Qualitative properties of stationary solutions of the NLS on the Hyperbolic space without and with external potentials

In this paper, we prove some qualitative properties of stationary solutions of the NLS on the Hyperbolic space. First, we prove a variational characterization of the ground state and give a complete characterization of the spectrum of the linearized operator around the ground state. Then we prove some rigidity theorems and necessary conditions for the existence of solutions in weighted spaces. Finally, we add a slowly varying potential to the homogeneous equation and prove the existence of non-trivial solutions concentrating on the critical points of a reduced functional. The results are the natural counterparts of the corresponding theorems on the Euclidean space. We produce also the natural virial identity on the Hyperbolic space for the complete evolution, which however requires the introduction of a weighted energy, which is not conserved and so does not lead directly to finite time blow-up as in the Euclidean case.


Introduction. Consider the following Nonlinear Schrödinger Equation on the Hyperbolic space (H d -NLS)
Here ∆ H d is the Laplace-Beltrami operator on the Hyperbolic space, d ≥ 2 is the spatial dimension, the nonlinearity has H 1 -subcritical exponent (1 < p < 1 + 4 d−2 for d ≥ 3 and 1 < p < +∞ for d = 2), the perturbation parameter is such that ≥ 0, the external potential h is bounded h ∈ L ∞ (H d ) and the wave function is defined by φ : R × H d → C and satisfies (1).
The study of the homogeneous ( = 0) nonlinear Schrödinger equation on the Hyperbolic space has received increasing interest in the recent years. People have tried to extend the theorems true in the flat space to the case of negatively curved space and results on local existence, blow-up, scattering and properties of standing waves have been proved. We refer to [6,4,5,7,9] and the references therein for an extended list of results on those important issues.
The focus of this paper is on the qualitative properties of standing waves, namely solutions of the form φ(t, x) = e iλt u(x), with u : H d → C being a solution of These solutions represent particles at rest and are usually present in the case of focusing nonlinearity, as it is our case. They play an important role for the dynamics as well, in particular for the dichotomy between scattering and blow-up of solutions. Equation (2) is variational and it is the Euler-Lagrange equation of the associated energy functional with dx the natural measure on H d . Therefore, the natural space in which this functional is well defined is H 1 (H d ), endowed with the norm As noted in [11], the lower bound on the spectrum , the norm · λ defined by solutions of equation (2) can be found by means of critical point theory. Least energy solutions are commonly called ground states and they minimize the energy, namely they satisfy where g := inf{E[u] : u is a non-trivial weak solution of (2)}. We denote by G λ the set of weak solutions to (2) satisfying (4).
The nondegeneracy of the ground state was originally proved by Ganguly and Sandeep [14], where the authors considered only real-valued functions, used the Poincaré disk model of H d and did not discuss the 0-Fredholm property of the linearized operator. These properties were proved by Selvitella in [15] using the polar model and extending the nondegeneracy to complex valued functions.
In this manuscript, we continue the study of qualitative properties of ground states of (1). Consider the case = 0. We first prove that the ground state is a mountain pass critical point of the energy functional E and that the Morse Index of the ground state is 1. Indeed, we have the following theorem. Theorem 1.1. Suppose = 0. We have the following properties.
• (Existence) There exists a mountain pass level c of E which is a critical level for E and a critical point u c of E such that u c > 0 and u c ∈ C 2 (H d , R). • (MP=GS) Consider the mountain pass critical level c, the Nehari critical level d (defined in Section 3 below) and the ground state level g. Then In particular, the mountain pass critical point u c is a ground state and any ground state is a mountain pass critical point. Then m(u c ) = 1.
Theorem 1.1 is the natural counterpart on the Hyperbolic space of the well known charactization of the ground state of the corresponding problem on the Euclidean space. See [2] for more details. Now, we pose ourselves in the weighted space Here (x 0 , x) = (cosh(r), sinh(r)ω) ∈ H d , with ω ∈ S d−1 . We have the following necessary conditions for existence of solutions to equation (2).
Theorem 1.2. Suppose that there exists a solution 0 = u ∈ H 1 (H d ) to equation (2) with = 0 such that Then, one of the following conditions need to be satisfied. For d ≥ 3, then: For d = 2, then: Remark 1. We should compare this theorem to the results of Mancini-Sandeep [11]. As proved in [11], there is no positive solution for λ < there is no positive solution in the energy space. So we do no treat those cases. For , the ground state u c does not satisfy the integrability conditions cosh 1/2 (r)u, cosh 1/2 (r)∇ H d u ∈ L 2 (H d ) and cosh 1/(p+1) (r)u ∈ L p+1 (H d ), so our result is not in contradiction with [11]. When λ > − (d−3)(d+1) 4 , the ground state is integrable and our result is in agreement with [11]. Nothing in [11] is said for what concerns the supercritical case p > 2 * − 1. Analogous argument for d = 2, where the ground state u c satisfies the integrability conditions only for λ > 3/4. Then, we consider the perturbed case, namely we take the parameter such that 0 < 1. The presence of the inhomogeneous term h(x) ≡ 0 breaks the symmetry of equation (2) and recovers compactness. Consider again the following problem: This type of problems is very well understood in the Euclidean setting (see for example Ambrosetti-Malchiodi [1]). We have the following theorem.
c (x)dx = 0 and smooth. Then problem (1) has a solution, provided is small enough.
We conclude the paper with a Virial Identity which formally mimics the one in the Euclidean case and some remarks on how that identity does not easily imply blow-up as in the Euclidean case. See Section 6 for more details.
The remaining part of this paper is organized as follows. In Section 2, we recall the polar model representation of the Hyperbolic space. In Section 3, we prove Theorem 1.1 on the variational characterization of the ground state. In Section 4, we prove Theorem 1.2 about necessary conditions for existence of solutions in weighted spaces. In Section 5, we prove Theorem 1.3 on concentration of solutions for the perturbed equation. In Section 6, we discuss the virial identity.
2. The polar model representation of the hyperbolic space. Recall the polar model representation of the Hyperbolic space: We endow H d with the Riemannian metric ds 2 := dr 2 + sinh 2 rdω 2 induced by the restriction to H d of the Lorentz metric ds 2 L = −x 2 0 + |x| 2 on R d+1 . The Laplace-Beltrami operator on the Hyperbolic space in polar coordinates can be rewritten as 3. Variational characterization of the ground state. In this section, we give the full proof of Theorem 1.1. We divide the argument into several steps. We recall the Embedding theorems, basic facts about the Mountain Pass theorem, the Nehari manifold and Palais' Symmetric Criticality principle. We collect these results to prove Theorem 1.1.

3.1.
Embeddings. In this subsection, we collect the Embedding theorems that we will use to prove Theorem 1.1. In Mancini-Sandeep [11], we can find the following Poincaré-Sobolev inequality.

QUALITATIVE PROPERTIES OF STATIONARY SOLUTIONS OF THE NLS ON H n 2667
As in the Euclidean space, the embedding for radial functions is compact, as shown in Bhakta-Sandeep [8].

E(γ(t)).
A sequence u n ∈ B is called Palais-Smale sequence on B, if E(u n ) is bounded and E (u n ) → 0, as n → +∞. If E(u n ) → c, as n → +∞, we will call u n ∈ B a Palais-Smale sequence at level c. We say that E satisfies the Palais-Smale condition at level c if every Palais-Smale sequence at level c has a converging subsequence. We have the celebrated Ambrosetti-Rabinowitz Theorem [3]. We know the Morse Index of a Mountain Pass critical point (see [2]). 3.3. The Nehari manifold. In this subsection, we construct the Nehari manifold.
Lemma 3.5 (Nehari Identity I). Suppose u ∈ H 1 (H d ) is a solution to (4), then u satisfies the following identity: Proof. Rewrite the equation using the polar model representation and multiply by u(r) sinh d−1 (r). Then integrate in θ ∈ S d−1 , r ∈ [0, +∞) by parts and use that

E(γ(t)).
We have the following So, there exists r > 0 as small as we want such that for v 2 This implies that E satisfies MP-1.
• MP-2 Take v ∈ H 1 r (H d ) not identically null. Take s > 0, we get: Then this γ is such that γ ∈ Γ.

QUALITATIVE PROPERTIES OF STATIONARY SOLUTIONS OF THE NLS ON H n 2669
If we take r < γ(1) H 1 r (H d ) ), then E satisfies MP-2. Since E satisfies both MP-1 and MP-2, then E has the mountain pass geometry. Now, consider a Palais-Smale sequence at level c, namely take u n ∈ H 1   (2) with = 0 such that cosh 1/2 (r)u, cosh 1/2 (r)∇ H d u ∈ L 2 (H d ) and cosh 1/(p+1) (r)u ∈ L p+1 (H d ). Then u satisfies the following identity: Proof. By density, we can assume that u ∈ C ∞ 0 (H d , R). Rewrite the equation using the polar model and multiply by u(r) cosh(r) sinh d−1 (r) and integrate in θ ∈ S d−1 , r ∈ [0, +∞). The result comes by integration by parts. Recall and integrate each term separately. .
Using E 1 + E 2 + E 3 = 0, we get: Then u satisfies the following identity: Proof. By density, we can assume that u ∈ C ∞ 0 (H d , R). Rewrite the equation using the polar model and multiply by u (r) sinh d (r) and integrate in θ ∈ S d−1 , r ∈ [0, +∞). The result comes by integration by parts. We use a similar decomposition as in the proof of the Nehari Identity II in Lemma 4.1.
Using E 1 + E 2 + E 3 = 0, we get: This concludes the proof of the lemma.
The result follows by joining the necessary conditions for the above identities to be satisfied.

5.
Concentration for the perturbed problem. In this section, we give a complete proof of Theorem 1.3. To do this, we first set our problem in the abstract setting of [1] and apply Theorem 2.16 from [1] (that we recall below). with 0 < 1, I 0 , G : H → R and H a Hilbert space. Suppose that I 0 has a smooth finite dimensional critical manifold Z (every z ∈ Z is a critical point of I 0 and dim(Z) < +∞). We have the following theorem.
Theorem 5.1 (Thm. 2.16 from [1]). Let I 0 , G ∈ C 2 (H, R) and suppose that I 0 has a smooth critical manifold Z which is non-degenerate. Letz ∈ Z be a strict local maximum or minimum of Γ := G| Z . Then for , such that | | 1, the functional I has a critical point u and ifz is isolated, then u →z as → 0.
Note that equation (2) with = 0 is invariant under phase shifts and hyperbolic rotations. Therefore, the orbit of the ground state u c of equation (2) with = 0 forms the manifold Z θ,d given by Z θ,d := e iθ L d u c (x), θ ∈ [0, 2π) and L d Q(x) an hyperbolic isometry of dimension d + 1. The definition of nondegeneracy for the ground state u c and so for the manifold Z θ,d is the following.