Multiple of Solutions for Nonlocal Elliptic Equations with Critical Exponent Driven by the Fractional -Laplacian of Order

Abstract

In this paper, we study the existence of infinitely many weak solutions for nonlocal elliptic equations with critical exponent driven by the fractional -Laplacian of order . We show the above result when is small enough. We achieve our goal by making use of variational methods, more specifically, the Nehari Manifold and Lusternik-Schnirelmann theory.

1. Introduction

This work is concerned with the existence of weak solutions of the following critical fractional -Laplacian problem:where is a smoothly bounded domain of , , , is the fractional critical Sobolev exponent, and is positive parameter.

denotes the fractional -Laplacian operator defined on smooth functions by This definition is consistent, up to a normalization constant depending on and , with the usual definition of the linear fractional Laplacian operator when Let us recall the weak formulation of problem (1). Let us set by the Gagliardo seminorm of the measurable function , and let be the fractional Sobolev space endowed with the norm where is the norm in We work in the closed linear subspace equivalently renormed by setting , which is uniformly convex Banach space. We note that the embedding is continuous for and compact for , where is the fractional critical Sobolev exponent (note that when the above exponent reduces to the classical Sobolev exponent ). As in the classical case the technical problem posed by such an exponent is that the Sobolev embedding is not compact. Denote by the best Sobolev constant of the immersion , that is,The dual space of is

Recently, a great attention has been focused on the study of the fractional Laplacian and nonlocal operators of elliptic; this type arises in both pure mathematical research and concrete applications, such as the thin obstacle problem [1, 2], minimal surfaces [3], phase transitions [4], crystal dislocation [5], Markov processes [6], and fractional quantum mechanics [7]. This is one of the reasons why nonlocal fractional problems are widely studied in the literature in many different contexts (see [8]).

When , our problem becomes a scalar quasilinear elliptic equation as follows:This has been widely studied by many authors. For example, when is bounded, the existence of nontrivial solution was studied (see, e.g., [9, 10]). The typical difficulty in dealing with (9) is that the Sobolev embedding is not compact, so the usual “Convex-Compact” method does not apply directly. As for the existence of infinitely many solutions to (9), by using Lusternik-Schnirelmans theory, J. G. Azorero and I. P. Aloson proved in [11] that if is bounded,, and is small, then (9) has infinitely many solutions. Also the main result of [11] was extended to the equation driven by the operator by G. Li and X. Liang in [12]. G. M. Figueiredo in [13] generalized the same result of [11] to the elliptic equation generated by the operator Several works have been devoted to study some existence and multiplicity results for fractional problems involving the -Laplacian operator of the type (1), generalizing therefore some classical results obtained in the scalar case. The reader can find a lot of papers in the literature involving this subject; we cite [14–17]. Our goal is to generalize the results of Garcia Azorero and Peral in [11] to the case of the fractional -Laplacian on a bounded domain.

Two major difficulties arise which have to be dealt with in order to reach the desirable conclusions.

First off, it is hard to prove the existence of infinitely many negative energy solutions for our equation by using the variational method because does not satisfy the (PS) conditions, more precisely, because the problem in question incorporates critical exponents.

Secondly, the functional is not bounded from below, so in order to comfortably follow through with our plan, we have to introduce an appropriate truncation to the problem, the choice of which is of utmost importance to the results we get in this paper.

Theorem 1. Assume that Then there exists such that, for each , problem (1) has infinitely many solutions with negative energy.

Theorem 1 is new as far as we know and it generalizes a similar result in [11] for the fractional -Laplacian type problem. We mainly follow the way in [11] to prove our main result. The paper is organized as follows. In Section 2, we show that the conditions hold for the related energy functional in certain critical levels. That is, we give in a precise range of compactness for the energy functional related. In Section 3, under the assumptions of Theorem 1 and by application of Ljusternik-Schnirelmann methods, we establish the existence of infinitely many solutions with small enough.

2. The Condition for the Associated Functional

We recall that a weak solution for problem (1) is a function , such that Now let us consider the functional defined byNote that the functional and its derivative at are given by for every Thus, the weak solutions of problem (1) are precisely the critical points of the energy functional . Since problem (1) has a variational structure, the proof of the main result (Theorem 1 and its consequences) reduces to finding critical points of the functional by using suitable abstract approaches. As usual in the critical case, the difficulty related to the variational formulation of (1) is the lack of compactness of the injection of the fractional Sobolev space in To overcome this difficulty in treating problem (1), we show that even if the functional does not verify globally the Palais-Smale condition, it satisfies such a condition in a suitable range related to the best fractional critical Sobolev constant noted in (8).

The Nehari manifold associated with is given by where denotes the Gâteaux derivative of

Definition 2. (i)For , a sequence is a for if and strongly in as , where is the dual of (ii) satisfies the condition in if any sequence for contains a convergent subsequence.

The first step for the sequence to hold is bounded.

Lemma 3. Let If is - sequence for , then is bounded in

Proposition 4. There exists a such that, for any and the functional satisfies condition.

Proof. Let be a sequence in such thatas , for all ThenFrom (17) and the Hölder inequality, it is implied that is bounded in . Up to a subsequence, this implies the following: , , , ,  Soas
Moreover So passing to the limit in (16) shows that is a weak solution of (1). Setting , we haveBy Brezis-Lieb’s Lemma [18], we getas Taking in (16) and Brezis-Liebs Lemma again, we haveSince is bounded in and converges to in , testing (16) with givesIt follows from (23) and (24) that We suppose that By the definition of the best constant given in (8), we have soIf , then the lemma is proved. If , then (28) implies thatFrom (29) and (22), we have We consider the following function This function obtains this absolute minimum for at point , that is,and let the constant be strictly positive because This leads to a contradiction with (9). Therefore and the proof is complete.

3. Proof of the Main Result

Under the hypothesis , using Sobolev’s inequality we obtainwhere and where is a positive constant independent of An easy computation shows that, for all , the real valued function has exactly two positive zeros denoted by and , and the point is where attains its nonnegative maximum and verifies

We now introduce the following truncation of the functional . Take the nonincreasing function and such thatLet We consider the truncated functionalSimilar to (33), we havewhere Clearly,for and if , , if and if , is strictly increasing and so , if ConsequentlyWe have the following result.

Lemma 5. This lemma can be expressed as three assertions:(1) is even.(2)If then Moreover, for all in a small enough neighborhood of (3)There exists , such that if , then verifies a local Palais-Smale condition for

Proof. Since and for near 0, and assertion (1) holds.
By taking , we can deduce from (37) thatand by (40) and (41) we haveand (2) holds.
For the proof of (3), let be a sequence , with Then we may assume that , By (8) in Lemma 5 there exists such that , , so and By Proposition 4, satisfies condition for , so there is a subsequence such that in Thus satisfies condition for

We will use the genus of symmetric set in , where the genus is the smallest integer , such that there exists an odd map where is a closed symmetric set in that does not contain zero (see [19]).

It is possible to prove the existence of level sets of with arbitrarily large genus, more precisely,

Lemma 6. such that

Proof. Let we consider to be subspaces of with being an n-dimensional subspace of Let with norm For we define andBy using the definitions of , , and inequality (45), we obtain Then, there exists and such that for and
Let , sotherefore, by the properties of the genus (see [19])

We are now in a position to prove our main result.

Proof of Theorem 1. For , we defineLet us setandand suppose , where is the constant given by Lemma 5.By Lemma 6 there exists such that , for all Because is continuous and even, , then for all n in But is bounded from below; hence for all n in
Let us assume that Note that ; therefore, verifies the Plais-Smale condition in , and it is easy to see that is a compact set.
If , there exists a closed and symmetric set U verifying , such that . By the deformation lemma (see [20]), we have an odd homeomorphism , such that , for some . By definition, There exists then , such that , i.e., , But , and
Then, . And this is a contradiction; in fact, implies
So we have proved that We are now ready to show that has infinitely many critical point solutions. Note that is nondecreasing and strictly negative. We distinguish two cases.
Case 1. Suppose that there are , satisfying In this case, we have infinitely many distinct critical points.
Case 2. We assume in this case that, for some positive integer , there is such that ; then which shows that contains infinitely many distinct elements. Since if , we see that there are infinitely many critical points of . The theorem is proved.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflicts of interest.