On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds

Given that(𝑀,𝑔)is a smooth compact and symmetric Riemannian 𝑛-manifold, 𝑛≥2, we prove a multiplicity result for antisymmetric sign changing solutions of the problem −𝜀2Δ𝑔𝑢


Introduction
Let M, g be a smooth compact connected Riemannian manifold without boundary of dimension n ≥ 2. Let us consider the problem where p > 2 if n 2, 2 < p < 2n/ n − 2 if n ≥ 3 and ε is a positive parameter.Here H 1 g M is the completion of C ∞ M with respect to

International Journal of Differential Equations
It is well known that any critical point of the energy functional J ε : H 1 g M → R constrained to the Nehari manifold N ε is a solution to 1.1 .Here 3 In 1 the authors show that the least energy solution of 1.1 , that is, the minimum of J ε on N ε is a positive solution with a spike layer, whose peak converges to the maximum point of the scalar curvature S g of M, g as ε goes to zero.Successively, in 2 see also 3, 4 the authors point out that the topology of the manifold M influences the multiplicity of positive solutions of 1.1 , that is, 1.1 has at least cat M nontrivial solutions provided that ε is small enough.Here cat M denotes the Lusternik-Schnirelman category of M. Recently, in 5-7 it has been proved that the existence of positive solutions is strongly related to the geometry of M, that is stable critical points of the scalar curvature S g generate positive solutions with one or more peaks as ε goes to zero.
As far as it concerns the existence of sign changing solutions to 1.1 , a few results are known.The first result has been obtained in 7 where it has been constructed solutions with one positive peak and one negative peak, which approach, as ε goes to zero, the minimum point and the maximum point of S g , provided the scalar curvature is not constant.In 8 the authors assume the following: In this paper we assume M satisfies S in the particular case τ −I.We look for solutions of the problem

1.5
We evaluate the number of solutions of problem 1.5 using Morse theory.Our main result reads as following.
Theorem 1.1.Assume that for ε small enough all the solutions to problem 1.5 with energy close to 2m ∞ are nondegenerate.Then there are at least P 1 M/G pairs u, −u of nontrivial solutions to 1.5 which change sign exactly once, where Here G {I, −I} and P 1 M/G is Poincaré polynomial P t M/g when t 1.

International Journal of Differential Equations 3
Concerning the assumptions of nondegeneracy of all the critical points with energy close to 2m ∞ , we think that it is true "generically" in some sense with respect to ε, g where ε is a positive parameter and g is a Riemannian metric.
We point out that problem 1.1 has been widely studied when the manifold M is replaced by an open bounded and smooth domain in R N with Dirichlet or Neumann boundary condition.In particular, it has been studied the effect of the domain topology or the domain geometry on the number of solutions.See, for example, 9-19 for the Dirichlet problem and 20-32 for the Neumann problem, The paper is organized as follows.In Section 2 we set the problem and we recall some known results; in Section 3 we give the proof of Theorem 1.1; in Section 4 we prove the technical Lemma 4.5, which is crucial for the proof of Theorem 1.1.

Setting of the Problem
First of all, we will recall some topological notions which are used in the paper.Definition 2.1 Poincaré polynomial .If X, Y is a couple of the topological spaces, the Poincaré polynomial P t X, Y is defined as the following power series in t: where H k X, Y is the kth homology group with coefficients in some fields.Moreover, we set If X is a compact manifold, we have that dimH k X < ∞ and in this case P t X is a polynomial and not a formal series.
Definition 2.2 Morse index .Let J be a C2-functional on a Banach space X and u ∈ X an isolated critical point of J with J u c.If J c : {v ∈ X : J v ≤ c} then the polynomial Morse index i t u of u is the following series: where , where μ u is the numerical Morse index of u and it is given by the dimension of the maximal subspace on which the bilinear form J u •, • is negatively definite.
It is useful to recall the following result see 33 .
Remark 2.3.Let X and Y be topological spaces.If f : X → Y and g : Y → X are continuous maps such that g • f is homotopic to the identity map on X then P t Y P t X Z t , where Z t is a polynomial with non negative coefficients.

International Journal of Differential Equations
Now, let us point out that the transformation τ −I : M → M induces a transformation on H 1 g M .We define the linear operator τ * as follows: The operator τ * is selfadjoint with respect to the following scalar product on H 1 g M , which is equivalent to the usual one: which induces the norm In particular, we have Here denotes the norm in L p M , which is equivalent to the usual one.Therefore, in virtue of the Palais Principle, the nontrivial solutions of 1.5 are the critical points of the restriction of J ε to the τ-invariant Nehari manifold where and let m ∞ be as in 1.6 .
It is easy to verify that J ε satisfies the Palais-Smale condition on N τ ε .Then, there exists v ε minimizer of m τ ε and v ε is a critical point of We recall that lim ε → 0 m ε m ∞ as it has been shown in 2, Remark 5.9 .
It is well known that there exists a unique positive spherically symmetric with respect to the origin function U ∈ H 1 R n minimizer of m ∞ .Obviously this fact implies that −ΔU U U p−1 in R n and for any ε > 0 we can define a family of functions U ε x : U x/ε satisfying the following equation On the tangent bundle of any compact connected Riemannian manifold M, it is defined the exponential map exp : TM → M which is a C ∞ -map.Then for ρ sufficiently small smaller than the injectivity radius of M the manifold M possesses a special set of charts given by exp x : B 0, ρ → B g x, ρ , where T x M is identified with R n for x ∈ M.Here B 0, ρ denotes the ball in R n centered at 0 with radius ρ and B g x, ρ denotes the ball in M centered at x with radius ρ with the distance given by the metric g.The system of coordinates corresponding to those charts are called normal coordinates.

The Main Ingredient of the Proof
Let us sketch the proof of our main result. Since 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 1 /Z 2 , which is obtained by identifying antipodal points in un unit sphere ∂Σ 1 in the space H τ g .
We are looking for pairs of nontrivial critical points u, −u if the functional J ε : H τ g → R, that is we are searching critical points for the functional J ε : H τ g \ {0}/Z 2 → R defined by J ε u : J ε u J ε −u .We use the same arguments as in 33 .The following relation can be proved as in 33, 34 see 33, Lemma 5.2 : where β • Φ ε is homotopic to the identity map and M d /G is homotopically equivalent to M g .Therefore by Remark 2.3 we get where Z t is a polynomial with nonnegative integer coefficients.

International Journal of Differential Equations
By our assumption we have that for ε small enough all the critical points u such that J ε u < 2 m ∞ δ are nondegenerate.Moreover the functional J ε satisfies the Palais-Smale condition.Then by Morse theory and relations 3.1 and 3.3 we get at least P 1 M/G pairs u, −u of nontrivial solutions for 1.5 .By Remark 4.7 these solutions change sign exactly once.That concludes the proof of Theorem 1.1.
Remark 3.1.By 33, Lemma 5.2 we deduce that Since P ∞ is homeomorphic to N τ ε /Z 2 we get P t N τ ε /Z 2 P t P ∞ .Provided the homology is evaluated with Z 2 -coefficients see, e.g., 35, Theorem 7.4 , we have P 1 P ∞ ∞.Then, if all the critical points are nondegenerate, we get infinitely many pairs u, −u of nontrivial solutions for 1.5 .

Technical Results
Let χ r be a smooth cut-off function such that

4.1
Fixing a point q ∈ M and ε > 0, let us define the function w ε,q on M as w ε,q x : U ε exp −1 q x χ r exp −1 q x if x ∈ B g q, r w ε,q x : 0 otherwise.

4.2
We choose r smaller than the injectivity radius of M and such that B g q, r ∩ B g −q, r ∅ for any q ∈ M. For any ε > 0 we can define a positive number t w ε,q such that Φ ε q : t w ε,q w ε,q ∈ H 1 g M ∩ N ε for any q ∈ M.

4.4
In 2, Proposition 4.2 the following lemma has been proved.
International Journal of Differential Equations 7 Now, fixing a point q ∈ M let us define the function Φ τ ε q : t w ε,q w ε,q − t w ε,τq w ε,τq .

4.5
It holds that By 4.4 and 4.6 , we deduce t w ε,q t w ε,τq .

4.7
The proof of the next results follows the same arguments as in 8 .

International Journal of Differential Equations
To get that , it is enough to prove that J ε Φ τ ε q 2J ε Φ ε q , because by Lemma 4.1 the statement will follow.Since the support of the function Φ τ ε q is B g q, r ∪ B g −q, r and B g q, r ∩ B g −q, r ∅, by 4.6 and the definition of the function Φ τ ε , we get Φ ε q p dμ g B g −q,r Φ ε τq p dμ g 2J ε Φ ε q .

4.13
That concludes the proof.

Lemma 4.3. One has that lim
Proof.By Lemma 4.2 and 4.12 we have that for any δ > 0 there exists ε 0 δ such that for any Since lim ε → 0 m ε 2m ∞ see 2, Remark 5.9 we get the claim.
For any function u ∈ N τ ε we can define a point β u ∈ R N by

4.16
Since J ε u ≤ 2 m ∞ δ , we have J ε u ≤ m ∞ δ and by 2, Proposition 5.10 we get the claim.
Proof.By Lemmas 4.2 and 4.4, I ε is well defined.In order to show that I ε is homotopic to the identity, we estimate the following difference: βΦ τ ε q − q M x − q Φ τ ε q p dμ g M Φ τ ε q p dμ g B 0,r y U y/ε χ r y p g q y 1/2 dy B 0,r U y/ε χ r y p g q y 1/2 dy ε B 0,r/ε z U z χ r |εz| p g q εz 1/2 dμ g B 0,r/ε U z χ r |εz| p g q εz 1/2 dμ g .

4.20
Hence |βΦ τ ε q − q|, |βΦ τ ε −q q| ≤ cε, because βΦ τ ε −q −βΦ τ ε q , for a constant c which does not depend on the point q.Therefore |I ε q − q| < cε; that concludes the proof.Remark 4.6.We have only to prove that any solution u of 1.5 such that J u < 2 m ∞ δ changes sign exactly once.In fact, assume that the set {u ∈ M : u x > 0} has h connected components M 1 , . . ., M h .Set u i x : u x if x ∈ M i ∪ −M i and u i x : 0 otherwise.We have u i ∈ N τ ε and

4.21
Then h 1.This concludes the proof.
with respect to τ, where τ : R N → R N is an orthogonal linear transformation such that τ / I and τ 2 I, I being the identity of R N .They prove problem 1.1 has at least G τ −cat M−M τ pairs of sign changing solutions which change sign exactly once.Here G τ −cat M−M τ denotes the G τ -equivariant Lusternik-Schnirelman category for the group G τ : {I, τ} and M τ : {x ∈ M : τx x}.