Nondegeneracy of Positive solutions of a Semilinear elliptic problem in the Hyperbolic space

In this article we will study the nondegeneracy properties of positive finite energy solutions of a semilinear elliptic equation in the N-dimensional Hyperbolic space. We will show that the degeneracy occurs only in an N dimensional subspace. We will prove that the positive solutions are nondegenerate in the case of geodesic balls.


Introduction
In this article we will study the degeneracy properties of positive solutions of the problem and H 1 (B N ) denotes the Sobolev space on the disc model of the Hyperbolic space B N and ∆ B N denotes the Laplace Beltrami operator on B N . Apart from its own mathematical interest, this equation arises in the study of Grushin operators( [2]), Hardy-Sobolev-Mazy'a equations( [5], [9]) and in some fluid dynamics models ( [10]). This equation is also the hyperbolic version of the well known Brezis Nirenberg problem ( [4]).
Our main aim in this paper is to study the degeneracy properties of positive solutions obtained in [9]. It is well known that the degeneracy properties are crucial in studying the stability of the solution and also to obtain various existence results, bifurcation results, etc using tools like finite dimensional reduction. In the case of bounded domains in Euclidean space degeneracy of solution is used to study the uniqueness and other qualitative properties of solutions (see [12], [6], [14]).
For some special values of λ and p, the degeneracy of the solution has been studied in [5]. In this special case the solution of (1.1) is explicitly known and this plays a major role in the proof as one can use informations on the solution of a hyper geometric ODE arising in the analysis. However in general we do not have explicit expression for the solution of (1.1) and the proof presented in the special case in [5] does not carry over.
In view of the existence and non-existence results obtained in [9] we assume: Let U be the unique positive radial solution of (1.1) under the assumption (1.2). Consider the linearized operator L given by We are interested in knowing whether 0 is in the spectrum of L and more generally to find the kernel of L. Degeneracy properties of similar equations in the Euclidean space R N are well known. In fact if u is a solution then a differentiation of the equation will tell us that ∂u ∂xi is a solution of the linearized equation. We show that in the hyperbolic case a different vector field will do the job ∂ ∂xi does in R N . Observe that the equation is invariant under hyperbolic isometries, i.e, if u solves (1.1) and τ is an isometry of B N then u • τ also solves (1.1). This fact indicates that the solutions has to be degenerate. However we see that the degeneracy happens only along an N dimensional subspace.
.., N be the vector fields in B N given by Now consider the problem in geodesic balls. Let B be a geodesic ball in B N , consider the problem We know from [15] that this problem has a solution under the assumptions (1.2). Using moving plane method (see [1], [9] and the references therein) we know that any solution of (1.5) is radial and the uniqueness of radial solution has been established in [9]. Let us denote by u the unique positive radial solution of (1.5) andL the linearized operator In the case of Euclidean ball, the nondegeneracy of the unique positive solution of semilinear elliptic equations has been studied by various authors (see [6], [12] and the references therein ). In the hyperbolic case we prove :

Notations and Preliminaries
In this section we will introduce some of the notations and definitions used in this paper and also recall some of the embeddings related to the Sobolev space in the hyperbolic space.
We will denote by B N the disc model of the hyperbolic space, i.e., the unit disc equipped with The hyperbolic gradient ∇ B N and the hyperbolic Laplacian ∆ B N are given by Let us recall the following Sobolev embedding theorem (see [9] for a proof): Poincaré-Sobolev inequality: For every N ≥ 3 and every p ∈ (1, N +2 is not compact, however for radial functions, using the point wise estimate we have the following compactness lemma (see [3] for a proof): For a positive function V on B N define: then we have the following compact embedding in the general case: The first integral converges to zero as n → ∞ thanks to Rellich's compactness theorem. Using Poincaré-Sobolev inequality the second integral can be estimated as follows: We also need some information on the isometries of B N . Below we recall the definition of a special type of isometries namely the hyperbolic translations. For more details on the isometry group of B N we refer to [13].

Radial Nondegeneracy
In this section we prove the radial nondegeneracy of the linearized operators. First consider the case of (1.1): Similarly for (1.5) we have: We will prove Theorem 3.1, the proof of Theorem 3.2 follows similarly with obvious modifications. As in [16] we use the information on the Morse index of a perturbed problem to establish the result.
then there exists an ǫ > 0 such that u = U is the unique critical point of I ǫ , U is a mountain pass critical point and hence of radial Morse index one.
We postponed the proof of this theorem to Section 5. Proof of Theorem 3.1. Assume that there exists a Φ = 0 such that L(Φ) = 0. We will show that this will contradict the fact that U is a critical point of I ǫ with radial Morse index of one. Recall the Morse Index of U denoted i(I ǫ , U ) is defined as we get for real numbers α, β (Φ, Φ) < 0 Moreover U and Φ are linearly independent and hence the radial Morse index of U is at least 2, which is a contradiction.

Proof of Theorems
In this section we will prove our main theorems. Proof of Theorem 1. 1 We divide the proof in to several steps.
Step 1 Let Φ i be as define in Theorem 1.1, then L(Φ i ) = 0 for all i.
Differentiating with respect to t and evaluating at t = 0 we see that L(Φ i ) = 0 where From the estimates on U (established in [9]) we get Φ i ∈ H 1 (B N ), i = 1, . . . , N and are linearly independent.
then there exists a constant c ∈ R such that ψ = cΦ 1 . Proof of Step 2.We know from Lemma 2.2 that the embedding Thus by standard compact operator theory and maximum principle the weighted eigenvalue problem −∆ B N ψ − λψ = µU p−1 ψ in B N + , ψ ∈ H 1 0 (B N + ) has a discrete set of eigenvalues µ 1 < µ 2 < ... → ∞, with µ 1 is simple. Moreover any eigenfunction corresponding to µ i , i > 1 change sign. Since Φ 1 | B N + is an eigenfunction with µ = p and it has a constant sign (since U ′ < 0) we conclude that µ 1 = p and hence Step 2 follows from the simplicity of µ 1 = p.
Step 3. Let Φ ∈ H 1 (B N ) and satisfies L(Φ) = 0, then for any unit vector ν ∈ R N there exists a constant c ∈ R such that where R ν and O ν denote the reflection with respect to the hyperplane {(ξ, ν) = 0} and an orthogonal transformation satisfying O ν (ν) = e 1 respectively. Proof of Step 3. Let us first consider the case when ν = e 1 = (1, . . . , 0) ∈ R N then ψ(x) = Φ(x) − Φ(R e1 x) satisfies (4.3) and hence from Step 2 we get the existence of c ∈ R such that Similar argument will give for any ν unit vector ∈ R N and O ν denote the orthogonal transformation such that O ν ν = e 1 Φ(x) − Φ(R ν x) = cΦ 1 (O ν x) This completes the proof of Step 3.
Step 4. Let Φ be in the kernel of L, then there exist c 1 , . . . , c N ∈ R such that the function Φ r defined by

Proof of Step 4 We have from Step 3 , given any unit vector
We earlier remark that for the orthogonal transformation is even in x 2 and of course, in x 1 as well. By iterating, we conclude that, for some c 1 , . . . , c N , for some c. Since by Hopf lemma for all x and for all ν. Hence Φ r is radial.
Step 5. It follows from Theorem 3.1 that Φ r = 0 and hence Φ = N i=1 c i Φ i . This completes the proof of Theorem 1.1 Proof of Theorem 1.2. Thanks to Theorem 3.2, we only have to show that ifL(Φ) = 0 then Φ is radial.
Let Φ ∈ H 1 0 (B) be such thatL(Φ) = 0, then from standard elliptic theory we know that Φ is Since the first eigen value ofL is positive we get ψ = 0 and hence φ is symmetric in the x 1 direction. Since the equation is invariant under rotation we get Φ is symmetric in all the directions and hence radial. This completes the proof of Theorem 1.2.

Perturbed problems
In this section we will prove Theorem 3.3. Let U be the unique solution of (1.1), define for Observe that for any ǫ > 0, u = U itself is a solution of (5.2). Our aim in this section is to prove that U is the only positive radial solution of this equation for small ǫ and U is a mountain pass critical point. Let λ satisfy (1.2), then there exists an ǫ 0 > 0 such that for all 0 < ǫ < ǫ 0 , I ǫ on H 1 r (B N ) has a mountain pass critical point. Proof. The inequality (2.1) easily establishes that I ǫ has the mountain pass geometry. Also for 1 < p < N +2 N −2 I ǫ satisfies the Palais Smale condition thanks to Lemma (2.1). Thus the theorem follows in the subcritical case. In the critical case, in general I ǫ does not satisfy the P.S. condition. However it follows from Theorem 3.3 of [3] that I ǫ does not satisfy the Palais Smale condition at level β if 0 < β < S N 2 N where S is the best constant in the Euclidean Sobolev inequality given by Thus in the critical case it remains to find a function u 1 ∈ H 1 r (B N ) such that I ǫ (u 1 ) ≤ 0 and sup 0≤t≤1 I ǫ (tu 1 ) < S u(x) we get Proof. First note that any positive radial solution of (5.2) is radially decreasing. Denote the solution by u(|x|). Normalizing the constant on the RHS to one and writing the equation in hyperbolic polar co-ordinates, i.e., |x| = tanh t 2 we have to prove the uniqueness of solutions of the ODE where λ ǫ = λ − ǫU p−1 . The term on the LHS of (5.4) is the H 1 (B N ) norm of u. As mentioned before, when ǫ = 0 uniqueness of solutions of (5.3)-(5.4) has been established in [9]. For ǫ > 0 also the proof follows in the same lines as in [9] with some modifications. So we will only outline the proof, modifying the details which are likely to differ in the case of ǫ > 0, and refer to [9] for details.
The main ingredient in the uniqueness proof is the following auxiliary energy Eû (see [9], [11])defined by : Proceeding exactly as in the proof of Lemma 3.4 of [9] we get: Let N ≥ 2, p > 1, λ < N −1 Using the estimates in (5.5) we have the following estimates on the auxiliary energy : as t → 0, and Sinceû satisfies Next we claim: Claim. There exists ǫ 0 > 0 and t 1 > 0 such that for all 0 ≤ ǫ < ǫ 0 Proof of Claim : By direct calculation we have To analyze the sign of G ′ ǫ , first recall that we have estimates on U given in 5.5. In the first case in (5.8) we have both A < 0 and B(β−2) β < 0. Since U and U ′ are bounded we get for ǫ small enough G ′ ǫ (t) < 0 for all t. In the fourth case, β = 2 and A = λ − N (N −2) 4 > 0 so for ǫ small enough G ′ ǫ (t) > 0 for all t. In the third case β < 2, B < 2 and A ≥ 0, again the conclusion follows. It remains to establish the claim in the second case in (5.8). In this case we have A < 0 and β < 2, B < 2. Hence G ′ ǫ (t0 for large enough t and G ′ ǫ (t) > 0 for t close to 0. Thus it remains to show that From (5.5) and the equation U satisfies, we get This completes the proof of claim.
has at most one solution.
The proof follows exactly like the case of ǫ = 0 and so we refer to [9]. Using this uniqueness result of geodesic balls, the rest of the uniqueness proof follows exactly as in [9].
Proof of Theorem 3.3. From Theorem 5.1 and Theorem 5.2 we know that I ǫ has a unique critical point and it is a mountain pass critical point. On the other hand U is a critical point of I ǫ , and hence it is the unique critical point and of mountain pass type and hence the radial Morse index is one (See [8]