Topological arguments for an elliptic equation involving the fractional Laplacian
© Al-Ghamdi et al.; licensee Springer. 2014
Received: 21 November 2013
Accepted: 21 July 2014
Published: 26 September 2014
In this paper, we consider a fractional Nirenberg-type problem involving σ-exponent of the Laplacian on the standard n-dimensional spheres . Using an algebraic topological method and the theory of the critical points at infinity, we provide a variety of classes of functions that can be realized as the σ-curvature on n-dimensional sphere.
MSC: 35M30, 35Q74, 74A15, 74A60, 74M25.
1 Introduction and main results
where be the conformal Laplacian, is the Laplace-Beltrami operator, and is the scalar curvature associated to the metric g.
Recently, several studies have been performed for classical elliptic equations similar to (2) and (1) but with the fractional conformal Laplacians instead of the Laplacian. This operator is introduced first in  where Graham, Jenne, Mason, and Sparling constructed a sequence of conformally covariant elliptic operators , on Riemannian manifolds for all positive integers k if n is odd, and for if n is even. Moreover, is the well-known conformal Laplacian used in (2), and is the well-known Paneitz operator. Up to positive constants is the scalar curvature associated to g and is the Q-curvature. Prescribing scalar curvature and Q-curvature problems on was studied extensively in last years, see for example –; see also –, and the references therein.
In , Chang and Gonzalez generalize the above construction, using the works of Graham-Zworski  and Caffarelli-Silvestre  for factional Laplacian on the Euclidean space , to define conformally invariant operators of non-integer order σ. These lead naturally to a fractional-order curvature , which will be called here σ-curvature. For the σ-curvatures on general manifolds we refer to , , , and references therein. Corresponding to the Yamabe problem, fractional Yamabe problems for σ-curvatures are studied in , , and  and fractional Yamabe flows on are studied in .
where is the fractional Laplacian operator (see, e.g., p.117 of ).
A sufficient condition was found recently in  for the two dimensional sphere, it is an Euler-Hopf-type criterium, namely the authors proved the following existence result.
Let K be a positive function satisfying the following (nd) condition:
then (3) has at least one solution. Here, is the Morse index of K at y.
Let k be an index, . We say that if it satisfies the following:
Now, we are ready to state our first main result.
where S denotes the set of solutions of (3).
Observe that the index . It follows that the above mentioned result (Theorem 1.1) is a corollary of our theorem. Actually, we will give in Remark 3.1 a situation where Theorem 1.1 does not give a result, while by Theorem 1.2, we derive the existence of many solutions.
Second, we consider the case and K close to a constant, i.e. K of the from , where and small. In order to state our result, we need the following assumption. We say that an integer if it satisfies the following:
(): For any , we have .
We shall prove the following result.
then, forsufficiently small, there exists a solution to the problem (3).
Observe that . We therefore have the following corollary.
then forsufficiently small, (3) has at least one solution.
The rest of this paper is organized as follows: we first recall some known facts about the variational structure of problem (3) and the associated critical points at infinity. In Section 3, we give the proofs of our main results and in Section 4, we give a more general result for and .
2 Preliminary results
2.1 The variational structure
defined in .
Motivated by the work of Caffarelli and Silvestre , several authors have considered an equivalent definition of the operator by means of an auxiliary variable; see  (see also –, and ). In fact, we realize problem (3), through a localization method introduced by Caffarelli and Silvestre on the Euclidean space , through which (3) is connected to a degenerate elliptic differential equation in one dimension higher by a Dirichlet to Neumann map. This provides a good variational structure to the problem. By studying this problem with classical local techniques, we establish existence of positive solutions. Here the Sobolev trace embedding comes into play, and its critical exponent .
defined in .
defined on Σ the unit sphere of . We set, . Problem (3) will be reduced to finding the critical points of J subjected to the constraint . The exponent is critical for the Sobolev trace embedding . This embedding being continuous and not compact, the functional J does not satisfy the Palais-Smale condition, which leads to the failure of the standard critical point theory. This means that there exist sequences along which J is bounded, its gradient goes to zero and which do not converge. The analysis of sequences failing PS condition can be analyzed along the ideas introduced in  and . In order to describe such a characterization in our case, we need to introduce some notations.
where i is an isometry from to ; the completion of , with respect to the Dirichlet norm.
If U is a function in , one can find an optimal representation, following the ideas introduced in , namely we have:
where v belongs toand satisfies the following condition:
The failure of the Palais-Smale condition can be characterized taking into account the uniqueness result of the corresponding problem at infinity; see e.g. Li-Zhu  following the ideas introduced in ,  as follows.
Assume that (3) has no solution and letbe a sequence satisfying, a positive number and. Then there exist an integer, a positive sequence () and an extracted subsequence of, again denotedsuch that.
such that remains in for .
such a critical point at infinity.
The following propositions characterize the critical points at infinity of the associated variational problem.
Such a critical point at infinity has a Morse index equal to. Its level is, whereis the best constant of Sobolev.
See Corollary 4.4 of . □
Arguing as in , we have the following.
The Morse index ofis equal to. Its level is. Hereis the best constant of Sobolev.
2.2 The unstable manifolds of critical points at infinity
At the beginning of this subsection, we give some basic definitions with will allow us to describe the unstable manifolds of the critical points at infinity in .
It is convenient to specify that the notations of stable or unstable manifolds, of flow lines, all are relative to the vector field , with respect to the standard Riemannian structure on . Recall the following generic hypothesis:
All stable and unstable manifolds intersect transversely and all such intersections are smooth regularly embedded submanifolds of .
where λ is a fixed constant large enough.
Observe that correspond to , the stable manifold of the critical point y along the flow lines of . Therefore it is easy to see that if , then behaves as .
The following lemma gives a sufficient condition to ensure that is included in .
It follows from . The idea is that a flow line in cannot go out from unless the concentration point of the flow line nearby a critical point z of K with (see proof of Proposition 4.3 of ), therefore it is the case when the critical point y is dominated by . Hence under the condition of the lemma such a situation cannot occur, it follows that every flow line in is indeed in and we then conclude to the result of the lemma using Remark 2.9. □
3 Proof of results
This section is devoted to the proofs of the main results of this paper. Our proofs uses algebraic topological arguments and the tools of the theory of critical points at infinity; see . In our case, the space of variation is contractible and has no topology. However, due to the non-compactness of the problem, there are critical points at infinity whose topological contribution can be computed; see . The main idea is to use the difference of topology of the critical points at infinity between the level sets of the associated Euler-Lagrange functional J, and the main issue is under our conditions on K, there remains some difference of topology which is not due to the critical points at infinity but due to the existence of solution of (3).
3.1 Proof of Theorem 1.2
Since satisfies (), we can find satisfying the following:
where λ is a fixed constant large enough and is the best constant of Sobolev.
where w is a solution of (3) dominated by . Here ≃ denotes retracts by deformation.
Such an equality contradicts the assumption of Theorem 1.2.
where S denotes the set of solutions of (3).
so by Theorem 1.2 we derive the existence of a solution of problem (3).
3.2 Proof of Theorem 1.3
We have the following lemma.
where is independent of u, tends to zero when ε tends to zero. Hence the lemma follows. □
Indeed, from (23), it is sufficient to prove that contractible set.
We drive then that is contractible since Z is a contractible set. Hence our claim follows.
where , h is continuous and a fixed point in .
Such equality contradicts the assumption of Theorem 1.3. This completes the proof of Theorem 1.3.
4 A general existence result
, Γ is the gamma function and is the Laplace-Beltrami operator on . We are now ready to state the following existence result.
where S denotes the set of solutions of (28).
The Morse index of such critical point at infinity is: . Now the remainder of the proof is identical to the proof of Theorem 1.2. □
We express our gratitude to the referees for their valuable criticisms of the manuscript and for helpful suggestions.
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