Existence, symmetry, and regularity of ground states of a nonlinear Choquard equation in the hyperbolic space

1 Introduction

In this study, we explore the following nonlinear Choquard problem in the hyperbolic space B N :

where Δ B N denotes the Laplace-Beltrami operator on B N , λ ≤ ( N − 1 ) 2 4 , 1 < p < 2 α * = N + α N − 2 , 0 < α < N , N ≥ 3 , and ℋ λ ( B N ) denotes the completion of C 0 ∞ ( B N ) with respect to the norm mentioned in (2.4). Moreover, ( − Δ B N ) − α 2 is the Green kernel of the fractional operator, ( N − 1 ) 2 4 is the bottom of the L 2 -spectrum of − Δ B N , and 2 α * is the critical exponent in the context of the Hardy-Littlewood-Sobolev (HLS) inequality.

The equation when posed in R N bears a notable resemblance to the Choquard equation, thoroughly explored by Moroz and Schaftingen [37]. The equation is given as follows:

where I α denotes the Riesz potential, α ∈ ( 0 , N ) , and p > 1 . The Riesz potential I α : R N → R is defined as

with Γ representing the Euler gamma function. To draw a connection between equation (1.1) and the one explored in this article, it is useful to view the Riesz potential as a representation of the inverse fractional Laplacian. In mathematical physics, the Newtonian potential of a function f ∈ S ( R N ) is defined as

where S ( R N ) denotes the Schwartz space of rapidly decreasing functions in R N . The convolution kernel 1 4 π N 2 Γ N − 2 2 1 ∣ x ∣ N − 2 appearing in I 2 ( f ) matches the fundamental solution

of − Δ where ω N − 1 is the surface area of the unit sphere in R N . Extending this idea, the generalization of the Newtonian potential indicates that for any function f ∈ S ( R N ) , in the distributional space S ′ ( R N ) , we have I α ( − Δ ) α ⁄ 2 f = ( − Δ ) α ⁄ 2 I α f = f , where ( − Δ ) α ⁄ 2 can be defined in several equivalent ways [11,25].

These partial differential equations (PDEs) are not just fascinating from a purely mathematical perspective but they also show up in various physical scenarios. For example, they appear in models like the Einstein-Klein-Gordon and Einstein-Dirac systems [20]. In practical applications like nonlinear optics and Bose-Einstein condensates, similar equations help describe how waves or particle densities evolve when both linear and nonlinear effects are at play. The Choquard equation is particularly interesting because it deals with nonlocal interactions, meaning the behavior at any point in the system is influenced by the entire field. Particularly, the following Choquard equation:

originally stems from the work of Frohlich and Pekar on polarons. They studied how free electrons in an ionic lattice interact with photons or the polarization they cause, essentially depicting an electron interacting with its own induced hole [14–16]. In 1976, Ph. Choquard adapted this model to explain a one-component plasma, showing its broader relevance in plasma physics. Interestingly, the Choquard equation is also known as the Schrödinger-Newton equation when it combines quantum mechanics with Newtonian gravity [24,35,41,42].

The Choquard equation in R N has been the focus of extensive research, particularly concerning the existence and qualitative properties of its solutions. Moroz and Schaftingen [37] established the existence of solutions to (1.1) by employing variational methods with the concentration-compactness lemma of P.L. Lions. The HLS inequality in R N [29] is heavily utilized in these techniques. Furthermore, in [38], they examined a more generalized form of the Choquard equation. In a subsequent study, Moroz and Schaftingen [40] considered the equation with a potential, establishing the existence of solutions for the lower critical exponent corresponding to the HLS inequality. Gao and Yang [19] also contributed by studying the Brezis-Nirenberg type Choquard equation within a bounded domain. For additional related works, refer [12,13,33,39,47–50].

Recent research has extensively aimed to generalize elliptic equations to non-Euclidean domains, driven by their relevance to various physical phenomena. This exploration seeks to understand how curvature impacts the behavior and properties of the solutions. We refer to [6,9,17,23,27] and references cited therein, without any claim of completeness. Notably, Sandeep and Mancini’s seminal work [34] demonstrated the existence and uniqueness of finite-energy positive solutions for the homogeneous elliptic equation

where λ ≤ ( N − 1 ) 2 4 , 1 < p ≤ N + 2 N − 2 if N ≥ 3 ; 1 < p < ∞ if N = 2 . They demonstrated that, for the subcritical case and p > 1 , equation (1.2) has a positive solution if and only if λ < ( N − 1 ) 2 4 , and that this solution is unique up to hyperbolic isometries, except potentially for N = 2 and λ > 2 ( p + 1 ) ( p + 3 ) 2 . Following their work, researchers of [6,17,18] examined the existence of sign-changing solutions, compactness, and nondegeneracy, while [5,7] focused on infinite energy solutions and their asymptotic behavior. Sandeep and Mancini also found that equation (1.2) emerges naturally when analyzing Euler-Lagrange equations related to Hardy-Sobolev-Maz’ya (HSM) inequalities. They derived a sharp Poincaré-Sobolev inequality (2.2) in hyperbolic space, which has spurred further interest in analogous HSM inequalities [36] involving higher-order derivatives [31,32]. Li et al. [27] have thereafter explored the existence, nonexistence, and symmetry of solutions for the higher-order Brezis-Nirenberg problem in hyperbolic space. This work relies on complex estimations of Green’s functions for fractional Laplacians, Helgason-Fourier analysis, and HLS inequality in the hyperbolic spaces. Specifically, the HLS inequality in hyperbolic space can be stated as follows: For 0 < λ < N and p = 2 N 2 N − λ , if f , g ∈ L p ( B N ) ,

where T y ( x ) denotes Möbius transformation, ρ denotes the hyperbolic distance (see Section 2), and

is the best HLS constant on R N . This constant is sharp, and there are no nonzero extremal functions. This result can be easily derived from the HLS inequality in R N through a conformal change in metric to the Euclidean ball [31]. Furthermore, this inequality parallels the HLS inequality in R N for the specific case when p = r = 2 N N − λ , as stated in Theorem 4.3 of [29]. The presence of sinh ρ 2 in the hyperbolic space context is attributed to the Green’s function associated with the conformal Laplacian − Δ B N − N ( N − 2 ) 4 [32]. Furthermore, Theorem 2.13 states a generalized form of the HLS inequality for any complete Riemannian manifold, under an additional assumption regarding the heat kernel.

In light of the aforementioned HLS inequality, He [23] examines an equation in the hyperbolic space that is analogous to the Choquard equation in R N . The study demonstrates the existence of a positive solution in the subcritical range, specifically for N + α N < p < N + α N − 2 under the condition λ ≤ ( N − 1 ) 2 4 , and also for p = N + α N − 2 when N ( N − 2 ) 4 < λ ≤ ( N − 1 ) 2 4 . Interestingly, the upper critical exponent N + α N − 2 resembles the critical Sobolev exponent 2 N N − 2 that appears in the Brezis-Nirenberg problem, while the exponent N + α N naturally emerges from the application of the HLS inequality.

Inspired by the aforementioned problems and the general HLS inequality (2.13) on a complete Riemannian manifold, we have investigated the problem ( F α , λ ) in this article. The derivation of this inequality relies on Stein’s maximal ergodic theorem. A significant challenge while analyzing problems like ( F α , λ ) is the loss of compactness, even in the subcritical range, as the embedding H 1 ( B N ) ↪ L p ( B N ) is noncompact for any 2 ≤ p ≤ 2 N N − 2 . This issue arises from the hyperbolic isometries in the infinite volume space B N . In Euclidean spaces, such challenges are typically addressed by dilating a given sequence, but this approach is ineffective in B N due to the equivalence of its conformal and isometry groups. To address this, we employ a concentration function approach near infinity. Another complexity arises from the nonlocal nature of the problem, driven by the Green kernel of the fractional operator. In addition to these, several technical challenges emerge in this context. These include the nonlinear nature of hyperbolic translations (as described in Section 2.1) and the complexity of the semigroup formula for the inverse fractional Laplacian, which involves the heat kernel in hyperbolic space. Unlike the Euclidean case, where the heat kernel is given by G t ( x ) = 1 ( 4 π t ) N ⁄ 2 e − ∣ x ∣ 2 ⁄ 4 t , x ∈ R N , t > 0 , the heat kernel in hyperbolic space is far more complex (refer Section 2.5). To the best of our knowledge, the problem described by ( F α , λ ) has not been studied in the literature to date.

The problem ( F α , λ ) can be analyzed using variational methods, thanks to the HLS inequality (2.14). To establish the existence of finite energy solutions, it is essential to examine the energy functional associated with (1.2), defined by

for N + α N ≤ p ≤ N + α N − 2 and the space ℋ λ ( B N ) defined in Section 2. We establish the following main existence result for p within the subcritical range.

To prove the above theorem, we demonstrate the existence of a minimizer for the energy functional on the associated Nehari manifold. This proof hinges on a concentration-compactness argument at infinity and the fact that the equation under consideration is invariant under the isometry group in hyperbolic space. This minimizer is referred to as a ground state. To establish the radial symmetry of the solutions, we utilize a symmetrization argument. This method is particularly effective in Choquard-type problems due to the presence of a Polya-Szego-like inequality, which implies a strong symmetrization effect on the nonlocal term. By using the minimality of the ground state, we derive a relationship between the function and its polarization, thereby establishing radial symmetry. Finally, we obtain certain regularity results through classical bootstrap arguments.

The study is structured as follows: Section 2 introduces key notations and geometric definitions, along with preliminary concepts related to the hyperbolic space. Section 3 presents the proof of Theorem 1.1. In Section 4, we state and prove the result concerning radial symmetry, while Section 5 addresses the regularity results. Numerous positive constants, whose specific values are irrelevant, are denoted by C .

2 Preliminaries

In this section, we will introduce some of the notations and definitions used in this study and also recall some of the embeddings related to the Sobolev space on the hyperbolic space.

We will denote by B N the disc model of the hyperbolic space, i.e., the Euclidean unit ball B ( 0 , 1 ) ≔ { x ∈ R N : ∣ x ∣ 2 < 1 } equipped with the Riemannian metric

where d x is the standard Euclidean metric and ∣ x ∣ 2 = ∑ i = 1 N x i 2 is the standard Euclidean length. The corresponding volume element is given by d V B N = ( 2 1 − ∣ x ∣ 2 ) N d x , where d x denotes the Lebesgue measure on B N . ∇ B N and Δ B N denote the gradient vector field and Laplace-Beltrami operator, respectively. Therefore, in terms of local (global) coordinates, ∇ B N and Δ B N take the form

where ∇ , Δ are the standard Euclidean gradient vector field and Laplace operator, respectively, and “ ⋅ ” denotes the standard inner product in R N .