Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds

1 Introduction

Let ( M , g ) be a n -dimensional smooth compact Riemannian manifold without boundary, where g denotes the metric tensor. We consider the following elliptic system:

where Δ g = div g ∇ is the Laplace-Beltrami operator on M , and h ( x ) is a C 1 -function on M , p , q > 0 .

The starting point on the study of system (1.1) is its scalar version

If p ∈ ( 1 , 2 * − 1 ) ( 2 * = 2 n n − 2 if n ≥ 3 and 2 * = + ∞ if n = 1 , 2), by the compact embedding H g 1 ( M ) ↪ L g p ( M ) , one can obtain a solution of (1.2). Micheletti and Pistoia [26] also considered the following singularly perturbed nonlinear elliptic problem:

where p ∈ ( 1 , 2 * − 1 ) , n ≥ 2 , and ε > 0 is a small parameter. By performing the Lyapunov-Schmidt reduction procedure, they obtained a single blowing-up solution for (1.3). Successively, multiple blowing-up solutions and clustered solutions are constructed in [27] and [9] respectively (see also [7] for solutions concentrating at a submanifold).

In the critical case p = 2 * − 1 , the situation is more complicated, and the existence of solutions for (1.2) is related to the position of the potential h with respect to the geometric potential

where Scal g is the scalar curvature of the manifold. Particularly, if h ( x ) ≡ h g , equation (1.2) is referred to as the Yamabe problem and it always has a solution (see, e.g., [1,32]).

The supercritical case p > 2 * − 1 is even more difficult to deal with. Using the Lyapunov-Schmidt reduction, Micheletti et al. [28] first constructed a single blowing-up solution for (1.2) in asymptotically critical case (i.e., p = 2 * − 1 ± ε with ε > 0 small enough). Here, a family of solutions u ε to (1.2) is said to blow up at ξ 0 ∈ M if there exists a family of points ξ ε ∈ M such that ξ ε → ξ 0 and u ε ( ξ ε ) → + ∞ as ε → 0 . Since then, equation (1.2) has been studied extensively. For instance, sign-changing blowing-up solutions were studied in [11,30], multiple blowing-up solutions in [10], clustered solutions in [5], and sign-changing bubble tower solutions in [29], among others. Of particular interest is the result by Ghimenti et al. [14], who established the existence of a single blowing-up solution concentrated along a minimal submanifold.

If M is either a smooth bounded domain or R n , and h ( x ) = 0 , system (1.1) reduces to the following elliptic system:

called the Lane-Emden system, where n ≥ 3 . In this case, the critical hyperbola

plays a similar role to the Sobolev exponent 2 * for the single equation. System (1.4) has received remarkable attention for decades. When Ω = R n , by applying the concentration compactness principle, Lions [25] found a positive least energy solution of (1.4)–(1.5), which is radially symmetric and radially decreasing. Wang [31] and Hulshof and Van der Vorst [20] independently proved the uniqueness of the positive least energy solution ( U 1,0 ( z ) , V 1,0 ( z ) ) to (1.4)–(1.5). Moreover, Frank et al. [13] established the non-degeneracy of (1.4)–(1.5) at each least energy solution. Using the Lyapunov-Schmidt reduction and the non-degeneracy result obtained in [13], Guo et al. [17] established the existence and non-degeneracy of multiple blowing-up solutions to (1.4)–(1.5) with two potentials. For more investigations about system (1.4) with Ω = R n , we can see [8,15].

If Ω is a smooth bounded domain, Kim and Pistoia [23] first built multiple blowing-up solutions for the following system:

where n ≥ 8 , α , β 1 , β 2 ∈ R , ε > 0 is a small parameter, 1 < p < n − 1 n − 2 , and ( p , q ) satisfies (1.5). Meanwhile, they also found solutions to the following asymptotically critical system:

where n ≥ 4 , α , β > 0 , ε > 0 is a small parameter, 1 < p < n − 1 n − 2 , and ( p , q ) satisfies (1.5). Moreover, using the local Pohozaev identity, Guo et al. [16] proved its non-degeneracy. Recently, Jin and Kim [21] studied the Coron’s problem for the critical Lane-Emden system and established the existence, qualitative properties of positive solutions. Guo and Peng [19] considered sign-changing solutions to the slightly supercritical Lane-Emden system with Neumann boundary conditions. It is worth emphasizing that Guo et al. [18] obtained positive solutions with boundary layers concentrating along one or several submanifolds for the asymptotically critical Lane-Emden system. For more classical results regarding Hamiltonian systems in bounded domains, we refer the readers to [4,22] and references therein.

Now, for any integer 0 ≤ m ≤ n − 1 , let

be the ( m + 1 ) st critical exponent. We remark that p n , m * = p n − m , 0 * is nothing but the critical exponent for the Lane-Emden system in dimension n − m . In particular, note that a point is a zero-dimensional manifold, it is natural to ask that does problem (1.1) have solutions blowing up and concentrating at a m-dimensional ( 1 ≤ m ≤ n − 1 ) submanifold of M when 1 p + 1 + 1 q + 1 → p n , m * ? In this case, it is easy to find that problem (1.1) is supercritical ( p n , m * > p n , 0 * ), and a major difficulty arises due to the lack of the Sobolev embedding, which makes variational methods and so on hard to apply. However, if the problem exhibits some symmetry or if the domain can be written as a product space, one can work in a lower-dimensional setting. This allows the avoidance of super-criticality issues in the ambient dimension. Therefore, in this article, we consider the case where ( M , g ) is a warped product manifold.

We recall the notion of warped product manifold introduced by Bishop and O’Neill in [3]. Let ( ℳ , g ) and ( K , κ ) be two Riemannian manifolds of dimensions N and m , respectively. Let ω ∈ C 2 ( ℳ ) , ω > 0 . The warped product M ≔ ℳ × ω K is the n ≔ N + m -dimensional product manifold ℳ × K furnished with metric g ≔ g + ω 2 κ . The function ω is called a warping function. If u ∈ C 2 ( M ) , it holds

Assume that h is invariant with respect to K , i.e., h ( x , y ) = h ( x ) for any ( x , y ) ∈ ℳ × K . We look for solutions of (1.1), which are invariant with respect to K , i.e., ( u ( x , y ) , v ( x , y ) ) = ( u 1 ( x ) , v 1 ( x ) ) , then by (1.6), we know ( u , v ) solves (1.1) if and only if ( u 1 , v 1 ) solves

or equivalently,

It is clear that if ( u 1 , v 1 ) is a pair of solutions to (1.7), which blows up and concentrates at ξ 0 ∈ ℳ , then ( u ( x , y ) , v ( x , y ) ) = ( u 1 ( x ) , v 1 ( x ) ) is a pair of solutions to (1.1), which blows up and concentrates along the fiber { ξ 0 } × K , which is an m -dimensional submanifold of M . Moreover, we can see that { ξ 0 } × K is a minimal submanifold of M if ξ 0 is a critical point of ω .

Moreover, to guarantee that the associated energy functional is well defined, we further require that 1 p + 1 + 1 q + 1 → ( p N , 0 * ) + . Therefore, we are led to study the following problem:

where a ∈ C 2 ( ℳ ) , min x ∈ ℳ a ( x ) > 0 , h ∈ C 1 ( ℳ ) , α , β > 0 are the real numbers, ε > 0 is a small parameter, and ( p , q ) ∈ ( 1 , + ∞ ) × ( 1 , + ∞ ) is a pair of numbers satisfying

Without loss of generality, we assume that 1 < p ≤ N + 2 N − 2 ≤ q .

To state our main result, we give the following definition.

Let us recall ( U 1,0 ( z ) , V 1,0 ( z ) ) , which is the least energy solution of (1.4)–(1.5) in R N given by Wang [31] and Hulshof and Van der Vorst [20]. Let L 1 , L 2 , … , L 5 be the positive numbers defined by

Our main result states as follows.

In particular, Theorem 1.1 applies to the case a = ω N , where ω is the warping function. For any j = 1 , 2 , … , k , let Γ j ≔ { ξ j 0 } × K , and

If ξ j 0 is a critical point of ω ( x ) , then Γ 1 × Γ 2 × ⋯ × Γ k is a m -dimensional minimal submanifold of M k . Moreover, by (1.6) and Theorem 1.1, we immediately have the following result.

The proof of our result relies on the well-known finite-dimensional Lyapunov-Schmidt reduction method, originally introduced in [2,12]. This article is structured as follows: In Section 2, we introduce the geometric framework and present some key preliminary results. The proof of Theorem 1.1 is provided in Section 3. In Section 4, we carry out the finite-dimensional reduction, and Section 5 is dedicated to analyzing the reduced problem. Throughout this article, C , C i ( i ∈ N + ) denote the positive constants that may vary from line to line.

2 Framework and preliminary results

We start with some properties of the least energy solution ( U 1,0 ( z ) , V 1,0 ( z ) ) of (1.4)–(1.5) in R N , which is given by Wang [31] and Hulshof and Van der Vorst [20].

Moreover, we have the following elementary inequality.

Now, we recall some definitions and results about the compact Riemannian manifold ( ℳ , g ) .

Fix such r 0 in this article with r 0 < min { i g , min j ≠ m { d g ( ξ j 0 , ξ m 0 ) } } , where i g denotes the injectivity radius of ( ℳ , g ) . For any 1 < s < + ∞ and u ∈ L s ( ℳ ) , we denote the L s -norm of u by

where d v g = det g d z is the volume element on ℳ associated with the metric g . We introduce the Banach space

equipped with the norm

where

Denote by ℐ * the formal adjoint operator of the embedding ℐ : X q , p ( ℳ ) ↪ L p + 1 ( ℳ ) × L q + 1 ( ℳ ) . By the Calderón-Zygmund estimate, the operator ℐ * maps L p + 1 p ( ℳ ) × L q + 1 q ( ℳ ) to X p , q ( ℳ ) . Then, we rewrite problem (1.8) as

where f ε ( v ) ≔ v p − α ε , g ε ( u ) ≔ u q − β ε . Moreover, by the Sobolev embedding theorem, we have

Let χ be a smooth cutoff function such that 0 ≤ χ ≤ 1 in R N , χ ( z ) = 1 if z ∈ B ( 0 , r 0 2 ) , and χ ( z ) = 0 if z ∈ R N \ B ( 0 , r 0 ) . For any ξ ∈ ℳ , δ > 0 , and η ∈ R N , we define the following functions on ℳ :

and