1. Introduction

Let (Sⁿ, δ_ij) be the unitary sphere with the standard metric. A natural question in Riemannian geometry is: given a function K : Sⁿ → R, is there a metric g conformally related to the standard metric δ_ij such that K is the scalar curvature of Sⁿ with respect to the metric g? This is equivalent to the problem of finding a positive smooth function u : Sⁿ → R which satisfies the equation

If we set g = u 4/n−2 δ_ij, where u is a solution of this problem, then the function K is the scalar curvature of Sn with respect to the metric g.

The problem of conformal deformation of metrics in Sⁿ have been extensively studied by many authors (for example, see [1], [2], [3], [5], [6], [7], [8], [9] and the references therein). An important feature of this problem is that it is a conformal invariant one. More precisely, if u is a solution of equation (1) then for any conformal map F : Sⁿ → Sⁿ the function αF (u) = |(F⁻¹)′ | n−2/2 u ◦ F⁻¹ is a solution to problem (1) with scalar curvature K ◦ F.

The problem of conformal deformation of metrics in Sⁿ can be approached using the so called Yamabe method, which consists in studying first the subcritical problem in the equation (1):

with p ∈ ­ (1, n+2/n−2) , and then consider the limit of the solutions when p ↑ n+2/n−2 .

Let E(u) be the energy norm associated with the linear part of (2), and let S be the set of non-negative functions u ∈ W^2,q(Sⁿ), (q > n/2 ) such that E(u) = E(1). Let us consider the open unit ball Bn+1 and the map Φ : Bⁿ⁺¹ → S defined by

where Fy : Sn → Sn is the restriction to Sn of a special conformal map Fy : Bn+1 → Bn+1 that satisfies Fy(0) = y and fix the points ± y/|y|; this function maps 0 to y and commutes with rotations about the line joining the origin and the point y. This map is referred to as a centered dilation.

For p ∈ ­ (1, n+2/n−2) and u ∈ S, let J_p(u) defined by J_p (u) = ʃSn Ku^p+1 dσ. If u is a critical point of Jp(·) on S, then a multiple of u satisfies problem (2). Let us define the function J_p = J_p ◦ Φ. In this paper, we will consider the equation

where K : Sⁿ → R is a nondegenerate function (Morse function) with ∆K ≠ 0 in its critical points, and Lu = ∆u − n(n−2)/4 u.

Let F : Sⁿ → Sⁿ be a conformal transformation and v = αF (u) : |(F ⁻¹)′ | n−2/2 u ◦ F ⁻¹. A straightforward calculation shows that u is solution of (3) if and only if the function η = v − 1 is a solution of an equation of the form

where a = vol(Sⁿ)(J_p(y))⁻¹K ◦ F ⁻¹|(F ⁻¹)′ | n−2/2 ^δ(1 + η)^−δ, L(η)=∆η + nη, Q(η) is a term which is quadratically small in η, and δ = n+2/n−2 − p. The linear operator L has an (n + 1) dimensional kernel consisting of the first order spherical harmonics. This obstruction to invert the linear operator L may be removed by replacing equation (4) by the projected equation T (y, η)=0, where

and P denotes the L²-orthogonal projection onto the orthogonal complement W of the first eigenspace of the laplacian on Sⁿ.

This work is motivated by the work of Schoen and Zhang in [8] on the prescribed scalar curvature problem on the n-dimensional sphere, n ≥ 3, and by the work of Escobar and García in [3] on the prescribed mean curvature on the n-dimensional unit ball, n ≥ 3. In fact our method parallels those of [8] and [3]. In this paper we will find in Section 3, using the inverse function Theorem, small solutions η_y of equation (5), where y is close to a critical point of J_p. In Section 4, we will find L^q and integral estimates of η_y and its first two derivatives.

In the last section, setting u_y = α_Fy (1 + η_y), we perturb the function uy and consider the function ũ_y = Λ_yu_y in order to achieve that E(ũ_y) = E(1). Next we define the map J_p(y) = J_p(ũy) and we show that the functions J_p(y) and J_p(y) are close in the C² norm, using the estimates of the functions η_y. The fact that the functions J_p(y) and J_p(y) are close implies that J_p(y) has a unique critical point y1 close to the critical point y₀ of J_p(y). This implies that ũ_y1 is a solution of the equation

Multiplying the function ũ_y1 by suitable constants, we find a solution of problem (2) and prove that u_y1 = α_Fy1 (1 + η_y1) is a solution of problem (3), respectively.

2. Preliminaries

Let y ∈ Bⁿ⁺¹. Up to a rotation we will assume that y = (0,..., 0, y_n+1), y_n+1 ≥ 0. In this case the centered dilation function F_y is given by F_y(x)=Σ−1 ◦ D_µ ◦ Σ(x), where the function

is the stereographic projection from the south pole of the sphere, the function

is the inverse of the stereographic projection, and the function D_µ : Rⁿ → Rⁿ is defined by D_µ(x) = µx, where x = (x, x_n+1) ∈ Sⁿ with x = (x₁,...,x_n) and µ = 1−|y|/1+|y| .

Since F_y = Σ⁻¹ ◦ D_µ ◦ Σ, then F_y(x) = B⁻¹(4µAx,(A² − 4µ²|x| ²) and F_y(0) = y, where

Note that F_y⁻¹ = F_−y.

If y ∈ B_β(1−|y0|)(y0) for some 0 <β< 1, then we have

The number µ satisfies the inequalities

And

Consider the map Φ : Bⁿ⁺¹ → S defined by Φ(y) = α_y := αF_y (1) = |(F_y⁻¹)′ | n−2/2 , where F_y : Sⁿ → Sⁿ is the conformal map that satisfies F_y(0) = y, and fix the points ± y/|y|. For p ∈ ­(1, n+2/n−2) and u ∈ S, let J_p(u) be defined by

If u is a critical point of Jp(·) on S, p ∈ ­ (1, n+2/n−2) , then a multiple of u satisfies problem (2). Let us define J_p = J_p ◦ Φ. The functions Jp are eigenfunctions of the laplacian on Bⁿ⁺¹ with the hyperbolic metric. In fact,

where δ = n+2/n−2 − p.

Let us define the function v_p(y) = ʃ_Sn (α_y(ξ))^p+1dσ(ξ), so that v_p(y) = vol(Sⁿ) for p = n+2/n−2 . The function v_p is positive and radially symmetric. Let us define the function ĴP = v_p⁻¹ Jp. For n ≥ 3 the functions ĴP are uniformly bounded in the C²(Bⁿ⁺¹) norm and they agree with K on Sⁿ. Using that all critical points of the function K are non-degenerate and △K ≠ 0 at each critical point, the following facts are proven in Proposition 2.1 in [8]. Since ĴP is C² in the closed ball, then ∂ĴP/∂r = 0 in the boundary of the ball. From here it can be seen that the critical points of ĴP near ∂Bⁿ⁺¹ actually lie on ∂Bⁿ⁺¹ and are the critical points of K. If y0 is a critical point of the function Jp near ∂Bⁿ⁺¹, then |∂vp/∂r (y0)| ≤ Cv_p(y₀)(1 − |y₀|). It is also proven that there exist constants C₁, C₂ > 0 such that

and consequently,

The estimates of the following proposition (see [4]) are very useful in this work.

Proposition 2.1. Let y₀ be a point near ∂Bⁿ⁺¹ which is the critical point of the function J_p and let y ∈ B_β(1−|y0|)(y0). Then,

The following propositions, which are useful to find a solution of problem (2), are respectively the Corollary 2.2 and Lemma 2.3 in [8].

Proposition 2.2. There is a number β < 1 such that, if we denote by y₀ one of the critical points of Jp near ∂Bⁿ⁺¹, then the following bound holds for y ∈ B_β(1−|y0|(y0):

Proposition 2.3. Suppose f,g are C² functions in the closed unit ball Bⁿ⁺¹ in Rⁿ⁺¹. Suppose there is a positive constant c such that

Assume f has a unique critical point y₀ in Bⁿ⁺¹, and g is close to f in the sense that

If ǫ is sufficiently small, then g has a unique critical point y₁ in Bⁿ⁺¹.

3. The projected equation

To begin with, we will do several transformations of equation (2). One of those transformations involves the definition of an operator

by setting T (η)(y) = T (y, η); this operator and the inverse function Theorem allow us to find a solution to problem (5).

After multiplying a solution u of equation (2) by a suitable constant, we can rewrite that equation as

where Lu = ∆u − n(n−2)/4 u. Let y₀ be a critical point of J_p which is one of the critical points of J_p near ∂Bⁿ⁺¹ given by Proposition 2.1 in [8]. Let y ∈ B_β(1−|y0|)(y0), with 0 <β< 1. To find a solution of equation (12), we will consider first the equation

where we have replaced J_p(u) by J_p(y).

A straightforward calculation shows that if u is solution of (13), F : Sⁿ → Sⁿ is a conformal transformation and v = αF (u) : |(F ⁻¹)′ | n−2/2 u ◦ F ⁻¹, then v is a solution of the problem

Setting v = 1+ η, and defining L(η)=∆η + nη, Q(η) = n(n−2)/4 ­ ((1 + η) n+2/n−2 − 1 − n+2/n−2 η), and a = vol(Sⁿ)(Jp(y))⁻¹K ◦ F⁻¹|(F⁻¹)′ | n−2/2 δ(1 + η)^−δ, if v is a solution of equation (14), then η is a solution of problem

Let {ξ₁, ξ₂,...ξ_n+1} a generator set of the first eigenfunctions of the laplacian of Sⁿ, that is,

This obstruction to invert the linear operator L may be removed by replacing equation (15) by the projected equation T (y, η)=0, where

and P denotes the L²-orthogonal projection onto the orthogonal complement W of the first eigenspace of Sⁿ, where we have used that (L(η), ξi)=0 implies P(L(η)) = L(η).

In order to keep track of the dependence on y, as in [8], we define a map

by setting T (η)(y) = T (y, η), where η is the map η(y) = ηy. We choose a norm on B^j,q which reflects the scales which appear in the problem:

Hence,

One of the main objectives of this work is to prove the existence of solutions of the projected equation (16). To reach it we will prove a similar result to Lemma 2.5 in [8].

Theorem 3.1. For p → n+2/n−2 and q ∈ (n/2, n), the function T is C¹ and satisfies the following bounds:

Moreover, ||(T′ (0))⁻¹|| ≤ C, where the constant C is independent on p. There exists η ∈ B^2,q with ||η|| ≤ Cǫ(p)µ^σ and T (η)=0. Furthermore η is the unique small solution of T (η)=0.

Proof. The bound for

follows from the following three lemmas.

Lemma 3.2. For any q ∈ ­ (n/2, n), ||T (y, 0)||_0,q ≤ Cµ^2−2w, where 0 <w< 1.

Proof. For η = 0 we have that

where a₀ = a(ξ, y, 0) = vol(Sⁿ)(J_p(y))⁻¹K ◦ F_y|F′_y| n−2/2 δ, and |F′_y| = 1−|y|²/|y+ξ|² , ξ ∈ Sn. It is easy to see that

To finish the lemma, in the following claims we will show that the terms in the right hand side of the previous inequality have the required bound.

Claim 1. ||F′_y| n−2/2 δ ⁻¹| ≤ Cµ^2−2w, with 0 < w < 1 and y ∈ B_β(1−|y0|)(y0).

Proof. Let us observe that |F′y| n−2/2 δ is of the form δ^δ. Taking 0 <w< 1 and using the L’Hôpital rule we get

Then, for δ small enough, |δ^δ − 1| ≤ Cδ^1−w ≤ Cµ^2−2w, and consequently,

Claim 2.|(vol(Sⁿ))⁻¹Jp(y) − K (y/|y|) ≤ Cµ^2−2w, where 0 <w< 1.

Proof. First observe that

Using Claim (1), we get

To find the bound of the second term in the right hand side, we consider the function Ĵ_p = Ĵ_p/v_p . By Taylor’s Theorem, there exists ζ between y and y/|y| such that

Since ∂Ĵ_­p/∂r (y/|y|) = 0 and Ĵ_p|Sⁿ = K, then

Therefore,

The inequality |T (y, 0)| ≤ Cµ^2−2w follows from Claims 1 and 2 and Proposition 2.1. Consequently,

Now, we will do the estimates of the first derivative of T (y, 0) in the y variable.

Lemma 3.3. For any q ∈ (n/2, n), ||∇_yT (y, 0)_0,q ≤ Cµ^1−w, with 0 < w < 1.

Proof. A calculation shows that

The proof of the following claims conclude the proof of the lemma.

Claim 3.

Proof. Since ∂ĴP/∂r = 0 in ∂Bⁿ⁺¹, the mean value Theorem implies |∂Ĵp/∂r (y)| ≤ C(1− |y0|).

Hence, |∂Ĵ_p/∂y_i| ≤ C(1 − |y₀). From Ĵ_p(y) = J_p(y)/v_p(y) and ∂(J_p(y))/∂y_i = v_p(y) ∂(Ĵ_p(y))/∂y_i + Ĵ_p(y) ∂(v_p(y))/∂y_i, we get

Therefore

Claim 4.

Proof. Since |F′_y|(ξ) = 1−|y|²/|y+ξ|² , a straightforward calculation shows that

Proposition 2.1 and Claims 3 and 4 yields to |∇_yT (y, 0)| ≤ Cµ^1−w, and therefore,

where w is a positive number less than one.

Now, we will estimate the second derivatives of T (y, 0) with respect to the y variable.

Lemma 3.4. For any q ∈ (n/2 , n) and 1− n/2q < r < 1/2 , we have ||∇y∇yT (y, 0)||_0,q ≤ Cµ^−2r.

Proof. Differentiating T (y, 0) twice with respect to the y variable we get

Let us estimate the first derivatives of A, B and D. Since

Claims 3 and 4 and Proposition 2.1 yield to ||∂A/∂y_i||_0,q ≤ C.

Now,

Hence, the inequality ||∂B/∂y_i||_0,q ≤ Cµ^−2r follows from the inequalities in Proposition 2.1 and Lemma 3.3. Finally, since

from Claims 3 and 4 and Proposition 2.1, we get ||∂D/∂y_i||_0,q ≤ C. The previous inequalities yield ||∇y∇yT(y, 0)||_0,q ≤ Cµ^−2r, as desired.

Using the previous lemmas, we reach the bound

where σ < 2 and ǫ(p) = µ^σ′ , with σ′ a small positive number.

Now we will estimate

Where

is given by

For this, consider φ ∈ B^2,q satisfying ||φ||B2,q ≤ 1. Let y ∈ B_{α(1−|y0|)(y0)}. Since

we have that

where a₀ = vol(Sⁿ)(Jp(y))⁻¹K ◦ F_y|F′_y| n−2/2 δ. Since q > n/2 , from the Sobolev embedding Theorem we get ||φ||L∞ ≤ C||φ||_2,q ≤ C||φ||_B2,q ≤ C. Therefore |L(φ)| ≤ C.

From this inequality and the estimates of Lemma 3.2, we obtain |T ′ (y, 0)(φ)| ≤ C, and ||T ′ (y, 0)(φ)||0,q ≤ C. Working similarly, and using the fact that φ, ∇yφ, ∇y∇yφ belong to H^2,q(Sⁿ) for q > n/2, we get ||T′ (0)|| ≤ C.

Now, we will show that the derivative of T′ is Lipschitz; that is,

For this, taking φ ∈ B_2,p such that ||φ||B_2,p ≤ 1, we get

where aη = a0(1 + η)^−δ. Since

using that ||η1||, ||η0|| ≤ 1/4 and the mean value Theorem, we get

and therefore

To finish the proof of Theorem 1, we need to show that T ′ (0) has a bounded inverse. Let φ ∈ B^2,q(Sⁿ) and Ψ ∈ B^0,q(Sⁿ). Consider the problem T′ (0)φ = Ψ. Let us recall that

Elliptic estimates shows that ||φ||_2,q ≤ C||L(φ)||_0,q. Since

from the estimates of Lemma 3.2 we get

then,

Taking µ^σǫ(p) small we get that 1 − kCǫ(p)µ^σ > 0 and ||φ||_2,q ≤ C||Ψ||_0,q. Working analogously, we have that

And

Therefore,

The rest of the proof follows from the inverse function Theorem.

4. Integral and L^q estimates of the function ηy

In this section, given the solution η_y, y ∈ B_β(1−|y0|), of the projected equation, we will find L^q estimates not only of the function η_y, but also of its first and second y derivatives; in addition, we will do also integral estimates of ∇_yη_y and ∇_y∇_yη_y.

Lemma 4.1. For q ∈ ­ (n/2, n), ||ηy||0,q ≤ Cǫ(p)µ^σ, with σ < 2, where ǫ(p) → 0 as p → n+2/n−2 .

Proof. From Theorem 3.1, T (y, ηy)=0. Then,

Setting a = a₀D, where D = (1 + η_y)−δ, we have

From the mean value Theorem it follows that

Using Hölder’s inequality,the estimates of Lemma 1, Theorem 1, q > n/2 and the Sobolev embedding Theorem, we have

Since ||ηy||_2,q,Sⁿ ≤ C||L(η_y)||_0,q,Sⁿ , then ||ηy||0,q,Sⁿ ≤ ||ηy||_2,q,Sⁿ ≤ Cǫ(p)µ^σ, as desired.

Lemma 4.2. For q ∈ ­ (n/2 , n), ||∇yηy||_0,q ≤ Cµ^1−w, with 0 < w < 1.

Proof. Differentiating the equation

we find that the terms of its derivative satisfy the inequalities

And

where we have used the estimates of Theorem 3.1 and δ = Cµ².

Hence,

Using Hölder’s inequality, the estimates in Theorem 3.1 and Lemma 4.1, we arrive to

and therefore ||η′_y||_2,q,Sⁿ ≤ Cµ^1−w for 0 < w < 1.

Differentiating twice the equation T (y, η)=0 and working as in Lemma 4.2, we get

Lemma 4.3. For q ∈ ­ (n/2, n), ||∇_y∇_yη_y||_0,q ≤ Cµ^−2r, with 1 – n/2q < r < 1/2 .

In what follows, we will estimate the integral of the function η′_y, y ∈ B_β(1−y0)(y0).

Lemma 4.4. For q ∈ ­ (n/2 , n) and y ∈ B_β(1−y0)(y0), ʃSⁿ ∇yηydσ|≤ Cǫ(p)µ^σ, with σ < 2.

Using that L(η_y)=∆η_y + nη_y, we obtain

Setting A = Vol(Sⁿ)Jp⁻¹ (y)K ◦ F_y|F′_y| n−2/2 δ, D = (1 + ηy)^−δ and E = (1 + η_y) n+2/n−2 , we get

Hence,

and therefore,

Writing (AD−1)E = (AD−1)(E−1)+A(D−1)+A−1, and observing that ­ ʃ_Sn Adσ = cte, we have

On the other hand,

Then,

Using the estimates on η_y,η′_y, the mean value Theorem and Hölder’s inequality, we arrive to

for s, s′ such that 1/s + 1/s′ = 1. Working similarly, we get

where we have used the mean value Theorem, Proposition 2.1, Lemma 4.1 and the estimates of Theorem 3.1. Using Lemma 4.2 and proceeding as before, we get

And

Consequently,

with σ < 2.

Finally, we will estimate the integral of η′′_y.

Lemma 4.5. For q ∈ ­ (n/2, n), ʃ_Sn ∇_y∇_yη_ydσ| ≤ Cǫµ^σ−2r, with r < 1/2.

Proof. Denoting ∂²η_y/∂_yj∂_yi by η′′y, and differentiating the terms on the right hand side of equation (17) with respect to y_j , we get

In what follows we will estimate the terms in the right hand side of this equality. Using Hölder’s inequality, Proposition 2.1 and the four previous lemmas, we have:

And

Putting together these inequalities, we obtain the desired bound for |ʃ_Sn ∇_y∇_yη_ydσ|.

5. Solutions of some nonlinear elliptic equations

In this section, using the estimates of Sections 3 and 4, we will prove that the functions J_p(y) and J_p(y) are close in the C2-norm. The fact this functions are close implies that J_p(y) has a unique critical point y1 close to the critical point y₀ of J_p(y). This implies that ũ_­y1 is a solution of equation (6).

Multiplying the function ũ_­y1 by the constant (Jp(ũ_­y1))^1−p we will find that u = (J_p(ũ­_y1))^1−pũ­_y1 is a solution of the subcritical problem (2). Recalling that ηy is a solution of the equation T (y, η)=0, if we let uy = α−1Fy (1 + ηy) we will prove that u_y1 = αF_y1^-1 (1 + η_y1) is a solution of the perturbed equation (3).

Consider the quotient

and define the functions γ_y = Λ_y (1 + η_y) and ũ­_y = αF_y (γ_y).

Recalling that S is the set of non-negative functions u ∈ W^2,q (Sⁿ), (q > n/2 ) such that E(u) = E(1), we get the following proposition:

Proposition 5.1. The function ũ_y belongs to the set S.

Proof. By Theorem 3.1, the function η_y satisfies the equation

Where

And

Summing the constant n− n(n+2)/4 in both side of the equation T (y, η)=0 and simplifying, we get

where ã = a(1 + η)^δ. Therefore,

Since

we have

Multiplying this equation by γ and integrating, we have

where we have used that ʃ_Sn P(f) = ʃ_Sn f for every integrable function f, and

Consequently,

Since ũ_y = αF_y(γ_y), the conformal invariance of the energy E implies that the function ũ_y ∈ S, as desired.

Let us define the function

Now, we will prove that the difference of the functions J_p(y) and J_p(y) = ʃ_Sn Kα_y^p+1 are very close in C² norm.

Proposition 5.2. Let y₀ be a critical point of the function J_p(y), and let y ∈ B_β(1−|y0|)(y₀). Then,

And

where σ < 2, 0 <w< 1, r < 1/2 and ǫ(p) goes to zero as p goes to n+2/n−2 .

Proof. A change of variables yields

To estimate this difference, we will do it for the terms in the right hand side separately. The mean value Theorem and Theorem 3.1 implies

And

To estimate (Λ_y^p+1 − 1), we make a change of variables to get

Since |Λy| ≤ 1 and Λ²_y − 1 = (Λ_y − 1)(Λ_y + 1), then

Where

Then,

From the previous estimates we get

Now, we need to estimate the difference of the first derivatives:

Let us write the first term in the right hand side as

where,

Since K is a Morse function, from the proof of Proposition 1.1 in [8] we have that ||1 − K ◦ F_y||_0,q ≤ Cǫ₀µ, where ǫ0 can be chosen as small as we want. From this fact, the mean value Theorem, Hölder’s inequality, Proposition 2.1, Theorem 3.1 and the integral and L^p estimates of the functions η_y and η′_y, we arrive to

Analogously,

A calculation shows that

and therefore

Consequently,

Writing the difference of the second derivatives as

and working as before, we obtain the desired estimate.

Proposition 5.3. The function J_p has a unique critical point y1 on B_β(1−|y0|)(y₀).

Proof. The inequalities in Proposition 5.2 imply that there exists ǫ > 0, sufficiently small, such that

For z ∈ Bⁿ⁺¹ we define

On one hand, by Proposition 2.2 we have

And

On the other hand, inequality (18) implies

Proposition 2.3 implies Proposition 5.3.

If we change, in the proof of Theorem 2.4 of [8], u_y1 for ũ_y1 = Λ_y1u_y1, and we follow the arguments in there, we get

Proposition 5.4. The critical point ũ_y1 of the function J_p in Proposition 5.3 is a solution of problem (6).

Corollary 5.5. The function u = (J_p(ũ_­y1))^1−pũ­_y1 is a solution of the subcritical problem (2) and the function u_y1 = Λ⁻¹_y1 ũ_­y1 = αF_y1^-1 (1 + η_y1 ) is a solution of the perturbated equation (3).

Acknowledgements

Gonzalo García and Liliana Posada thanks the Universidad del Valle for its support with the 7920 Project. The authors are very grateful to the reviewers for their suggestions and comments that really improved the paper.

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