Abstract
The aim of this paper is to investigate the existence of two positive solutions to subcritical and critical fractional integro-differential equations driven by a nonlocal operator . Specifically, we get multiple solutions to the following fractional -Laplacian equations with the help of fibering maps and Nehari manifold. . Our results extend the previous results in some respects.
1. Introduction
In this work, we are concerned with the existence of solutions for a nonlocal integro-differential equation where is a bounded smooth domain in with , the exponents and fulfill with the critical fractional Sobolev exponent , and is a kind of nonlocal integro-differential operator defined by:
and is a measurable function with the following property:
In recent years, the existence and multiplicity of solutions of elliptic equations in nonlinear analysis have attracted the attention of many scholars. In particular, problems with regular nolinearities like and singular nonlinearities . At the same time, elliptic problems can be divided into two categories according to their order: integer order and fractional order.
On the one hand, when in [1], the authors considered a class of semilinear problems with singular nonlinearities. Many results on the existence and multiplicity of solutions for singular problems have appeared in the literature [2–6]. For example, authors have investigated a singular problem with the kind of critical growth in [6], where They obtained the existence of solutions by means of the Nehari manifold method in a suitable range of .
On the other hand, in [7], Mukherjee and Sreenadh considered the following critical fractional Laplace operator equations with a singular nonlinearity They showed the existence and multiplicity of positive solutions with respect to the parameter for above equation by using variational methods. Furthermore, in [8], they studied a class of critical fractional problems with a lower order perturbation by means of variational and topological methods; precisely, they proved that the number of nontrivial weak solutions is at least twice the multiplicity of the eigenvalue. More details on the critical case of fractional -Laplace equations can be referred to [9]. In subcritical case, the existence of positive solutions to the following quasi-linear problem is studied by means of truncation and comparison techniques in [10]. Zuo et al. [11] investigated a superlinear fractional elliptic equations; the existence of infinity many solutions is obtained by the fountain theorem in subcritical case. Moreover, they also get at least two solutions for a fractional -Laplace system by the Nehari manifold method in [12]. We will adopt a new technique, considering both subcritical and critical cases in a more general operator context (see (1)).
In order to state our results, let us introduce some notations. The space where with The space is endowed with the norm, and we define the closed linear subspace with the norm
Let fulfill condition (3). We have that and is a reflexive Banach space (see [13]). Moreover, where is the usual fractional Sobolev space endowed norm and the embedding is continuous, and there exists a positive constant such that, for any
Definition 1. We say that is a weak solution of problem (1), if fulfills for all where .
The main results of this article are as follows.
Theorem 2. Set , fulfilling condition (3), if , then there exists , such that for , equation (1) has at least two positive solutions.
Theorem 3. Set , and be an open bounded domain in with Lipschitz boundary. fulfilling the condition (3), if , assumes that there exists with almost everywhere in , such that where will be introduced in Section 2. Then, there exists such that for problem (1) admits least two solutions.
2. Preliminaries
We define the energy functional associated to problem (1) as with
We can see that and
for any .
Now, we give the Nehari manifold where denotes the duality between and its dual space. Thus, if and only if
The Nehari manifold is closely related to the following function for defined by
Remark 4. Set , then .
Moreover,
According to (25) and Remark 4, for , we have
The is divided into three sets, which are local minimum, local maximum, and local inflection point, respectively, i.e.,
To prove our result, we should start to show the following auxiliary lemmas.
Lemma 5. If is a local minimizer of on and then is a critical point of .
Similar to Theorem 2.3 in [14], we can get this conclusion.
About fibering maps and the Nehari manifold, considering the function defined by
Obviously, for any if and only if
Moreover,
and moreover, we know that then
So, if and only if (or ). Assume and In view of (29), fulfills the following properties:
(i) has a unique critical point at (ii) on and on Further, it follows from that (30) has no solutions if fulfills
According to (25) and (30) if fulfills (34), then It seems as is sufficiently large. Therefore, for any Moreover, if fulfills
then there exist and with such that
combining (25) and (30), which imply that It follows from (32) that , which mean that the fibering map admits a local minimum and a local maximum at
3. The Subcritical Case:
Firstly, we prove the following lemmas.
Lemma 6. There exists such that for any we have .
Proof. Using the inverse method, if for any Then,
for .
Namely,
Thus,
Using the Hölder inequality and Remark 4, there exist two positive constants such that
It yields that If is small enough, then it is impossible. Thus, assuming no, the original set is empty.
Lemma 7. is coercive and bounded from below on for .
Proof. Let (19) and (23) we get
Using Remark 4 and Hölder inequality, we get
Prove complete due to
By Lemmas 6 and 7, for any we get , and so, is coercive and bounded from below on and . Therefore, we define
We have the following result.
Proposition 8. If then the functional has a minimizer in and satisfies (1).(2) is a solution of problem (1).
Proof. Since the bounded from below of on there exists a minimizing sequence , such that
We know that the sequence is bounded in by Lemma 7. is a Hilbert space (see Lemma 7 in [15]); thus, there exists such that, up to a subsequence,
when Further, by Lemma 8 in [15], we have
as and by ([16], TheoremIV − 9), there exists such that
for any It follows from the dominated convergence theorem that
So, there exists such that and . Therefore, we get .
In order to prove that strongly in Still use the arc method if not, then Hence, for , we get
That is, for large enough. Since we infer that for and for all So, must be In addition because is decreasing on , and so,
Obviously, the above equation is a contradiction. Therefore, strongly in It means that
i.e., is a minimizer if on By Lemma 5, is a solution to problem (1).
Proposition 9. If then admits a minimizer in and satisfies (1).(2) is a solution to problem (1).
Proof. Since the bounded from below of on , there exists a minimizing sequence , such that
Similar to Proposition 8, there exists such that
as and
as Moreover, from the nature of the fibering maps we infer that there exist with such that and .
Next, we show that strongly in . If not, then Thus, for we have for all , and
in a similar way, we still can get a contradiction. Thus, strongly in It means that
Namely, is a minimizer if on is a solution to problem (1) according to Lemma 5.
Proof of Theorem 10. We obtain that problem (1) has two solutions and in due to the Propositions 8 and 9 and Lemma 5; moreover, we know that two solutions are distinct since
4. The Critical Case:
For the critical case, since the embedding is not compact, then the energy functional does not satisfy the Palais-Smale condition globally, but it is true for the energy functional in a suitable range related to the best fractional critical Sobolev constant in the embedding For this, we define fractional Sobolev best constant as
Before we give the Proof of Theorem 13, we start by some auxiliary results. Firstly, using the same proofs of Lemma 6, we deduce that there exists such that for each . Also, it is clear that is coercive and bounded from below on for by Lemma 7. So, for any we also obtain that , and is coercive and bounded from below on and We define
Proposition 11. be a sequence for with then there exists a subsequence of which converges strongly in where is defined in (57) and is defined by
Proof. It follows from is bounded in that there exists such that weakly in that is as Moreover, using the same arguments as lemma 9 ([17]), we get that as and by ([16], TheoremIV − 9], there exists such that for any Then, using dominated convergence theorem, we have that Also, by the same method as in ([18], Lemma 1.32), we get as Then, By and as we know that If is clearly true. If in view of the definition of in 17, we get Thus, we have That is, On the other hand, we have By the assumption that we have In particular, and Then, Moreover, by Hlder inequality, we have Thus, where So, attains its minimum at and Therefore, which is a contradiction. Therefore, and we obtain that strongly in
Proposition 12. There exists and such that for In particular
Proof. We suppose there exists such that for all By condition (17) we have that there is such that
Let . Therefore, for we obtain that
Then, we have (77) holds.
Finally, let we obtain that
for by the nature of fibering maps
Proof of Theorem 13. There exist two sequences and in such that as because of Propositions 11 and 12. From related properties of fibering maps we have Similar to the Proof of Theorem 10, problem (1) admits two solutions and in Moreover, these two solutions are distinct since
Conflicts of Interest
The authors declare that they have no competing interests.