Clustered solutions for supercritical elliptic equations on Riemannian manifolds

Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 5. We are concerned with the following elliptic problem: −∆gu + a(x)u = u n+2 n−2+ε , u > 0 in M, where ∆g = divg(∇) is the Laplace–Beltrami operator onM, a(x) is a C2 function onM such that the operator −∆g + a is coercive, and ε > 0 is a small real parameter. Using the Lyapunov–Schmidt reduction procedure, we obtain that the problem under consideration has a k-peaks solution for positive integer k ≥ 2, which blow up and concentrate at one point in M.


Introduction
Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 5, where g denotes the metric tensor. We are interested in the following supercritical elliptic problem: −∆ g u + a(x)u = u n+2 n−2 +ε , u > 0 in M, (1.1) where ∆ g = div g (∇) is the Laplace-Beltrami operator on M, a(x) is a C 2 function on M, and ε > 0 is a real parameter with ε → 0.
There are many results about the existence and properties of solutions for nonlinear elliptic equations on compact Riemannian manifolds. Let us mention the following problem: −ε 2 ∆ g u + u = |u| p−2 u in M, (1.2) where (M, g) is a compact, connected, Riemannian manifold of class C ∞ with Riemannian metric g, with dimension n ≥ 3, 2 < p < 2n n−2 and ε is a positive parameter. The existence and multiplicity of solutions to problem (1.2) was considered in [2,4,25]. Moreover, the existence of peak solutions for (1.2) was obtained by Dancer, Micheletti and Pistoia [6,14,15].
The asymptotically critical case on a Riemannian manifold was studied by Micheletti, Pistoia and Vétois in [16]. They proved that problem (1.1) has blowing-up families of positive solutions, provided the graph of a(x) is distinct at some point from the graph of n−2 4(n−1) Scal g . Moreover, the existence of multi-peak solutions that are separate from each other for (1.1) was considered by Deng in [7]. Pistoia and Vétois [18] discovered the existence of sign-changing bubble towers for (1.1).
In the case a ≡ n−2 4(n−1) Scal g , equation (1.1) is intensively studied as the Yamabe equation whose positive solutions u are such that the scalar curvature of the conformal metric u 2 * −2 g is constant (see [1,21,24]).
It is important to recall some results for the following linear perturbation of the Yamabe problem: −∆ g u + ( n − 2 4(n − 1) Scal g + ε)u = u n+2 n −2 in (M, g), (1.3) where (M, g) is a non-locally conformally flat compact Riemannian manifold. Druet in [8] proved that problem (1.3) does not have any blowing-up solution when ε < 0 and the dimension of the manifold is n = 3, 4, 5 (except when the manifold is conformally equivalent to the round sphere). In case ε > 0, if n = 3, there is no blowing-up solutions to problem (1.3) as proved by Li and Zhu [13]. Esposito, Pistoia and Vétois in [9] showed that there exist blowing-up solutions for n ≥ 6, and they built solutions which blow up at non-vanishing stable critical points ξ 0 of the Weyl tensor, i.e., |Weyl g (ξ 0 )| g ̸ = 0. Recently, Robert and Vétois [20], Pistoia and Vaira [17], Thizy and Vétois [23] provided several geometric and analytic settings in which linear perturbations to (1.3) have positive clustered bubbles that are non-isolated blowing-up solutions. In particular, Pistoia and Vaira in [17] investigated the existence of cluster solutions for problem (1.3). More precisely, they proved that for any point ξ 0 ∈ M, which is non-degenerate and a non-vanishing minimum point of the Weyl tensor, and for any integer k, there exists a family of solutions developing k peaks collapsing to ξ 0 as ε goes to zero. Moreover, Thizy and Vétois [23] constructed clustering positive solutions for a perturbed critical elliptic equation on a closed manifold of dimension four and five.
Motivated by the previous consideration, in the present paper, we construct a family of cluster solutions for equation (1.1) with ε small enough.

Remark 1.2.
We can get the same result for the subcritical case. Precisely, if (M, g) is a smooth compact Riemannian manifold of dimension n ≥ 5, a(x) is a C 2 function on M such that the operator −∆ g + a is coercive, and ξ 0 is a nondegenerate minimum point of the function φ(ξ ) with φ(ξ 0 ) > 0, then for any ε < 0 small enough, problem (1.1) has a clustered solution u ε , which blows up at k points which collapse to ξ 0 as ε → 0. We will not give the details of the proof in this case.
The proof of our result relies on a very well known finite dimensional Lyapunov-Schmidt reduction procedure, introduced in [10,19] and used in many of the quoted papers. In particular, we refer to [6,14,15] for nonlinear elliptic problems on Riemannian manifolds, [7,16] for asymptotically critical elliptic equations on Riemannian manifolds, [9,17] for linear perturbations on the Yamabe problems, and recently this method has been used to study the fractional Yamabe problem by Choi and Kim in [5], and Kim, Musso and Wei in [12]. This paper is organized as follows. In Section 2, we introduce the framework and present some preliminary results. The proof of the main result is given in Section 3. Section 4 contains the asymptotic expansion of the energy functional.

The framework and preliminary results
Let r be a positive real number less than i M , where i M is the injectivity radius of M, and χ r be a smooth cutoff function such that 0 where B(0, r) denotes the ball in T ξ M centered at 0 with radius r. For any point ξ in M and any positive real number λ, we define the function W λ,ξ on M by It is clear that the embedding i : for some positive constant C independent of w. By standard elliptic estimates [11], given a real number s > 2n n−2 , that is, ns n+2s > 2n n+2 , for any w ∈ L ns n+2s (M), the function i * (w) belongs to L s (M) and satisfies for some positive constant C independent of w. For ε small, we set Taking into account that ns ε n + 2s ε = s ε 2 * − 1 + ε , and by (2.2), we can write problem (1.1) as where f ε (u) = u n+2 n−2 +ε + and u + = max{u, 0}.
It is known, see [3,19], that every solution of the linear equation is a linear combination of the functions Let us define on M the functions Let ξ 0 be a nondegenerate local maximum point of Scal g (ξ ), is the exponential map whose base point is ξ 0 and α = 1 2 and β = n − 4 2n .
We introduce the functional J ε : where 2 * = 2n n−2 denotes the Sobolev critical exponent. It is well known that any critical point of J ε is a solution to problem (1.1). We define the functionalJ ε : where Wd ,τ is as (3.1) and ϕ ε,d,τ is given by Proposition 3.1.
The next result, whose proof is postponed to Section 4, allows us to solve equation (3.3), by reducing the problem to a finite dimensional one.
For each small ε > 0, let alsod ε = (d ε1 , . . . , d εk ) ∈ (η −1 , η) k be an element attaining the maximum value of the map ∑ k j=1 h(d j ). Observe that d εi → d 0 as ε → 0 for all i = 1, . . . , k. From (3.6) and (3.7), we get an upper bound of the maximum of F ε as follows: However, given M 1 > 0 large enough, the above estimate does not make sense, since we can derive at the same time. Thus, (d, τ) ∈ O k η,R . Then problem (1.1) has a solution of the form u ε = Wd ,τ + ϕ ε,d,τ for ε > 0 sufficiently small, which we call a solution blowing-up at k points that collapse to ξ 0 as ε → 0. By taking into account the definition of the approximate solution ϕ ε,d,τ and (3.4), the proof is completed.
where ω n is the volume of the unit n-sphere. We have We will estimate each term in the following lemmas.
as ε → 0, C 0 -uniformly with respect toτ in (ℝ n ) k and tod in compact subsets of (ℝ + ) k , where C n is the positive constant as ε → 0, C 0 -uniformly with respect toτ in (ℝ n ) k and tod in compact subsets of (ℝ + ) k .
For the third term in (4.17), if l ̸ = h, using Hölder's inequality and the fact that the estimate for the second integral is bounded by Cε n(α−β) , then we can get that the third term is bounded by Cε n(α−β) . Moreover, for j ̸ = h and l = h, we have where we used the fact that for j ̸ = h and x ∈ B h . Thus, we have |I h | ≤ Cε n(α−β) . Therefore, (4.14) follows from (4.15), (4.16) and the last inequality.