2}-c_{n}\hspace{.167em}{{\sum_{i,j}}}\hspace{.167em}h_{ij}D_{i}uD_{j}u\right) \right \right) \right dx.\hspace{.167em}\end{equation*}\]

*ɛ*will become clear later on, in the discussion before Eq. 11 below.^{2}G_{2}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. Letdenote the unique (up to dilations and translations) positive solution to the problemand consider the Clearly, every

\[ \begin{equation*}z_{0}(x)={\kappa}_{n}{\cdot}\frac{1}{(1+{\vert}x{\vert}^{2})^{\displaystyle\frac{n-2}{2}}},\hspace{1em}{\kappa}_{n}=(4n(n-1))^{\frac{n-2}{4}}\end{equation*}\]

\[ \begin{equation*}-\hspace{.167em}\frac{4(n-1)}{(n-2)}{\Delta}u=u^{(n+2)/(n-2)},\hspace{1em}u\hspace{.167em}{\in}\hspace{.167em}D^{1,2}({\mathbb{R}}^{n})\end{equation*}\]

*n + 1*dimensional manifold\[ \begin{equation*}Z= \left \left\{z_{{\mu},{\xi}}={\mu}^{-\frac{n-2}{2}}z_{0} \left \left(\frac{x-{\xi}}{{\mu}}\right) \right {\vert}{\mu}>0,\hspace{.167em}{\xi}\hspace{.167em}{\in}\hspace{.167em}{\mathbb{R}}^{n}\right) \right .\end{equation*}\]

*z*is also a solution of Eq. 10 and hence is a critical point of_{μ,ξ}∈ Z*f*. It turns out that the restriction of_{0}*f"*on_{0}(z)*(T*is invertible for all_{z}Z)^{⊥}*z ∈ Z*. Let*L*denote this inverse. The following lemma is a sort of Lyapunov–Schmidt reduction for_{z}*f*._{ɛ}### Lemma 2.1.

*For ɛ small there exists*

*w = w*

_{ɛ}(μ, ξ) : Z → D^{1,2}(ℝ^{n})*such that*

*∇f*.

_{ɛ}(z + w) ∈ T_{z}Z*Moreover, let*

*φ*,

_{ɛ}: 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 of*

*φ*.

_{ɛ}*Then*

*u**

_{ɛ}= z_{μ*,ξ*}+ w_{ɛ}(μ*,ξ*)*is a critical point of*

*f*.

_{ɛ}According to Then, setting It is clear that we have to control the behaviour of Γ at infinity and at the boundary of

*Lemma 2.1*, it suffices to study the critical points of the finite dimensional functional*φ*. This is accomplished by taking the expansion of_{ɛ}*φ*with respect to ɛ. Roughly, because_{ɛ}*G*on_{1}(z) ≡ 0*Z*, the leading term of φ_{ɛ}also depends on*G*. More precisely, let_{2}*b*be the constant value of*f*on_{0}*Z*, and let*Γ : Z → ℝ*be defined by\[ \begin{equation*}{\Gamma}(z)=G_{2}(z)-\frac{1}{2}\hspace{.167em}(L_{z}{\nabla}G_{1}(z),{\nabla}G_{1}(z)).\end{equation*}\]

*Γ(μ, ξ) := Γ(z*there results_{μ,ξ})\[ \begin{equation*}{\phi}_{{\varepsilon}}({\mu},\hspace{.167em}{\xi})=b+{\varepsilon}^{2}{\Gamma}({\mu},\hspace{.167em}{\xi})+o({\varepsilon}^{2}).\end{equation*}\]

*Z, ∂Z = {(0, ξ) : ξ ∈ ℝ*. This is done in the following two lemmas.^{n}}### Lemma 2.2.

Γ

*can be extended smoothly to**∂Z**by setting**Γ(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*

\[ \begin{equation*}\frac{{\partial}{\Gamma}}{{\partial}{\mu}}\hspace{.167em}(0,\hspace{.167em}{\xi})=0,\hspace{1em}\frac{{\partial}^{2}{\Gamma}}{{\partial}{\mu}^{2}}\hspace{.167em}(0,\hspace{.167em}{\xi})=0,\hspace{1em}\frac{{\partial}^{3}{\Gamma}}{{\partial}{\mu}^{3}}\hspace{.167em}(0,\hspace{.167em}{\xi})=0,\hspace{1em}{\forall}{\xi}\hspace{.167em}{\in}\hspace{.167em}{\mathbb{R}}^{n}.\end{equation*}\]

*∂*\( \begin{equation*}_{{\mathit{{\mu}}}}^{{\mathit{4}}}\end{equation*}\)

*Γ(0, ξ)*depends on the Weyl tensor

*W*. Precisely, let us expand the Weyl tensor

_{g}*W*with respect to ɛ. One finds there exists a tensor

_{g}*W**depending on

_{h}*h*such that

\[ \begin{equation*}W_{g}={\varepsilon}W{^\ast}_{h}+o({\varepsilon}).\end{equation*}\]

*W**is not identically zero, then

_{h}*g = δ + ɛh*is not locally conformally flat.

### Lemma 2.4.

*If*

*W**,

_{h}≢ 0*then there results*

\[ \begin{equation*} \left \left\{\begin{matrix}\frac{{\partial}^{4}{\Gamma}}{{\partial}{\mu}^{4}}\hspace{.167em}(0,\hspace{.167em}{\xi})<0,\enskip\hfill &if\hspace{.167em}n>6\\ -{\infty}\enskip\hfill &if\hspace{.167em}n=6.\end{matrix} \right \end{equation*}\]

*n ≥ 6*and

*W**, then Γ achieves the minimum at some

_{h}≢ 0*(μ*, ξ*)*with

*μ* > 0*. From Eq. 11 it follows that

*(μ*, ξ*)*is also a minimum of φ

_{ɛ}. Finally, according to

*Lemma 2.1*, we infer that

*z*is a solution of Eq. 3.

_{μ*,ξ*}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 where α, β have compact support. The functionals Γ corresponding to

*h*such that\[ \begin{equation*}h(x)={\alpha}(x)+{\beta}(x-x_{0}),\end{equation*}\]

*h, α, β*will be distinguished by adding the corresponding index. One can show the following lemma.### Lemma 2.5.

*Letting*

*|x*,

_{0}| → ∞*there results*

\[ \begin{equation*}{\Gamma}_{h}({\mu},\hspace{.167em}{\xi})={\Gamma}_{{\alpha}}({\mu},\hspace{.167em}{\xi})+{\Gamma}_{{\beta}}({\mu},\hspace{.167em}{\xi}-x_{0})+o(1).\end{equation*}\]

*Lemma 2.5*it readily follows that

*Γ*has two distinct strict minima on

_{h}*Z*provided

*|x*is sufficiently large. In conclusion, another application of

_{0}|*Lemma 3.1*yields

*Theorem 2.6*.

### Theorem 2.6.

*Let*

*g = δ + ɛ[α + β(⋅ − x*,

_{0})]*where α*,

*β have compact support. Suppose that*

*W**,

_{α}*W**

_{β}*are not identically =*

*0 and let n ≥ 6*.

*Then for*

*|x*

_{0}|*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 φ_{ɛ}and hence have a Morse index = 1, as critical points of*f*._{ɛ}## 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 φ

_{ɛ}found in*Theorem 2.6*to prove that φ_{ɛ}has a Mountain-Pass critical point on*Z*. Working directly with φ_{ɛ}one needs to sharpen*Lemma 2.2*by showing that not only Γ but also φ_{ɛ}can be extended to μ = 0, and there results*φ*. This permits us to prove:_{ɛ}(0,ξ) ≡ b### Theorem 3.1 (theorem 3.1 in ref. 2).

*Suppose the same assumptions as in*Theorem 2.6

*hold true. Then, for*

*|x*

_{0}|*large and*

*|ɛ|*

*small, Eq.*

**3**

*has a third solution with Morse index greater than or equal to*2.

#### Remark 3.2:

Because

*φ*and_{ɛ}(0, ξ) ≡ b*φ*as_{ɛ}(μ, ξ) → b*μ*, one infers that φ^{2}+ |ξ|^{2}→ ∞_{ɛ}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*S*the standard metric^{n}*g*. In other words, the preceding approach provides (in the perturbative case) a unified proof of the existence results by refs. 4, 5, and 7._{0}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*(μ*. Such a solution will be called a_{i}, ξ_{i}), i = 1, 2*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*f*, near a manifold of_{ɛ}*pseudocritical points*obtained by gluing together two functions such as*z*. Dealing with the Yamabe problem, one needs more careful estimates, because the functions_{μ,ξ}∈ Z*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*

*|x*

_{0}|*large and |ɛ| small, Eq.*

**3**

*has a two-bump solution*

*u*

_{ɛ}*close to*

*z*.

_{1}+ z_{2}, z_{i}∈ Z*Moreover, the Morse index of*

*u*

_{ɛ}*is greater than or equal to*2

*and*

*f*.

_{ɛ}(u_{ɛ}) ∼ 2b#### Remark 3.4:

More in general, when

*h = Σ*\( \begin{equation*}_{{\mathit{1}}}^{{\mathit{m}}}\end{equation*}\)*α*and_{i}(x−x_{i})*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 where

*g*of the form\[ \begin{equation*}g={\delta}+{\varepsilon}\hspace{.167em}{{\sum^{{\infty}}_{1}}}\hspace{.167em}{\sigma}_{i}{\alpha}_{i}(x-x_{i})\end{equation*}\]

*W**_{αi}*≢ 0*. Taking*σ*with_{i}= i^{−b}*b > 2/n*, and letting*|x*, one can show that the functional φ_{i}| ≫ 1_{ɛ}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*(S*, some further restriction on the dimension^{n}, g)*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*C*norm. One can prove:^{2}### Theorem 3.5 (theorem 1.3 in ref. 2).

*Let*

*k ≥ 2*

*and*

*n ≥ 4k + 3*.

*Then there exists a family of*

*C*

^{k}*metrics*

*g*

_{ɛ}*such that:*

(

*i*)*g*_{ɛ}*converges in**C*^{k}(S^{n})*to the standard metric**g*_{0}*as**ɛ → 0*,*and*(

*ii*)*for ɛ small, the Yamabe problem for**(S*^{n}, g_{ɛ})*has a sequence**v*_{ɛ,i}*of solutions such that**∥v*_{ɛ,i}∥_{L∞}*→ ∞**as**i → ∞*.#### Remark 3.6:

*Theorem 3.5*shows that the Schoen compactness result cannot be extended to

*C*manifolds of arbitrary dimension.

^{k}## 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.

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Copyright © 2002, The National Academy of Sciences.

#### Submission history

**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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