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International Journal of Differential Equations
Volume 2010 (2010), Article ID 432759, 11 pages
On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds
1Dipartimento di Matematica Applicata “U.Dini”, Università di Pisa, via F. Buonarroti 1/c, 56100 Pisa, Italy
2Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, via Antonio Scarpa 16, 00161 Roma, Italy
Received 1 October 2009; Accepted 7 December 2009
Academic Editor: Thomas Bartsch
Copyright © 2010 Anna Maria Micheletti and Angela Pistoia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Given thatis a smooth compact and symmetric Riemannian -manifold, , we prove a multiplicity result for antisymmetric sign changing solutions of the problem in . Here if and if .
Let be a smooth compact connected Riemannian manifold without boundary of dimension . Let us consider the problem where if if and is a positive parameter. Here is the completion of with respect to It is well known that any critical point of the energy functional constrained to the Nehari manifold is a solution to (1.1). Here In  the authors show that the least energy solution of (1.1), that is, the minimum of on is a positive solution with a spike layer, whose peak converges to the maximum point of the scalar curvature of as goes to zero. Successively, in  (see also [3, 4]) the authors point out that the topology of the manifold influences the multiplicity of positive solutions of (1.1), that is, (1.1) has at least nontrivial solutions provided that is small enough. Here denotes the Lusternik-Schnirelman category of Recently, in [5–7] it has been proved that the existence of positive solutions is strongly related to the geometry of , that is stable critical points of the scalar curvature generate positive solutions with one or more peaks as 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  where it has been constructed solutions with one positive peak and one negative peak, which approach, as goes to zero, the minimum point and the maximum point of , provided the scalar curvature is not constant. In  the authors assume the following:(S)the manifold is a regular submanifold of invariant with respect to , where is an orthogonal linear transformation such that and being the identity of
They prove problem (1.1) has at least pairs of sign changing solutions which change sign exactly once. Here denotes the -equivariant Lusternik-Schnirelman category for the group and
In this paper we assume satisfies () in the particular case We look for solutions of the problem 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 small enough all the solutions to problem (1.5) with energy close to are nondegenerate. Then there are at least pairs of nontrivial solutions to (1.5) which change sign exactly once, where Here and is the Poincaré polynomial when
Concerning the assumptions of nondegeneracy of all the critical points with energy close to , we think that it is true “generically” in some sense with respect to where is a positive parameter and is a Riemannian metric.
We point out that problem (1.1) has been widely studied when the manifold is replaced by an open bounded and smooth domain in 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.
2. 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 is a couple of the topological spaces, the Poincaré polynomial is defined as the following power series in : where is the th homology group with coefficients in some fields. Moreover, we set If is a compact manifold, we have that and in this case is a polynomial and not a formal series.
Definition 2.2 (Morse index). Let be a -functional on a Banach space and an isolated critical point of with If then the (polynomial) Morse index of is the following series: where is the th homology group of the couple If is a nondegenerate critical point of then where is the (numerical) Morse index of and it is given by the dimension of the maximal subspace on which the bilinear form is negatively definite.
It is useful to recall the following result (see ).
Remark 2.3. Let and be topological spaces. If and are continuous maps such that is homotopic to the identity map on then where is a polynomial with non negative coefficients.
Now, let us point out that the transformation induces a transformation on We define the linear operator as follows: The operator is selfadjoint with respect to the following scalar product on which is equivalent to the usual one: which induces the norm In particular, we have Here denotes the norm in 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 to the -invariant Nehari manifold where
In fact, since and is a selfadjoint operator, we have and so if
Let us set and let be as in (1.6).
It is easy to verify that satisfies the Palais-Smale condition on Then, there exists minimizer of and is a critical point of on Thus and belong to then We recall that as it has been shown in [2, Remark ].
It is well known that there exists a unique positive spherically symmetric (with respect to the origin) function minimizer of Obviously this fact implies that in and for any we can define a family of functions satisfying the following equation in .
On the tangent bundle of any compact connected Riemannian manifold , it is defined the exponential map which is a -map. Then for sufficiently small (smaller than the injectivity radius of ) the manifold possesses a special set of charts given by where is identified with for Here denotes the ball in centered at with radius and denotes the ball in centered at with radius with the distance given by the metric The system of coordinates corresponding to those charts are called normal coordinates.
3. The Main Ingredient of the Proof
Let us sketch the proof of our main result.
Since (see Lemma 4.3), given for small enough, we have Thus is not a critical value of for any Fixed if the number of critical points of is finite in we can choose such that is not a critical value of
Let be the set obtained by identifying antipodal points of the Nehari manifold It is easy to check that the set is homeomorphic to the projective space which is obtained by identifying antipodal points in un unit sphere in the space
We are looking for pairs of nontrivial critical points if the functional that is we are searching critical points for the functional defined by We use the same arguments as in . The following relation can be proved as in [33, 34] (see [33,Lemma ]):
By our assumption we have that for small enough all the critical points such that are nondegenerate. Moreover the functional satisfies the Palais-Smale condition. Then by Morse theory and relations (3.1) and (3.3) we get at least pairs 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 ] we deduce that Since is homeomorphic to we get Provided the homology is evaluated with -coefficients (see, e.g., [35, Theorem ]), we have Then, if all the critical points are nondegenerate, we get infinitely many pairs of nontrivial solutions for (1.5).
4. Technical Results
Let be a smooth cut-off function such that Fixing a point and , let us define the function on as We choose smaller than the injectivity radius of and such that for any For any we can define a positive number such that Namely, verifies In [2, Proposition ] the following lemma has been proved.
Lemma 4.1. Given the map is continuous. Moreover, given there exists such that if then
Lemma 4.2. Given the map is continuous. Moreover, given there exists such that if then
Proof. Since is a radially symmetric function, we set Moreover, since we have
because by (4.7) we have Hence
To get that it is enough to prove that because by Lemma 4.1 the statement will follow. Since the support of the function is and by (4.6) and the definition of the function , we get That concludes the proof.
Lemma 4.3. One has that
For any function we can define a point by
Lemma 4.4. There exists such that for any for any (as in Lemma 4.2), and for any function , it holds that where
Proof. Let Since we set and It is easy to see that Then we have Since we have and by [2, Proposition ] we get the claim.
Lemma 4.5. There exists such that for any the map is well defined, continuous, and homotopic to the identity map.
Proof. By Lemmas 4.2 and 4.4, is well defined. In order to show that is homotopic to the identity, we estimate the following difference: Hence because for a constant which does not depend on the point Therefore ; that concludes the proof.
Remark 4.6. We have only to prove that any solution of (1.5) such that changes sign exactly once. In fact, assume that the set has connected components Set if and otherwise. We have and Then This concludes the proof.
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