In this paper, we are concerned with the equation
where , , , and . Here stands for the Riesz potential of order γ defined as for any .
where stands for the principal value of the integral and is a normalizing constant. The operator is referred to as the infinitesimal generator of the Levy stable diffusion process. The function is required to satisfy one (or both) of the following conditions:
For all , the set has finite Lebesgue measure.
Note that condition (V2) is weaker than , as for instance satisfies (V2) but has no limit as .
In the last few decades, problems involving the fractional Laplacian and nonlocal operators have received considerable attention. These kinds of problems arise in various applications such as continuum mechanics, phase transitions, population dynamics, optimization, finance, and many others.
The prototype model of (1.1) is the fractional Choquard equation
studied by d’Avenia, Siciliano and Squassina in  in the case where V is a positive constant. They obtained the existence of groundstate and radially symmetric solutions with diverging norm and diverging energy levels.
Equation (1.3) for was first introduced by Pekar  in 1954 in quantum mechanics. In 1996, Penrose [21, 22] used equation (1.3) in a different context as a model in self-gravitating matter (see also [11, 16]). Since then, the Choquard equation has been investigated in various settings and in many contexts (see, e.g., [1, 10, 14, 19]). For a most up to date reference on the study of the Choquard equation in a standard Laplace setting, the reader may consult .
For , , , , and , equation (1.1) becomes
In this paper, we shall be interested in the study of groundstate solutions and least energy sign-changing solutions to (1.1). To this aim, we denote by the completion of with respect to the Gagliardo seminorm
Also, denotes the standard fractional Sobolev space defined as the set of satisfying with the norm
Let us define the functional space
endowed with the norm
Throughout this paper, we shall assume that p and q satisfy
A crucial tool to our approach is the Hardy–Littlewood–Sobolev inequality
for , and such that
We shall first be concerned with the existence of ground state solutions for equation (1.1) under the assumption that V satisfies (V1). This will be achieved by a minimization method on the Nehari manifold associated with , which is defined as
The groundstate solutions will be obtained as minimizers of
Our main result in this sense is stated below.
Our approach relies on the analysis of the Palais–Smale sequences for . Using an idea from [3, 4], we show that any Palais–Smale sequence of is either converging strongly to its weak limit or differs from it by a finite number of sequences which further are the translated solutions of (1.2). The novelty of our approach is that we shall rely on several nonlocal Brezis–Lieb results as we present in Section 2.
We now turn to the study of least energy sign-changing solutions of (1.1). In this setting, we require V to fulfill both conditions (V1) and (V2). By the result in [25, Lemma 2.1] (see also [23, 24]), the embedding is compact for , where .
Our approach in the study of least energy sign-changing solutions of (1.1) is based on the minimization method on the Nehari nodal set defined as
The solutions will be obtained as minimizers for
In this situation, the problem is more delicate as some of the usual properties of the local nonlinear functional do not work. For instance, since
we have in general that
Therefore, the standard local methods used to investigate the existence of sign-changing solutions do not apply immediately to our nonlocal setting.
Our second main result in this regard is stated below.
The remaining of the paper is organized as follows: In Section 2, we collect some nonlocal versions of the Brezis–Lieb lemma, which will be crucial in investigating the groundstate solutions of (1.1). Further, Sections 3 and 4 contain the proofs of our main results.
2 Preliminary results
Let . There exists a constant such that for any we have
Lemma 2.2 ([2, Proposition 4.7.12]).
Let . Assume is a bounded sequence in that converges to w almost everywhere. Then weakly in .
Lemma 2.3 (Local Brezis–Lieb lemma).
Let . Assume is a bounded sequence in that converges to w almost everywhere. Then for every we have
Fix . Then there exists such that for all a, we have
Using (2.1), we obtain
Now using the Lebesgue dominated convergence theorem, we have
Therefore, we get
where . Further, letting , we conclude the proof. ∎
Lemma 2.4 (Nonlocal Brezis–Lieb lemma [17, Lemma 2.4]).
Let and . Assume is a bounded sequence in that converges almost everywhere to some . Then
For , we observe that
Using all the above arguments and passing to the limit in (2.2), we conclude the proof. ∎
Let and . Assume is a bounded sequence in that converges almost everywhere to u. Then for any we have
By using , it is enough to prove our lemma for . Denote and observe that
Apply Lemma 2.3 with and by taking and then , respectively. We find
Using now the Hardy–Littlewood–Sobolev inequality, we obtain
Also, by Lemma 2.2 we have
By Hölder’s inequality and the Hardy–Littlewood–Sobolev inequality, we have
On the other hand, by Lemma 2.2 we have weakly in , so
Thus, from (2.7) we have
3 Proof of Theorem 1.1
In this section, we discuss the existence of groundstate solutions to (1.1) under the assumption . For , we have
So, for we have
Since , the equation
has a unique positive solution , and the corresponding element is called the projection of u on . The following result presents the main properties of the Nehari manifold , which we use in this paper.
is bounded from below by a positive constant.
Any critical point u of is a free critical point.
(i) By using the continuous embeddings and together with the Hardy–Littlewood–Sobolev inequality, for any we have
Therefore, there exists such that
Using the above fact, we have
(ii) Let for . Now, for , from (3.1) we get
Assuming that is a critical point of and using the Lagrange multiplier theorem, there exists such that . In particular, . As , this implies , so . ∎
For all , we have
Also, consider the Nehari manifold associated with as
Let be a sequence of , that is, is bounded and strongly in . Then there exists a solution of (1.1) such that, by replacing with a subsequence, one of the following alternatives holds:
strongly in .
weakly in , and there exists a positive integer and k functions , which are nontrivial weak solutions to ( 1.2 ), and k sequences of points , such that the following conditions hold:
and if , ;
Since is bounded in , there exists such that, up to a subsequence, we have
Now, assume that does not converge strongly to u in , and set . Then converges weakly to zero in , and
By Lemma 2.4, we have
Further, for any , by Lemma 2.5 we have
From Lemma 2.4 we deduce that
We need the following auxiliary result.
Assume by contradiction . By Lemma 2.1, we deduce that strongly in . Then by the Hardy–Littlewood–Sobolev inequality we get
Using this fact together with (3.6), we get strongly in . This is a contradiction. Hence, . ∎
Now, we return to the proof of Lemma 3.2. Since , we may find such that
Consider the sequence . Then there exists such that, up to a subsequence, we have
Next, passing to the limit in (3.7), we get
Similarly to before, we have
Then, using Lemma 2.4, we deduce that
So, by (3.5) one can get
Using the above techniques, we also obtain
Now, if converges strongly to zero, then we finish the proof by taking in the statement of Lemma 3.2. If weakly and not strongly in , then we iterate the process. In k steps one could find a set of sequences , , with
and k nontrivial solutions of (1.2) such that, denoting
As is bounded and , we can iterate the process only a finite number of times, which concludes our proof. ∎
For , any sequence of is relatively compact.
In order to finish the proof of Theorem 1.1 we need the following result.
Let be a groundstate solution of (1.2); we know that such a groundstate exists, and for this we refer the reader to . Denote by tQ the projection of Q on , that is, is the unique real number such that . Set
As and , we get
From the above equalities one can easily deduce that . Therefore, we have
as desired. ∎
Further, using the Ekeland variational principle, for any there exists such that
4 Proof of Theorem 1.2
In this section, we discuss the existence of a least energy sign-changing solution of (1.1).
Assume and . Then for any and there exists a unique pair such that . Furthermore, if , then for all we have .
We shall follow an idea developed in . Denote
Let us define the function by
Note that Φ is strictly concave. Therefore, Φ has at most one maximum point. Also
and it is easy to check that
The energy level is achieved by some .
Let be a minimizing sequence for . Note that
where is a positive constant. Therefore, for some constant we have
which implies that is bounded in . So, and are also bounded in , and, by passing to a subsequence, there exists such that
Moreover, by the Hardy–Littlewood–Sobolev inequality, we estimate
Since , we can deduce
By Lemma 4.1, there exists a unique pair such that . By the weakly lower semi-continuity of the norm , we deduce that
Letting now , we finish the proof. ∎
is a critical point of , that is,
Assume by contradiction that v is not a critical point of . Then there exists such that
Since is continuous and differentiable, there exists small such that
Let D be the open disc in of radius centered at . We define a continuous function as
Further, we define a continuous map as
Since the mapping is continuous in , it follows that L is continuous. If , that is, if we are on the boundary of D, then by definition. Then and, using Lemma 4.1, we get
Therefore, the Brouwer degree is well defined and . Then there exists such that . Thus, and, using the definition of , we get
Using equation (4.5), we deduce that
If , then by definition and we deduce that
If , then, using Lemma 4.1, we have
which is a contradiction to equation (4.6). ∎
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About the article
Published Online: 2017-08-05
This work is part of the author’s PhD thesis and has been carried out with the financial support of the Research Demonstratorship Scheme offered by the School of Mathematics and Statistics, University College Dublin.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 694–706, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0126.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0