On a fourth-order elliptic equation involving the critical Sobolev exponent: The effect of the graph topology
Introduction
The main purpose of this paper is to establish the existence of solutions for the following fourth-order equation involving the critical Sobolev exponent under Navier boundary conditions: where is a small nonnegative number, is a bounded domain in , with smooth boundary , is a smooth positive function in and is the critical exponent for the Sobolev embedding .
The interest in this problem grew up from its resemblance to some fourth-order type of equations arising in conformal geometry. A well-known example is the problem of prescribing Paneitz curvature (so-called -curvature): given a function defined in Riemannian manifold of dimension , we ask whether there exists a metric conformal to such that is the Paneitz curvature of the new metric . We can refer to [1], [2], [3], [4], [5], [6], [7], [8], [9] and the references therein for details.
In contrast to the concentration phenomena for the second order elliptic problems with the critical Sobolev exponent which has been extensively investigated in the recent years, (see for example [10], [11], [12], [13], [14], [15], [16], [17]), fewer results are available to the concentration phenomena for the fourth-order elliptic problems (see for example [18], [19]).
In the special case where , a first result concerning (1.1) has been obtained by Van der Vorst [20], who proved the existence of the solution of (1.1) for , where denotes the first positive eigenvalue of on , generalizing to the fourth-order elliptic problem, the famous result of Brezis–Nirenberg [11]. Van der Vorst proved also in [21] a nonexistence result of (1.1) when and is a star shaped domain. Such nonexistence result has been proved by Pohozaev [22] for the second order elliptic problems. In the same case, ( and ), F. Ebobisse and M. Ould Ahmedou [23] investigated the effect of the topology of the domain on the existence of the solution. They proved that (1.1) has a solution provided that some homology group of is nontrivial. This result generalizes to (1.1) the famous result of Bahri–Coron [24] initially obtained for the second order elliptic problem. In [18], El Mehdi and Selmi have constructed a solution of (1.1) which concentrates around a critical point of Robin’s function as . Such concentration result extends to fourth order equations some results obtained by O. Rey (see [25], [16]) for the second order elliptic equations.
However, as far as the author knows, the case where has not been considered before.
Pushing further the resemblance of the two problems, the second order elliptic type problems from one part and the fourth order elliptic type problems from another part, we investigate the effect of the graph topology of the function on the existence of a solution for (1.1), extending the results obtained by J. Chabrowski and S. Yan [26], to the case of the fourth-order elliptic problem. Namely, in the first part of this paper, we will present the effect of the topological structure of the graph of the function on the existence of the solution of (1.1) as . In the second part, we construct solutions of (1.1) which blow up and concentrate around a boundary or interior local maximum point of as tends to zero. Then, we estimate the number of such solutions using the Lusternik–Schnirelmann category of the set of local maxima of . In the last part, we will prove the nonexistence of solutions which concentrate around a non degenerate critical point of . Compared with the second order case, further technical problems arise. In fact, in our case we cannot use, as in the Laplacian case, the method of moving planes in order to show that blow up points are away from the boundary of the domain. This is due to the fact that the Navier boundary condition is not invariant under the Kelvin transformation of a biharmonic operator. To overcome such a difficulty, we perform refined expansions of the Euler functional associated to (1.1) and its gradient near a neighborhood of highly concentrated functions. Such expansions, which are of independent interest, are highly nontrivial and use the techniques developed by A. Bahri [27] and O. Rey [16] in the framework of the “Theory of critical points at infinity”.
To state our results, we need to introduce the following notations. Let Using the topological structure of the global maximum set of , we have the following first result.
Theorem 1.1 For , assume the following. is not contractible in a small neighborhood of itself, but is contractible in for some constant belonging tosuch that . For each critical point of such that , we have
Then, for each ,(1.1) has a solution converging (up to subsequence) strongly in as .
The proof of Theorem 1.1 goes along the idea of [28] to construct a solution whose energy is strictly greater than , where is the best Sobolev constant (see (1.3)). Above this energy level, the corresponding functional does not satisfy the Palais–Smale condition. To overcome this difficulty, we first perturb the original problem suitably and construct a solution for this perturbed problem, whose energy is strictly greater than . Then we prove that the solution for the perturbed problem (see (3.1) below) converges strongly in to a solution of the original problem (1.1).
Unlike Theorem 1.1, where the set of global maxima of was used, the next two results use only local information. Namely, we will give the concentration and multiplicity results for problem (1.1) using the topological structure of the local maximum set of the function . For this purpose, we recall that the Lusternik–Schnirelmann category of the subset denoted by is the least integer such that can be covered by closed subsets of such that for all is contractible in . If no such integer exists, then by definition . If we write . We need also to fix the following notations. For and , let It is well known (see [29]) that are the only solutions of with and . They are also the only minimizers of the Sobolev inequality on the whole space; that is, We denote by the projection of the ’s onto , defined by and we set We denote by Green’s function of , that is, for all , where denotes the Dirac mass at and . We also denote by the regular part of , that is, The space is equipped with the norm and its corresponding inner product defined by For , , let where the is the -th component of .
In the following two results, we construct a solution for (1.1) which concentrates at a local maxima of as .
Theorem 1.2 For , let be a connected closed set in satisfyingLet be a positive function satisfying: , ,
where and are some positive constants. Then, for each , problem (1.1) has at least solutions of the formwhere and as , we have
Theorem 1.3 For , let be a connected closed set in satisfyingLet be a connected closed set compactly contained in such that and satisfying , , , , for each and for all ,
where is some positive constant. Then for each , problem (1.1) has at least solutions of the formwhere and as , we have
Remark 1.4 Similar conditions to those given in Theorem 1.2, Theorem 1.3, have been imposed on a global maximum point in [30], in order to obtain the existence of one solution in the case of the Laplacian problem.
In contrast with the previous results where we gave existence and multiplicity results, we give next a nonexistence result. Namely, we show that this problem has no solutions that concentrate around a non degenerate critical point of as approaches 0.
Theorem 1.5 For , let be a critical point of such that . Then, there is no solution of (1.1) of the form (1.6) satisfying (1.7), (1.8), (1.9), (1.10).
Remark 1.6 We believe our results, (Theorem 1.1, Theorem 1.3, Theorem 1.5), to be true for . More careful estimates similar to those developed by Rey [31] in the case of Laplacian, should include the case . However, in Theorem 1.2 we should assume that in order to make the conditions of this result meaningful. The assumption is sharp, since for , (1.1) has no solution when is a ball for small [32].
We organize the remainder of the present paper as follows. In Section 2 we set up the variational structure and give some careful expansion of the Euler functional associated to problem (1.1). Sections 3 Proof of, 4 Proof of will be devoted to the proof of the existence results. The nonexistence result will be proved in Section 5. To prove our results, we need some technical lemmas whose proofs are collected in the Appendix.
Section snippets
Variational structure and technical framework
Problem (1.1) enjoys a variational structure. Indeed, for , solutions of (1.1) correspond to positive critical points of the functional defined on by: We define the following subset of where is a small positive constant. Let us define the functional It is well known that is a critical point of if and only if is a critical point of , (see for
Proof of Theorem 1.1
Let be a small constant. For each fixed small , we consider the following perturbed problem The solutions of (3.1) are critical points on of the functional
It is easy to check that the energy of the solution of (3.1) in the case is at least . Following the idea of [28], to construct a solution for (3.1) whose energy is strictly greater than , we have the following
Proof of Theorem 1.2
First, we define Let with small enough. It is easy to see that Let where is a small fixed constant, and are large constants. Without loss of generality, we may assume . Let where is a large constant to be determined later. For , we denote Using Proposition 2.1,
Proof of Theorem 1.5
Arguing by contradiction, we suppose that (1.1) has a solution of the form satisfying (1.7), (1.8), (1.9), (1.10). First, multiplying (1.1) by and integrating over , we get where is a constant. Observe that, Using Lemma A.1, it is easy to check that
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