Multiplicity results for the Yamabe problem on Sⁿ

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 R_g 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 R_g′ is constant, say R_g′ ≡ 1. Hereafter, we will always deal with dimension n ≥ 3. Letting the conformality factor ρ be of the form ρ = u^4/(n−2), u > 0, R_g′ and R_g are related by the formula

where Δ_g denotes the Laplace–Beltrami operator. Hence solving the Yamabe problem amounts to finding a positive solution u ∈ H¹(M) of

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

The difficulty here is related to the fact that the embedding of H¹(M) into L^2*(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 u_m is a minimizing sequence such that

Then u_m 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/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_g is not identically zero and Eq. 2 is proved by taking appropriate functions that concentrate on points where W_g ≠ 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, where S 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

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 = (h_ij) is smooth and has compact support and consider a metric g = (g_ij) of the form

Expanding Δ_g and R_g with respect to ɛ, Eq. 3 becomes of the form

Above, the function ψ depends on x through h.

Solutions of Eq. 5 are the stationary points u ∈ D^1,2(ℝⁿ) of a functional such as

where

and G is related to ψ and is a perturbation term, in the sense that G(0, u) ≡ 0. More precisely, if we write

there results

where

It follows that here f_ɛ has the form

where

and

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

denote the unique (up to dilations and translations) positive solution to the problem

and consider the n + 1 dimensional manifold

Clearly, every z_μ,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 (T_zZ)^⊥ is invertible for all z ∈ Z. Let L_z denote this inverse. The following lemma is a sort of Lyapunov–Schmidt reduction for f_ɛ.

Lemma 2.1.

For ɛ small there exists w = w_ɛ(μ, ξ) : Z → D^1,2(ℝⁿ)such that ∇f_ɛ(z + w) ∈ T_zZ. Moreover, let φ_ɛ : Z → ℝ,

and let (μ*,ξ*) be a critical point ofφ_ɛ. Thenu*_ɛ = z_μ*,ξ* + w_ɛ(μ*,ξ*) is a critical point off_ɛ.

According to Lemma 2.1, it suffices to study the critical points of the finite dimensional functional φ_ɛ. This is accomplished by taking the expansion of φ_ɛ with respect to ɛ. Roughly, because G₁(z) ≡ 0 on Z, the leading term of φ_ɛ also depends on G₂. More precisely, let b be the constant value of f₀ on Z, and let Γ : Z → ℝ be defined by

Then, setting Γ(μ, ξ) := Γ(z_μ,ξ) there results

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

Lemma 2.3.

Let g be of the form of Eq. 4. Then there results

It turns out that the fourth derivative ∂Γ(0, ξ) depends on the Weyl tensor W_g. Precisely, let us expand the Weyl tensor W_g with respect to ɛ. One finds there exists a tensor W*_h depending on h such that

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

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 φ_ɛ. Finally, according to Lemma 2.1, we infer that z_μ*,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

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

From Lemma 2.5 it readily follows that Γ_h 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 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_ɛ(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 φ_ɛ(0, ξ) ≡ b and φ_ɛ(μ, ξ) → b as μ² + |ξ|² → ∞, one infers that φ_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 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 (μ_i, ξ_i), 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 f_ɛ, near a manifold of pseudocritical points obtained by gluing together two functions such as z_μ,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 solutionu_ɛclose to z₁ + z₂, z_i ∈ Z. Moreover, the Morse index of u_ɛis greater than or equal to2 and f_ɛ(u_ɛ) ∼ 2b.

Remark 3.4:

More in general, when h = Σα_i(x−x_i) 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

where W*_αi ≢ 0. Taking σ_i = i^−b with b > 2/n, and letting |x_i| ≫ 1, one can show that the functional φ_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 (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 C^kmetrics g_ɛsuch that:

• (i)  g_ɛconverges inC^k(Sⁿ) to the standard metricg₀as ɛ → 0, and

• (ii)  for ɛ small, the Yamabe problem for(Sⁿ, g_ɛ) has a sequencev_ɛ,iof solutions such that∥v_ɛ,i∥_L^∞ → ∞ asi → ∞.

Remark 3.6:

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