Abstract

We study a class of semilinear nonlocal elliptic systems posed on settings without compact Sobolev embedding. By employing critical point theory and concentration estimates, we prove the existence of infinitely many solutions for values of the dimension , where provided

1. Introduction and Main Result

Nonlocal problems and operators have enjoyed much attention from mathematicians in recent years probably due to their interesting analytical structure and their numerous physical applications in many fields such as flame propagation, conservation laws, ultrarelativistic limits of quantum mechanics, quasi-geostrophic flows, and the thin obstacle problem (see [1ā€“3]). In this paper, we are concerned with the study of the infinitely many solutions for the following system: where is an open bounded domain in , and , satisfy , and . When , , and , problem (1) reduces to the BrĆ©zis-Nirenberg type problem with fractional Laplacian:In [4], BrĆ©zis and Nirenberg considered the existence of positive solutions for problem (2) with . Such a problem involves the critical Sobolev exponent for , and it is well known that the Sobolev embedding is not compact even if is bounded. Hence, the associated functional of problem (2) does not satisfy the Palais-Smale condition, and critical point theory can not be applied directly to find solutions of the problem. However, it is found in [4] that the functional satisfies the condition for , where is the best Sobolev constant and is the least energy level at which the Palais-Smale condition fails. So a positive solution can be found if the mountain pass value corresponding to problem (2) is strictly less than . In [5], a concentration-compactness principle was developed to treat noncompact critical variational problems. In the study of the existence of multiple solutions for critical problems, to retain the compactness, it is necessary to have a full description of energy levels at which the associated functional does not satisfy the Palais-Smale condition. A global compactness result is found in [6], which describes precisely the obstacles of the compactness for critical semilinear elliptic problems. This compactness result shows that above a certain energy level it is impossible to prove the Palais-Smale condition. For this reason, to obtain infinitely many solutions for the critical problem, it is essential to find a condition that can replace the standard Palais-Smale condition. In [7], Devillanova and Solimini considered (2) with and obtained infinitely many solutions for every if . They proved this latter result by employing the concentration estimates and lower bound of augmented Morse index on minā€“max points (see [8]), which seem unapplicable to the case of (1) directly. This work was extended to an analogous problem involving p-Laplacian for by Cao et al. [9]. They proved that if , the following problem, where and , has infinitely many solutions. Recently, Yan et al. extended the result in [10] to problem (2) and obtained infinitely many solutions for every if , where . Similar problems to (1) for the Laplacian operator have been studied extensively in recent years; see [6, 11ā€“14] and the references therein. In particular, Alves et al. [11] studied the p-Laplacian system with critical growth and obtained the existence of infinitely many solutions if An important cornerstone for these works has been laid out in a remarkable paper [15], where Caffarelli and Silvestre gave a new formulation of the fractional Laplacian through Dirichlet-Neumann maps. This is extensively used in the recent literature since it allows transforming nonlocal problems to local ones, which permits the use of variational methods. This will come in handy for this work. A more general form of the fractional operator has been studied and a multiplicity of solutions has been shown in several cases. For further reading, we refer the reader to these recently published papers on the fractional Laplacian [16ā€“22].

In this paper, we prove that (1) has infinitely many solutions under the following conditions: , where , by studying system (1) in the critical case . We adapt the original idea used in [7] which was described in the above paragraph (see also [9ā€“11, 13]). After perturbing problem (1) into a subcritical case with a (PS) functional in all energy levels, estimates on the set of the solution sequences to the subcritical case of (1), a global compactness argument, and a (local) Pohozaev identity are used to establish the strong convergence and finally use minā€“max theorems on a genus homotopic class to produce infinitely many critical values. The paper is organized as follows: in Section 2, some notations and preliminary results are established. In Section 3, we establish a local Pohozaev identity which allows us to prove the -strong convergence of solutions for the subcritical case of problem (18), and, finally, we show how this technique allows the application of classical minā€“max arguments to (1) and to prove, in this way, the existence of infinitely many solutions. Our main result is the following.

Theorem 1. Let and , with . Then problem (1) has infinitely many solutions when

2. Notation and Preliminaries

The powers of the positive Laplacian operator , in , with zero Dirichlet boundary conditions are defined via its spectral decomposition, namely, where is the sequence of eigenvalues and eigenfunctions of the operator in under zero Dirichlet boundary data and are the coefficients of for the base in In fact, the fractional Laplacian is well defined in the space of functions and

A pair of functions is said to be a weak solution of problem (1) if for all Solutions to problem (1) will be obtained as critical points of the corresponding energy functional The functional is well defined in , and, moreover, the critical points of the functional correspond to solutions of (1). We now conclude the main ingredients of a recently developed technique use in order to deal with fractional powers of the Laplacian operator. Motivated by the work of Caffarelli and Silvestre [15], several authors have considered an equivalent definition of the operator in a bounded domain with zero Dirichlet boundary data means of auxiliary variable, see [2, 23]. Associated with the bounded domain , let us consider the cylinder . Now, for a function , we define the s-harmonic extension to the cylinder as the solution of the problem: where is a normalization constant. The extension function belongs to the space endowed with the norm The extension operator is an isometry between and ; namely, With this extension, we can reformulate (1) as the following local problem: where and are the s-harmonic extension of , respectively. Let be equipped with the norm An energy solution to this problem is a function satisfying for all If satisfies (12), then , defined in the sense of traces, belongs to the space and it is a solution of the original problem (1). The associated energy functional to problem (12) is denoted by Critical points of in correspond to the critical points of Since the problem is critical, the functional does not satisfy the Palais-Smale condition. Thus the minā€“max theorems can not be applied directly to obtain infinitely many solutions for (1). Following the original idea in [7], employed in a closer setting in [10], we deal first with the following perturbed problem: where and .

The functional corresponding to (18) becomes where Now is even and satisfies the Palais-Smale condition at all energy levels. It follows from the symmetric mountain pass lemma [24] that (18) has infinitely many solutions. More precisely, it follows from [25, Theoremā€‰ā€‰6.1] that there are positive numbers , with , as and a solution for (18), satisfying To obtain the existence of infinitely many solutions for (1), the first step is to investigate whether converges strongly in as That is, we need to study the compactness of the set of solutions for (18), with small enough.

3. -Strong Convergence of Solutions for the Subcritical Case of Problem (18)

In this section, we establish the strong convergence of solutions for the perturbed problem (18). Throughout this paper, we denote the norms of by , , respectively, and positive constants (possibly different) by .

Proposition 2. Suppose that with and satisfy Then for any sequence () which is a solution of (18) with , and for satisfies for some constant independent of , then has a subsequence which converges strongly in as .

Before giving the proof of Proposition 2, we give some estimates for Let be a bounded domain such that and in We choose large enough so that, for all , the following is verified: for all Let () with be a solution ofBy choosing and , we findMultiplying (22) by and integrating by part, we see thatSimilarly, for , we haveThen by (23) and (24) we haveSo in the following sections we can consider just the estimates of in .

Let Then we have (see Lemmaā€‰ā€‰2.5 [23]) where is the best Sobolev constant defined bywhich is achieved if and only if by functions are the s-harmonic extensions of Now we introduce the ā€œproblem at infinityā€:Here , , and . Set Then problem (30) can be rewritten as

Lemma 3. Let be a solution of (32). Then for all

Proof. Let be a solution of (32). Then Multiplying (33) by and integrating by part, we obtain which implies that Note that , , in . It follows from (30) that in Hence

From Proposition 6 in the Appendix, we have the global compactness result on (1). Let () be a solution of problem (18) with as , satisfying for some constant independent of Then replacing the solution if necessary with a subsequence, there exist sequences mutually diverging scaling with respective concentration points such that, as ,where is a weak solution of problem (12), as , achieves the constant , which is given in (28), and

As in [7, 9, 10] we shall introduce the following facts which are essentials to prove the strong convergence of in Among all the bubbles in (28), we can choose the slowest concentration rate, denoted by , which concentrates in the slowest rate. That is, the corresponding is the lowest order infinity among the ones appearing in the bubbles. Note that the number of the bubbles in is finite; we can always choose a constant , independent of , such that the region, does not contain any concentration point of for every We set two thinner subsets as follows:

Then the following integral estimates hold (see Propositionsā€‰ā€‰4.1 andā€‰ā€‰4.2 in [10]).

Lemma 4. Let be solution of (18) with . Then, there exists a constant independent of , such that, for all , one hasMoreover, one has

Proof. For the proof of this lemma, we use inequality (25) and we refer to [10, Propositionsā€‰ā€‰4.1, 4.2].

Note that , and we have the following local Pohozaev identity.

Lemma 5. Let , , and as and be any bounded set in Then one has the local Pohozaev identity on ā€‰ā€‰associated for equations of (18):where is the outward unit normal to and , with being a point in

Proof. Using the divergence theorem, we getwhere is the outward unit normal to and , with being a point in .
Since is solution of (18), we have the following equations:We obtain, from (43) and (44),Noting that the dot product , on , we obtainMoreover, we haveTherefore, from (45), (46), and (47), we infer that (44) holds.

Proof of Proposition 2. Let be a bounded sequence in composed of solutions for (18). Thus, in order to prove the -strong convergence in (36), we just need to show that the bubbles in (36) will not appear in the decomposition of and Since the proof is similar to that of Lemmaā€‰ā€‰6.1 in [9], here we only give a sketch of it. From Lemma 5, for the solution concentrating sequence of (18) with , we have the local Pohozaev identity on where is the outward unit normal to and , We decompose where and
We consider two different cases:(i)(ii)In case (i), we take with and in , where is the outward unit normal to
Since on , we findIn case (ii), ; we take a point
Since on , and, for large enough , we obtainBy assumption, and ; we may assume that Set . Recalling that, by (36), we have the decomposition , where , , and . as . Then we deduce that, for large enough, After a direct calculation, we haveNote that , as Inserting (54) into (53), we get for large enoughBy the choice of , as in [10], we only need to consider the right-hand side of (52) on By applying Hƶlder inequality and using Lemma 4, we getInserting (56) and (57) into (52), we obtain which is a contradiction for large enough due to

4. Proof of Theorem 1

For any positive integer , define the -homotopy class as follows: where the genus is the smallest integer , such that there exists an odd map For , we define the minā€“max value as (e.g., see p. 134 in [26]) It follows from Corollaryā€‰ā€‰7.12 in [26] that, for each small is critical value of , since satisfies the Palais-Smale condition. Thus problem (18) has a solution such that Note that uniformly with respect to as ; hence, is uniformly bounded with respect to for each fixed . By a direct calculation, we find uniformly with respect to for each fixed . So now, we can apply Proposition 2 and obtain a subsequence of , such that, as strongly in for some and Then is solution of (1) and We are now ready to show that has infinitely many critical point solutions. Note that is nondecreasing in By an argument similar to the one used in the proof of Theoremā€‰ā€‰1.1 in [9] we distinguish two cases.

Case 1. Suppose that there are , satisfying In this case, we have infinitely many distinct critical points and, therefore, infinitely many solutions.

Case 2. We assume, in this case, that for some positive integer , for all Suppose that, for any has a critical point with and In this case, we are done. So from now on we assume that there exists such that has no critical point with In this case, using the deformation argument, we can prove that where As a consequence, has infinitely many critical points. Thus we can obtain infinitely many solutions for problem (1).

Appendix

In this section, we give a global compactness result in the following proposition.

Proposition 6. Suppose that is a solution of (18) with , satisfying for that is the constant independent of . Then (i) can be decomposed asā€‰where is a weak solution of problem (12),ā€‰, as achieves the constant , which is given in (28); and ā€‰for , with , ;(ii)for , if , then, as ,

Proof. The proof follows without difficulty by modifying the proof of the concentration compactness result for (2) (see [10, 15]) and using Lemma 3. We omit the details for the sake of simplicity.

Competing Interests

The authors declare that they have no competing interests.