This paper is devoted to the study of a class of Kirchhoff-type problems driven by a nonlocal fractional operator and involving a singular term and a critical nonlinearity. More precisely, we consider
where Ω is an open bounded subset of with continuous boundary, dimension with parameter , is the fractional critical Sobolev exponent, is a real parameter, , while . Here is the fractional Laplace operator defined, up to normalization factors, by the Riesz potential as
where Δ denotes the classical Laplace operator while, just for a general discussion, for any . In literature, problems like (1.1) and (1.2) are called of Kirchhoff type whenever the function models the Kirchhoff prototype, given by
In particular, when for any , Kirchhoff problems are said to be non-degenerate and this happens for example if in the model case (1.3). While if but for any , Kirchhoff problems are called degenerate. Of course, for (1.3) this occurs when .
This kind of nonlocal problems has been widely studied in recent years. We refer to [17, 18, 19, 21, 20] for different Kirchhoff problems with M like in (1.3), driven by the Laplace operator and involving a singular term of type . In , Liu and Sun study a Kirchhoff problem with a singular term and a Hardy potential by using the Nehari method. The same approach is used in  for a singular Kirchhoff problem with also a subcritical term. In , strongly assuming in (1.3), Lei, Liao and Tang prove the existence of two solutions for a Kirchhoff problem like (1.2) by combining perturbation and variational methods. While in , Liao, Ke, Lei and Tang provide a uniqueness result for a singular Kirchhoff problem involving a negative critical nonlinearity by a minimization argument. By arguing similarly to , Liu, Tang, Liao and Wu  give the existence of two solutions for a critical Kirchhoff problem with a singular term of type .
Problem (1.1) has been studied by Barrios, De Bonis, Medina and Peral  when , namely without a Kirchhoff coefficient. They prove the existence of two solutions by applying the sub/supersolutions and Sattinger methods. In , Canino, Montoro, Sciunzi and Squassina generalize the results of [4, Section 3] to the delicate case of the p-fractional Laplace operator . While in the last section of , Abdellaoui, Medina, Peral and Primo provide the existence of a solution for nonlinear fractional problems with a singularity like and a fractional Hardy term by perturbation methods. Concerning fractional Kirchhoff problems involving critical nonlinearities, we refer to [2, 9, 13, 14, 16, 23] for existence results and to [5, 12, 24, 25, 29] for multiplicity results. In particular, in [9, 13, 14, 23] different singular terms appear, but are given by the fractional Hardy potential.
Inspired by the above works, we study a multiplicity result for problem (1.1). As far as we know, a fractional Kirchhoff problem involving a singular term of type has not been studied yet. We can state our result as follows.
Let , , , and let Ω be an open bounded subset of with continuous. Then there exists such that for any problem (1.1) has at least two different solutions.
The first solution of problem (1.1) is obtained by a suitable minimization argument, where we must pay attention to the nonlocal nature of the fractional Laplacian. Concerning the second solution, because of the presence of , we can not apply the usual critical point theory to problem (1.1). For this, we first study a perturbed problem obtained by truncating the singular term . Then by approximation we get our second solution of (1.1).
The paper is organized as follows: In Section 2, we discuss the variational formulation of problem (1.1), and we introduce the perturbed problem. In Section 3, we prove the existence of the first solution of (1.1), and we give a possible generalization of this existence result at the end of the section. In Section 4, we prove the existence of a mountain pass solution for the perturbed problem. In Section 5, we prove Theorem 1.1.
2 Variational setting
Throughout this paper, we assume without further mentioning that , , , and Ω is an open bounded subset of with continuous. As a matter of notations, we denote with and respectively the positive and negative part of a function φ.
Problem (1.1) has a variational structure, and the natural space where to find solutions is the homogeneous fractional Sobolev space . In order to study (1.1) it is important to encode the “boundary condition” in in the weak formulation, by considering also that the interaction between Ω and its complementary in gives a positive contribution in the so-called Gagliardo norm, given as
The functional space that takes into account this boundary condition will be denoted by and it is defined as
We refer to  for a general definition of and its properties. We also would like to point out that when is continuous, by [15, Theorem 6] the space is dense in , with respect to the norm (2.1). This last point will be used to overcome the singularity in problem (1.1).
In we can consider the norm
From now on, in order to simplify the notation, we will denote and by and , respectively, and by for any .
In order to present the weak formulation of (1.1) and taking into account that we are looking for positive solutions, we will consider the following Kirchhoff problem:
We say that is a (weak) solution of problem (2.2) if u satisfies
for any . Problem (2.2) has a variational structure and , defined by
is the underlying functional associated to (2.2). Because of the presence of a singular term in (2.2), the functional is not differentiable on . Therefore, we can not apply directly the usual critical point theory to in order to solve problem (2.2). However, it is possible to find a first solution of (2.2) by using a local minimization argument. In order to get the second solution of (2.2) we have to study an associated approximating problem. That is, for any , we consider the following perturbed problem:
For this, we say that is a (weak) solution of problem (2.4) if u satisfies
for any . In this case, solutions of (2.4) correspond to the critical points of the functional , set as
It is immediate to see that is of class .
We conclude this section by recalling the best constant of the fractional Sobolev embedding, which will be very useful to study the compactness property of the functional . That is, we consider
which is well defined and strictly positive, as shown in [10, Theorem 1.1].
3 A first solution for problem (1.1)
In this section, we prove the existence of a solution for problem (1.1) by a local minimization argument. For this, we first study the geometry of the functional .
There exist numbers , and such that for any , with , and for any . Furthermore, set
where . Then for any .
Let . From the Hölder inequality and (2.7) for any we have
Since , the function
admits a maximum at some small enough, that is, . Thus, let
Then for any with and for any , we get .
Furthermore, fixed with , for sufficiently small we have
since . ∎
We are now ready to prove the existence of the first solution of (1.1).
Fix and let ρ be as given in Lemma 3.1. We first prove that there exists such that . Let be a minimizing sequence for , that is, such that
Since , by the Hölder inequality, for any we have
which yields, by (3.3),
Let ; by [7, Theorem 2] it holds true that
and from this, since , for k sufficiently large we have
since . Hence, is a local minimizer for , with , which implies that is nontrivial.
Now, we prove that is a positive solution of (2.2). For any , with a.e. in , let us consider a sufficiently small so that . Since is a local minimizer for , we have
From this, by dividing by and passing to the limit as , it follows that
We observe that
with and a.e. in Ω as . Thus, by the Fatou lemma, we obtain
for any with a.e. in .
Since and by Lemma 3.1, we have . Hence, there exists such that for any . Let us define . Since is a local minimizer for in , the functional has a minimum at , that is,
For any and any , let us define . Then by (3.9) we have
We observe that, for a.e. , we obtain
from which we immediately get
From the last inequality it follows that
by the symmetry of the fractional kernel and (3.15), we get
where we set
Clearly , so that for any there exists sufficiently large such that
Also, by the definition of , we have and as . Thus, since , there exist and such that for any ,
Therefore, for any ,
from which we get
Finally, considering in (2.3) and using (3.12), we see that , which implies that is nonnegative. Moreover, by the maximum principle in [28, Proposition 2.17], we can deduce that is a positive solution of (2.2), and so also solves problem (1.1). This concludes the proof. ∎
We end this section by observing that the result in Theorem 3.2 can be extended to more general Kirchhoff problems. That is, we can consider the problem
where , with and , while the Kirchhoff coefficient M satisfies the following condition:
is continuous and nondecreasing. There exist numbers and ϑ such that for any ,
The main operator is the fractional p-Laplacian which may be defined, for any function , as
where . Then, arguing as in the proof of Theorem 3.2 and observing that we have not used yet the assumption that is continuous, we can prove the following result.
Let , , , and let Ω be an open bounded subset of . Let M satisfy . Then there exists such that for any problem (3.18) admits a solution.
4 A mountain pass solution for problem (2.4)
In this section, we prove the existence of a positive solution for the perturbed problem (2.4) by the mountain pass theorem. For this, throughout this section we assume without further mentioning. Now, we first prove that the related functional satisfies all the geometric features required by the mountain pass theorem.
Let , and be given as in Lemma 3.1. Then, for any and any with , one has . Furthermore, there exists , with , such that .
Since , by the subadditivity of , we have
for any and any . Thus, we have for any and the first part of the lemma directly follows by Lemma 3.1.
For any , with , and , we have
since . Hence, we can find , with sufficiently large, such that . ∎
We discuss now the compactness property for the functional , given by the Palais–Smale condition. We recall that is a Palais–Smale sequence for at level if
We say that satisfies the Palais–Smale condition at level c if any Palais–Smale sequence at level c admits a convergent subsequence in .
Before proving this compactness condition, we introduce the following positive constants, which will help us for a better explanation:
Let . Then the functional satisfies the Palais–Smale condition at any level verifying
with , given as in (4.3).
Thus, by inequality (3.12) we deduce that as , which yields
Hence, we can suppose that is a sequence of nonnegative functions.
as , with for a fixed . If , then immediately in as . Hence, let us assume that .
Since , by (4.5) it follows that
so by the dominated convergence theorem and (4.5) we have
Therefore, we have proved the crucial formula
Noting that (4.8) implies in particular that
by using (4.9), it follows that
From this we obtain
Considering , we have
Indeed, the restriction follows directly from the fact that . By (4.1), considering that , for any we have
We now give a control from above for the functional under a suitable situation. For this, we assume, without loss of generality, that . By , we know that the infimum in (2.7) is attained at the function
that is, it holds true that
Let us fix such that , where , and let us introduce a cut-off function such that
For any , we set
Then we can prove the following result.
There exist and such that for any ,
with , given as in (4.3).
Let . Let and be as in (4.11) and (4.13), respectively. By (2.6), we have as , so that there exists such that . By Lemma 4.1, we know that . Hence, by the continuity of there exist two numbers , such that .
Now, since is independent of ε, by [27, Proposition 21] we have
from which, by the elementary inequality
with , it follows that, as ,
Hence, by the last inequality, (2.6) and since , for any sufficiently small, we have
with a suitable positive constant . We observe that
Thus, by (4.14) it follows that
Now, let us consider a positive number q, less than 1, satisfying
that is, since , and , such that
By the elementary inequality
with , considering sufficiently small, with r given by (4.12), and since , we have
Thus, let us consider such that
and let us set
which concludes the proof. ∎
We can now prove the existence result for (2.4) by applying the mountain pass theorem.
There exists such that, for any , problem (2.4) has a positive solution with
Hence, by Lemma 4.2 the functional satisfies the Palais–Smale condition at level . Thus, the mountain pass theorem gives the existence of a critical point for at level . Since
we obtain that is a nontrivial solution of (2.4). Furthermore, by (2.5) with test function and inequality (3.12), we can see that , which implies that is nonnegative. By the maximum principle in [28, Proposition 2.17], we have that is a positive solution of (2.4), concluding the proof. ∎
5 A second solution for problem (1.1)
In this last section, we prove the existence of a second solution for problem (1.1), as a limit of solutions of the perturbed problem (2.4). For this, here we need the assumption that is continuous in order to apply a density argument for the space .
Proof of Theorem 1.1.
We want to prove that in as . When , by (5.1) we have in as . For this, we suppose . We observe that
so by the Vitali convergence theorem and (5.1) it follows that
For any , by an immediate calculation in (2.4) we see that
Now, let with . By (5.4), we have
so that by (5.1) and the dominated convergence theorem we obtain
If , then in as since .
which is the desired contradiction, thanks to (4.3).
Therefore, in as , and by (2.3) and (2.5) we immediately see that is a solution of problem (2.2). Furthermore, by (4.19) we have , which also implies that is nontrivial. Reasoning as at the end of the proof of Theorem 4.4, we conclude that is a positive solution of (2.2), and so also solves problem (1.1). Finally, is different from since . ∎
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About the article
Published Online: 2017-08-03
Funding Source: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Award identifier / Grant number: PNPD–CAPES 33003017003P5
The author is supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior through the fellowship PNPD–CAPES 33003017003P5. The author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica “G. Severi” (INdAM).
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 645–660, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0075.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0