1 Introduction and Main Results
We are concerned with the fractional nonlinear Schrödinger–Poisson system
where and is the fractional Laplacian operator for . The fractional Schrödinger equation was introduced by Laskin  in the context of fractional quantum mechanics for the study of particles on stochastic fields modeled by Lévy processes. The operator can be seen as the infinitesimal generator of Lévy stable diffusion processes (see Applebaum ). If , then (1.1) reduces to the nonlinear fractional scalar field equation
This equation is related to the standing waves of the time-dependent fractional scalar field equation
which is a physically relevant generalization of the classical nonlinear Schrödinger equation. In fact, up to replacing with , the operators in the above equations converge to , in a suitable sense, due to the results in Bourgain, Brezis and Mironescu . Here, i is the imaginary unit and t denotes the time variable. For power-type nonlinearities, the fractional Schrödinger equation (1.3) was derived in  by replacing the Brownian motion in the path integral approach with the so-called Lévy flights (see, e.g., Metzler and Klafter ). Hence, the equation we want to study appears as a perturbation of a physically meaningful equation. Also, Frank and Lenzmann [21, 22] obtained deep results on the uniqueness and the non-degeneracy of ground states for (1.2) in the case when for subcritical p; see also Secchi and Squassina , where the soliton dynamics for (1.3) with an external potential was investigated. In , Giammetta studied the evolution equation associated with the one-dimensional system
In this case, the diffusion is fractional only in the Poisson equation. Our system is more general and contains this as a particular case. If , the equation
for a field (see also Fröhlich, Jonsson and Lenzmann ). The square root of the Laplacian also appears in the semi-relativistic Schrödinger–Poisson–Slater system (see Bellazzini, Ozawa and Visciglia  and also the model studied in D’Avenia, Siciliano and Squassina ).
Observe that if we formally take , then (1.1) reduces to the classical Schrödinger–Poisson system
which describes systems of identically charged particles interacting with each other in the case when magnetic effects can be neglected (see Benci and Fortunato ). In recent years, the Schrödinger–Poisson system (1.5) has been widely studied by many researchers. Here, we would like to cite some related results, for example, Cerami and Vaira  for positive solutions, Azzollini and Pomponio  for ground state solutions, D’Aprile and Wei  for semi-classical states, and Ianni  for sign-changing solutions. See also Ambrosetti  and the references therein. In , Azzollini, d’Avenia and Pomponio were concerned with (1.5) under the Berestycki–Lions conditions (H2)–(H4) with (see below). They proved that (1.5) admits a positive radial solution if small enough. For the critical case, we refer to  and to the recent work  by the authors of the present work.
1.1 Main Results
In this paper, we are mainly concerned with positive solutions of (1.1). First, we consider the subcritical case with the Berestycki–Lions conditions. More precisely, we assume the following hypotheses on g.
, where .
There exists such that .
Our first result is the following theorem.
The hypotheses (H2)–(H4) are the so-called Berestycki–Lions conditions, which were introduced in Berestycki and Lions  for the derivation of the ground state of (1.2) with . Under (H1)–(H4), Chang and Wang  proved the existence of ground state solutions to (1.2) for . The hypothesis (H1) is only used to get the better regularity of solutions to (1.2), which guarantees the Pohožaev identity. By the Pohožaev identity, (H4) is necessary.
The hypothesis is just used to guarantee that the Poisson equation makes sense, due to the fact that . For details, see Section 2 below.
In the variational approach to the study of elliptic problems, the Palais–Smale condition ((PS) condition for short) plays a crucial role. To verify the (PS) condition, the so-called Ambrosetti–Rabinowitz condition
has been frequently used in the literature. The main role of ((AR)) is to guarantee the boundedness of the (PS) sequence in some suitable Sobolev space. More recently, Pucci, Xiang and Zhang  considered fractional p-Laplacian equations of Schrödinger–Kirchhoff type
With the use of ((AR)), they established the existence of multiple solutions to (1.6) via the Ekeland variational principle and the mountain pass theorem. In fact, ((AR)) is a technical assumption. Many mathematicians have tried to remove or weaken it. In , Berestycki and Lions considered the autonomous scalar field equation. Without using ((AR)), they proved the existence of ground state solutions by the constraint variational method. However, it is not easy to use the idea in  in order to deal directly with non-autonomous problems. In , Jeanjean introduced a monotonicity trick to overcome the difficulty due to the lack of ((AR)) in the non-autonomous case. In , without ((AR)), the authors of the present work considered the existence and the concentration of positive solutions to (1.1) in the critical case for . It is natural to wonder if similar results can hold for the critical fractional case. This is just our second goal of the present paper. In the critical case, we assume the following hypotheses on g.
There exists and such that for all .
Our second result is the following theorem.
In the case , the hypotheses (H2)’–(H4)’ were introduced in Zhang and Zou  (see also Alves, Souto and Montenegro ) to obtain the ground state of the scalar field equation in . In , Shang and Zhang considered the fractional problem (1.2) in the critical case (see also Shang, Zhang and Yang ). With the help of the monotonicity of , the ground state solutions were obtained by using the Nehari approach. To the best of our knowledge, there are few results in the literature about the ground states of the critical fractional problem (1.2) with a general nonlinearity, particularly without the Ambrosetti–Rabinowitz condition and the monotonicity of . Theorem 1.4 seems to be the first result in this direction.
Without loss generality, from now on, we assume that .
We conclude by fixing some notation that we will use throughout the paper. We define the norm
and we let denote the Fourier transform of u.
In the rest of the paper, we use the perturbation approach to prove Theorem 1.1 and Theorem 1.4. Similar arguments can also be found in . The paper is organized as follows. In Section 2, we introduce the functional framework and some preliminary results. In Section 3, we construct the min-max level. In Section 4, we use a perturbation argument to complete the proof of Theorem 1.1 and we give the proof of Theorem 1.4.
2 Preliminaries and Functional Setting
2.1 Fractional-Order Sobolev Spaces
The fractional Laplacian with of a function is defined by
where is the Fourier transform, i.e.,
where i is the imaginary unit. If ϕ is smooth enough, it can be computed by the singular integral
where is a normalization constant and stands for the principal value.
For any , we consider the fractional-order Sobolev space
endowed with the norm
and with the inner product
It is easy to see that the inner products
on are equivalent (see ). The homogeneous Sobolev space is defined by
which is the completion of under the norm
and the inner product
For a further introduction on fractional-order Sobolev spaces, we refer the interested reader to Di Nezza, Palatucci and Valdinoci . Let
Now, we introduce the following Sobolev embedding theorems.
For any , is continuously embedded into for and compactly embedded into for . Moreover, is compactly embedded into for .
2.2 The Variational Setting
Then, for , by Lemma 2.2, the linear operator defined by
is well defined on and is continuous. Thus, it follows from the Lax–Milgram theorem that there exists a unique such that . Moreover, for , we have
where we have set
We define the energy functional by
Obviously, the critical points of are the weak solutions of (2.2).
we summarize some properties of and which will be used later.
If and , then, for any , the following hold.
is continuous and maps bounded sets into bounded sets.
, , and for some .
for any and .
If weakly in , then weakly in .
If weakly in , then .
If u is a radial function, so is .
The proof is similar to that in , so we omit the details here. ∎
3 The Subcritical Case
3.1 The Modified Problem
It follows from Lemma 2.4 that is well defined on and is of class . Since we are concerned with positive solutions of (2.2), similarly to  (see also ), we modify our problem first. Without loss of generality, we assume that
where ξ is given in (H4). Let
and define a function by
and for . If is a solution of (2.2), where g is replaced by , then, by the maximum principle (see Cabré and Sire ), we get that u is positive and for any , i.e., u is a solution of the original problem (2.2) with g. Thus, from now on, we can replace g by , but still use the same notation g. In addition, for , let
Then, we have for ,
and, for any , there exists such that
3.2 The Limit Problem
In , Chang and Wang proved that, with the same assumptions on g as in Theorem 1.1, there exists a positive ground state solution of (3.4). Moreover, each such solution U of (3.4) satisfies the Pohožaev identity
Under the assumptions in Theorem 1.1, S is compact in .
As shown in Cho and Ozawa , for general , we do not have a similar radial lemma in . So the Strauss compactness lemma (see ) is not applicable here. Before we prove Proposition 3.1, we begin with the following compactness lemma which is a special case of [12, Lemma 2.4.]
(Chang and Wang )
Assume that satisfies
and there exists a bounded sequence for some with
Then, up to a subsequence, we have strongly in as .
Proof of Proposition 3.1.
Obviously, is bounded. It follows from Lemma 2.2 that is bounded. By (3.3), as we can see in , is bounded, which yields that is bounded in . Without loss of generality, we can assume that there exists such that weakly in , strongly in for , and a.e. .
In the following, we adopt some ideas from  to prove that strongly in . For , let
Then, we know that is a minimizer of the constrained minimizing problem
Then, by Fatou’s Lemma,
which implies that . Meanwhile, it is easy to see that . Similarly to , we know that satisfies
which yields that
By Fatou’s Lemma, we know that as . Thus, strongly in . ∎
3.3 The Min-Max Level
Take and let
Then, by the definition of , we know that and
By the Pohožaev identity, we have
Thus, there exists such that for . Set
By virtue of Lemma 2.4, we have . Note that since , we get that as .
Moreover, similarly to , we can prove the following lemma, which is crucial in defining the uniformly bounded set of the mountain paths (see below).
There exist and such that, for any , we have
Now, for any , we define a min-max value as
Obviously, for , we have
Then, we can define so . Moreover, we have
We have .
It suffices to prove that
Now, we give the mountain pass value
It follows from [12, Lemma 3.2] that L satisfies the mountain pass geometry. As we can see in Jeanjean and Tanaka , b agrees with the least energy level of (3.4), i.e., . Note that for . Then, for any and it follows that , which concludes the proof. ∎
3.4 Proof of Theorem 1.1
Now, for , define
In the following, we will find a solution of (2.2) for sufficiently small and some . The following proposition is crucial for obtaining a suitable (PS) sequence for and plays a key role in our proof.
Let be such that and with
Then, for d small enough, there is , up to a subsequence, such that in .
For convenience, we write λ for . Since and S is compact, we know that is bounded in . Then, by Lemma 2.4, we see that
It follows from [12, Lemma 3.3] that there is , up to a subsequence, such that strongly in . Obviously, for d small. This implies that , , and . Thus, , i.e., , which completes the proof. ∎
By Proposition 3.5, for small , there exist , such that
Similarly to , we have the following proposition.
There exists such that, for small ,
where for .
From Lemma 2.4 and the Pohožaev identity, we have
and the conclusion follows. ∎
For small enough, there exists such that as .
Proof of Theorem 1.1.
It follows from Proposition 3.7 that there exists such that, for , there exists with as . Noting that S is compact in , we get that is bounded in . Assume that weakly in . Then . It follows from the compactness of S that and for n large. So, for small . By Lemma 2.4, we have
it follows from (3.3) that, for some ,
4 The Critical Case
In this section, we consider the Schrödinger–Poisson system (1.1) in the critical case. First, we establish the existence of ground state solutions to the fractional scalar field equation (1.2) with a general critical nonlinear term. Then, by a perturbation argument, we seek solutions of (1.1) in some neighborhood of the ground states of (1.2).
4.1 The Limit Problem
We recall that U is called a ground state solution of (1.2) if and only if , where
The existence of ground states is reduced to looking at the constraint minimization problem
and eventually removing the Lagrange multiplier by some appropriate scaling. Now, we state the main result in this subsection.
and g is odd, i.e., for .
Then, (1.2) admits a positive ground state solution.
Moreover, similarly to Theorem 4.1, a similar result in for can be also obtained.
Proof of Theorem 4.1.
The proof follows the lines of that in . For completeness, we give the details here.
for any , where , are fixed constants. Let be a cut-off function with support such that on and on , where . Let . From , it follows that
by we have
In the following, we will show that
By , we know that . Then, it is easy to see that there exist such that
Then, we obtain that
Noting that , it is easy to verify that (4.3) is true. Thus, it follows that for small . By a scaling, we get that .
Next, obviously, . For small , we have and
If , then for all . From (4.3), it follows that
for small , which yields . Then,
Step 2. Here, we show that M can be achieved. Noting that g is odd and using the fractional Pólya–Szegő inequality (see Park ), without loss of generality, we can assume that there exists a positive minimizing sequence such that and as . By Lemma 2.2, it is easy to see that is bounded in . By Lemma 2.1 we can assume that weakly in , strongly in , and a.e. in . Setting , we have and
where as . Letting , it follows from Lemma 3.2 that
Next, we prove that is the minimizer for M. Setting , , , and , we have and . Under a scale change, we get that
which is a contradiction. On the other hand, if , then , which implies that . Then,
which is again a contradiction. Then, we conclude that , i.e., M is achieved by .
Finally, letting , where
we have that U is a ground state solution of (1.2). ∎
Furthermore, similarly to Chang and Wang , if we additionally assume that , then U satisfies the Pohožaev identity
Under the assumptions of Theorem 4.1, is compact in .
4.2 Proof of Theorem 1.4
Then, there exists such that for . Setting
there exist and such that, for any ,
Then, for any , we define a min-max value as
Similarly to Section 3, we have the following proposition.
We have , where m is the least energy of (1.2).
Now for , define
Similarly to Section 3, for small and some , we will find a solution of (2.2) in the critical case. Also, similarly to , we can get the following compactness result, which can yield the gradient estimate of .
Let be such that and with
Then, for d small enough, there is , up to a subsequence, such that in .
For convenience, we write λ for . Since and is compact, we know that is bounded in . Moreover, up to a subsequence, there exists such that weakly in , a.e. in , and for i large. Then, by Lemma 2.4, we see that
Then . Obviously, if d small. So, . Meanwhile, thanks to Lemma 3.2, we have
Then, by Lemma 2.2, for d small enough, strongly in . ∎
By Proposition 4.6, for small , there exist , such that
Similarly to Section 3, we have the following proposition.
There exists such that, for small ,
where for .
Proof of Theorem 1.4.
With the help of (4.5) and Proposition 4.7, similarly to , for small enough, there exists such that as . As above, there exists with for small . Moreover, up to a subsequence, weakly in , a.e. in , and for n large. Furthermore, . By Lemma 2.4, we have
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About the article
Published Online: 2016-01-27
Published in Print: 2016-02-01
Research partially supported by INCTmat/MCT/Brazil. The first author was partially supported by CAPES/Brazil and CPSF (grant no. 2013M530868). The second author was supported by CNPq and CAPES/Brazil.