Non degeneracy for solutions of singularly perturbed nonlinear elliptic problems on symmetric Riemannian manifolds

Given a symmetric Riemannian manifold (M, g), we show some results of genericity for non degenerate sign changing solutions of singularly perturbed nonlinear elliptic problems with respect to the parameters: the positive number {\epsilon} and the symmetric metric g. Using these results we obtain a lower bound on the number of non degenerate solutions which change sign exactly once.


Introduction
Let (M, g) be a smooth connected compact Riemannian manifold of finite dimension n ≥ 2 embedded in R N . Le us consider the problem −ε 2 ∆ g u + u = |u| p−2 u in M u ∈ H 1 g (M ) Recently there have been some results on the influence of the topology (see [3,12,23]) and the geometry (see [5,7,16]) of M on the number of positive solutions of problem (1). This problem has similar features with the Neumann problem on a flat domain, which has been largely studied in literature (see [6,8,10,11,13,18,19,24,25,26]).
Concerning the sign changing solution the first result is contained in [15] where it is showed the existence of a solution with one positive peak and one negative peak when the scalar curvature of (M, g) is non constant.
Moreover in [9] the authors give a multiplicity result for solutions which change sign exactly once when the Riemannian manifold is symmetric with respect to an orthogonal involution τ using the equivariant Ljusternik Schnirelmann category.
In this paper we are interested in studying the non degeneracy of changing sign solutions when the Riemannian manifold (M, g) is symmetric.
We consider the problem where τ : R N → R N is an orthogonal linear transformation such that τ = Id, τ 2 = Id (Id being the identity on R N ). Here the compact connected Riemannian manifold (M, g) of dimension n ≥ 2 is a regular submanifold of R N invariant with respect to τ . Let M τ = {x ∈ M : τ x = x}. In the case M τ = ∅ we assume that M τ is a regular submanifold of M . In the following H τ g = u ∈ H 1 g (M ) : τ * u = u where the linear operator τ * : We obtain the following genericity results about the non degeneracy of changing sign solutions of (2) with respect to the parameters: the positive number ε, and the symmetric metric g (i.e. g(τ x) = g(x)). Theorem 1. Given g 0 ∈ M k , the set D = (ε, h) ∈ (0, 1) × B ρ : any u ∈ H τ g0 solution of −ε 2 ∆ g0+h u + u = |u| p−2 u is not degenerate is a residual subset of (0, 1) × B ρ . Remark 2. By the previous result we prove that, given g 0 ∈ M k and ε 0 > 0, the set D * = h ∈ B ρ : any u ∈ H τ g0 solution of −ε 2 ∆ g0+h u + u = |u| p−2 u is not degenerate is a residual subset of B ρ .
In the following we set Theorem 3. Given g 0 ∈ M k and ε 0 > 0. If there exists µ > m τ ε0,g0 which is not a critical level of the functional J τ ε0,g0 , then the set Here the set B ρ is the ball centered at 0 with radius ρ in the space S k , where ρ is small enough and S k is the Banach space of all C k , k ≥ 3, symmetric These results can be applied to obtain a lower bound for the number of non degenerate solutions of (2) which change sign exactly once when M is invariant with respect to the involution τ = − Id and 0 / ∈ M . We get the following propositions.
Proposition 5. Given g 0 ∈ M k and ε 0 > 0, if there exists µ > m τ ε0,g0 not a critical value of J ε0,g0 in H τ g0 , then the set is an open dense subset of B ρ .
The paper is organized as follows. In Section 2 we recall some preliminary results. In Section 3 we sketch the proof of the results of genericity (theorems 1 and 3) using some technical lemmas proved in Section 4. In Section 5 we prove propositions 4 and 5.

Preliminaries
Given a connected n dimensional C ∞ compact manifold M without boundary endowed with a Riemannian metric g, we define the functional spaces L p g , L p ε,g , H 1 g and H 1 ε,g , for 2 ≤ p < 2 * and a given ε ∈ (0, 1). The inner products on L 2 g and H 1 g are, respectively while the inner products on L 2 ε,g and H 1 ε,g are, respectively Finally, the norms in L p g and L p ε,g are We define also the space of symmetric L p and H 1 functions as As defined in the introduction, S k is the space of all C k symmetric covariants 2-tensor h(x) on M such that h(x) = h(τ x) for x ∈ M . We define a norm · k in S k in the following way. We fix a finite covering is an open coordinate neighborhood. If h ∈ S k , denoting h ij the components of h with respect to local coordinates (x 1 , . . . , x n ) on V α , we define The set M k of all C k Riemannian metrics g on M such that g(x) = g(τ x) is an open set of S k . Given g 0 ∈ M k a symmetric Riemannian metric on M , we notice that there exists ρ > 0 (which does not depend on ε if 0 < ε < 1) such that, if h ∈ B ρ the sets H 1 ε,g0+h and H 1 ε,g0 are the same and the two norms · H 1 ε,g 0 +h and · H 1 ε,g 0 are equivalent. The same for L 2 ε,g0+h and L 2 ε,g0 . If h ∈ B ρ and ε ∈ (0, 1) we set We introduce the map A ε h which will be used in the following section. Remark 6. If h ∈ B ρ and 0 < ε < 1, there exists a unique linear operator is the adjoint of the compact embedding i ε,g0 : H τ ε,g0 (M ) → L p ′ τ ε,g0 (M ) with 2 ≤ p < 2 * . We recall that, if h ∈ B ρ with ρ small enough and ε > 0, then H 1 ε,g0 and H 1 ε,g0+h (as well as L p ε,g0 and L p ε,g0+h ) are the same as sets and the norms are equivalent. This is the reason why we can define A ε h on L p ′ ,τ g0 with values in H τ g0 . We summarize some technical results contained in lemmas 2.1, 2.2 and 2.3 of [14].
Lemma 7. Let g 0 ∈ M k and ρ small enough. We have We recall two abstract results in transversality theory (see [20,21,22]) which will be fundamental for our results.
is a dense open subset of V Theorem 9. If F satisfies (i) and (ii) and (iv) The map π • i is σ-proper, that is F −1 (0) = ∪ +∞ s=1 C s where C s is a closed set and the restriction π • i |Cs is proper for any s then the set θ is a residual subset of V 3 Sketch of the proof of theorems 1 and 3.
. By the regularity of the map A (see 3 of Lemma 7) we get the map F is of class C 1 . We are going to apply transversality Theorem 8 to the map F , in order to prove Theorem 1. In this case we have Assumptions (i) and (iv) are verified in Lemma 10 and in Lemma 11. Using Lemma 12 we can verify (ii).

Indeed, we have to verify that for
We recall that the operator and is a Fredholm operator of index 0. Then By Lemma 12 we get that the linear functionals f i are independent. Therefore assumption (ii) is verified. At this point by transversality theorems we get that the set On the other hand we observe that 0 is a non degenerate solution of −ε 2 ∆ g0+h u + u = |u| p−2 u, for any ε > 0 and any h ∈ B ρ . Then, we complete the proof of Theorem 1.
The proof of Remark 2 is analog to the proof of Theorem 1 using Corollary 13.
We now formulate the problem for Theorem 3. Given g 0 ∈ M k and ε 0 > 0, we assume that there exists µ > m τ ε0,g0 which is not a critical level for the functional J ε0,g0 . It is clear that any µ 0 ∈ (0, m τ ε0,g0 ) is not a critical value of J ε0,g0 . We set We are going to apply transversality theorem 9 to the map H. In this case It is easy to verify assumptions (i) and (ii) for the map H using Lemma 10, Lemma 12 and Corollary 13. Using Lemma 14 we can verify assumption (iii) so we are in position to apply Theorem 9 and to get the following statement: the set is an open dense subset of B ρ . Nevertheless 0 is a non degenerate solution of −ε 2 0 ∆ g0+h u + u = |u| p−2 u for any h, and there is no solution u ≡ 0 with J ε0,g0 (u) < µ 0 , so we get the claim.

Technical lemmas
In this section we show some lemmas in order to complete the proof of the results of genericity of non degenerate critical points.
is a Fredholm map of index zero.
Proof. By the definition of the map A, we have We will verify that K : H τ ε0,g0 → H τ ε0,g0 is compact. Thus K : H τ g0 → H τ g0 is compact and the claim follows. In fact, in v n is bounded in H τ g0 , v n is also bounded in H τ ε0,g0+h0 because h 0 ∈ B ρ . Then, up to subsequence, v n converges to v in L t ε0,g0+h0 for 2 ≤ t < 2 * . So we havê We had to prove that π • i : , and F (ε n , h n , u n ) = 0, then, up to a subsequence, the sequence {u n } converges to u 0 ∈ I g0 (0, s) I g0 0, 1 s .

Lemma 12. For any
Proof.
Step 3. Conclusion of the proof.

By
Step 2, we have that, for any h ∈ S k Here g = g 0 + h 0 . Moreover it holds We choose h(ξ) = α(ξ)g(ξ) for any α ∈ C ∞ (M ) with α(τ ξ) = α(ξ), so, by (12), the function u 0 1 − |u 0 | p−2 w is antisymmetric with respect to the involution τ . Furthermore u 0 1 − |u 0 | p−2 w is also symmetric, so By contradiction we assume that w does not vanish indentically in M . Since By (14) we have that w ≡ 0 on the open subset M + ∪ τ M + . Also, we notice that M 0 and M 1 are disjoint sets because u 0 is a continuous funcion. By this, and by (13), we have that −ε 0 ∆ g w + w = 0 on M 0 and w = 0 on ∂M 0 . By the maximum principle, we conclude that w = 0 on M 0 . So we have that, by (13), −ε 0 ∆ g w + w = (p − 1)w on the whole M .
On the other hand, by [1], we have that µ g ({x ∈ M : w(x) = 0}) = 0. A contradiction arises and that concludes the proof With the same argument we can prove the following corollary.
Lemma 14. Given g 0 ∈ M k and ε 0 , if there exists a number µ > m ε0,g0 not a critical level of the functional J ε0,g0 , then, for ρ small enough, the map Proof. Let {u n } ⊂ D, where D = u ∈ H τ g0 : µ 0 < J ε0,g0 (u) < µ , and µ 0 is an arbitrary number in (0, m τ ε0,g0 ). It is sufficient to prove that if u n satifisfies −ε 2 0 ∆ g0+hn u n + u n = |u n | p−2 u n with h n → h 0 ∈ B ρ , then the sequence {u n } has a subsequence convergent in D. First we show that {u n } is bounded in H τ g0 . Since the sets H 1 g0+h (M ) and H 1 g0 (M ) are the same in h ∈ B ρ and the norms · H 1 sign exactly once. This estimate is formulated also in [17]. In the cited paper this result is proved under an assumption on non degeneracy of critical points that we do not need. We sketch the proof of propositions 4 and 5 showing how we use the results of genericity for non degeneracy of critical points to obtain the same estimate.
We recall that there exists a unique positive spherically symmetric function U ∈ H 1 (R n ) such that \begin −∆U + U = U p−1 in R n . Also, it is well known that for any ε > 0, U ε (x) := U x ε is a solution of −ε 2 ∆U ε + U ε = U p−1 ε in R n . Let g 0 be in M k and h be in B ρ for some ρ > 0. Let us define a smooth cut off real function χ R such that χ R (t) = 1 if 0 ≤ t ≤ R/2, χ R (t) = 0 if t ≥ R and |χ ′ (t)| < 2/R. Fixed q ∈ M and ε > 0 we define on M the function where B g (q, R) is the geodesic ball of radius R centered at q. We choose R smaller than the injectivity radius of M and such that B g (q, R)∩B g (−q, R) = ∅.
Lemma 15. For any δ > 0 there exists ε 2 = ε 2 (δ) such that, if ε < ε 2 theñ Moreover we have that For a proof of this result we refer to [3]. For any function u ∈ N τ ε,g0+h we define Lemma 16. There existsδ such that ∀δ <δ there existsε =ε(δ) and for any ε <ε the map is continuous and homotopic to identity, for all For a proof of this result we refer to [3]. Let us sketch the proof of Proposition 4. We are going to find an estimate on the number of pairs non degenerate critical points (u, −u) for the functional J ε,g : H τ g → R with energy close to 2m ∞ with respect to the parameters (ε, h) ∈ (0,ε) × B ρ forε, ρ small enough.
Let N τ ε /Z 2 be the set obtained by identifying antipodal points of the Nehari manifold N τ ε . It is easy to check that the set N τ ε /Z 2 is homeomorphic to the projective space P ∞ = Σ/Z 2 obtained by identifying antipodal points in Σ, Σ being the unit sphere in H τ g . We are looking for pairs of nontrivial critical points (u, −u) of the functional J ε : H τ g → R, that is searching for critical points of the functionalJ ε,g : H τ g {0} /Z 2 → R; J ε,g ([u]) := J ε,g (u) = J ε,g (−u).
By Lemma 15 and Lemma 16 we have thatβ g •Φ ε,g : M/G → M d /G is a map homotopic to the identity of M/G and that M d /G is homotopic to M/G. Therefore we have were Z(t) is a polynomial with non negative coefficients. Since the functional J ε,g satisfies the Palais Smale condition by the compactness of M , and the critical points of J ε,g in J 3m∞ ε,g are non degenerate (because (ε, h) ∈ D(ε, ρ)), by Morse Theory and relations (19) and (20) we get at least P 1 (M/G) pairs (u, −u) of non trivial solutions of −ε 2 ∆ g u + u = |u| p−2 u with J ε,g (u) = J ε,g (−u) < 3m ∞ . So, these solutions change sign exactly once. That concludes the proof of Proposition 4.
Remark 17. In the same way we obtain that, given g 0 ∈ M k and ε 0 > 0, the set The proof of Proposition 5 can be obtained with similar arguments.