We study the following nonlinear Choquard equation:
where , , V is a continuous real function and F is the primitive function of f. Under some suitable assumptions on the potential V, which include the case , that is, as , we prove the existence of a nontrivial solution for the above equation by the penalization method.
1 Introduction and main results
In this paper, we study the existence of nontrivial solutions for the following nonlinear Choquard equation:
where , , Δ is the Laplacian operator, V is a nonnegative continuous real function and F is the primitive function of f.
This problem arises when one looks for standing wave solutions for a nonlinear Schrödinger equation of the form
Here W is the external potential and Q is the response function that possesses information on the mutual interaction between the bosons. This type of nonlocal equation is known to describe the propagation of electromagnetic waves in plasmas  and plays an important role in the theory of Bose–Einstein condensation (see ). It is clear that solves the evolution equation (1.1) if and only if u solves
When the response function is the Dirac function, i.e., , the nonlinear response is local and we are lead to the Schrödinger equation
The aim of this paper is to study the existence of solutions for a class of nonlocal Schrödinger equation, that is, the response function Q in (SNE) is of Coulomb type, for example , then we arrive at the Choquard–Pekar equation,
The case and goes back to the description of the quantum theory of a polaron at rest by Pekar in 1954  and the modeling of an electron trapped in its own hole (in the work of Choquard in 1976), in a certain approximation to Hartree–Fock theory of one-component plasma . The equation is also known as the Schrödinger–Newton equation, which was introduced by Penrose in his discussion on the selfgravitational collapse of a quantum mechanical wave-function. Penrose suggested that the solutions of (1.4), up to reparametrization, are the basic stationary states which do not spontaneously collapse any further, within a certain time scale.
As far as we know, most of the existing papers consider the existence and properties of the solutions for the nonlinear Choquard equation (SNE) with constant potential. In , Lieb proved the existence and uniqueness, up to translations, of the ground state to equation (1.4). Later, in , Lions showed the existence of a sequence of radially symmetric solutions to this equation. Involving the properties of the ground state solutions, Ma and Zhao  considered the generalized Choquard equation (1.4) for , and they proved that every positive solution of (1.4) is radially symmetric and monotone decreasing about some point, under the assumption that a certain set of real numbers, defined in terms of and q, is nonempty. Under the same assumption, Cingolani, Clapp and Secchi  gave some existence and multiplicity results in the electromagnetic case, and established the regularity and asymptotic decay at infinity of the ground states of (1.4). In , Moroz and Van Schaftingen eliminated this restriction and showed the regularity, positivity and radial symmetry of the ground states for the optimal range of parameters, and derived an asymptotic decay at infinity for them as well. Moreover, Moroz and Van Schaftingen in [28, 30] also considered the existence of ground states for critical growth in the sense of the Hardy–Littlewood–Sobolev inequality and under an assumption of Berestycki–Lions type.
Involving the problem with nonconstant potentials, that is,
where V is a continuous periodic function with , noticing that the nonlocal term is invariant under translation, it is possible to prove easily an existence result by applying the mountain pass theorem, see , for example. For a periodic potential V that changes sign and 0 lies in the gap of the spectrum of the Schrödinger operator , the problem is strongly indefinite, and in , the authors proved the existence of nontrivial solution with and by reduction methods. For a general class of response function Q and nonlinearity f, Ackermann  proposed an approach to prove the existence of infinitely many geometrically distinct weak solutions.
As far as we know, the only papers that considered the generalized Choquard equation with vanishing potential are [29, 33]. Replacing Δ by , Secchi  obtained the existence result for small ε when and for some by a Lyapunov–Schmidt type reduction. In , the authors prove the existence and concentration behavior of the semiclassical states for the problem with for small ε. There, they developed a penalization technique by constructing supersolutions to a linearization of the penalized problem in an outer domain, and then estimated the solutions of the penalized problem by a comparison principle. Besides these two papers, in our mind, for constant and general nonlinearities, the existence of nontrivial solution for the Choquard equation with vanishing potential is still not known.
Motivated by the above facts and a recent paper due to Alves and Souto , in the present paper we intend to study the Choquard equation (SNE) with potentials vanishing at infinity, that is, . This class of problems goes back to the work by Berestycki and Lions in , where the authors studied the Schrödinger equation with zero mass and showed that the problem
has no ground state solution if . However, they also proved that if f behaves like for small t, and like for large t when , then the problem has a ground state solution. For example, these conditions are verified by the function
and studied the existence and nonexistence of ground state solutions.
Motivated by the above papers and their hypotheses, we assume the following conditions on the nonlinearity f:
. Assume that and
for some . Note that interval is not empty because of the above condition on μ. Note that by (f1), and since , we have that
The nonlinearity f is also supposed to verify the Ambrosetti–Rabinowitz type superlinear condition for the nonlocal problem, that is, there exists such that
where . Without loss of generality, we will assume that in the rest of the paper.
Related to the potential , we assume that it is a nonnegative continuous function, and we fix the following notations:
and we define the function by
One of the main results of this paper is the following existence result.
where is a continuous, which makes V continuous. Then, we know that
If the potential V is a radial function, that is,
then we define the function by
and we can apply the arguments in Theorem 1.1 to treat the homogeneous case with nonlinearity
Involving this situation, we have the following existence result.
Assume that , V is a radial function and condition holds. There exists a constant such that if for some , then problem (SNE) has a positive solution.
As an application, we can use Theorem 1.3 to study
where V is a potential as below.
We have that
and so for R large enough,
where is a continuous, which makes V continuous. For the above potential, we have that
and so for all .
We now briefly outline the organization of the contents of this paper. In Section 2, we introduce an auxiliary problem, by using a penalization method due to del Pino and Felmer . Section 3 is devoted to the proof of Theorem 1.1. Finally, in Section 4, we consider the case where the potential V is a radial function and the nonlinearity is homogeneous. By adjusting some arguments in the proof of Theorem 1.1, we can prove the result in Theorem 1.3.
We fix the following notations, which will be used from now on:
C, denote positive constants.
denote the open ball centered at the origin with radius .
, , denotes the Lebesgue space with the norm
If u is a mensurable function, we denote by and its positive and negative part respectively, i.e.,
denotes the space of infinitely differentiable functions with compact support in .
We denote by the Hilbert space
endowed with the inner product
and the norm
S is the best constant that verifies
From the assumptions on V, it follows that the subspace
is a Hilbert space with its norm defined by
Let E be a real Hilbert space and be a functional of class . We say that is a Palais–Smale sequence at level c for I, sequence for short, if satisfies
Furthermore, we say that I satisfies the condition, if any sequence possesses a convergent subsequence.
2 The penalized problem
We would like to make some comments on the assumptions involving the nonlinearity f. The first point is that since we are looking for positive solutions, we may suppose that
The second one is that due to the application of variational methods, we must have
To observe the above integrability, it is important to recall the Hardy–Littlewood–Sobolev inequality, which will be frequently used in the paper.
Proposition 2.1 (The Hardy–Littlewood–Sobolev inequality [23, Theorem 4.3])
Let and with . If and , then there exists a sharp constant , independent of , such that
Thereby, by the Hardy–Littlewood–Sobolev inequality,
if for and
However, by (2.2),
shows that for all .
From the above commentaries, the Euler–Lagrange functional , associated to (SNE) and given by
is well defined and belongs to with its derivative given by
Thus, it is easy to see that the solutions of (SNE) correspond to critical points of the energy functional I. However, because I does not verify in general the condition, there are some difficulties to prove the existence of nontrivial critical points for it.
In order to overcome the lack of compactness of I, we adapt the penalization method introduced by del Pino and Felmer in . For and to be determined later, we set the functions
where denotes the characteristic function of the ball . Using the previous notations, let us introduce the auxiliary problem
where . A direct computation shows that, for all , the following hold:
The Euler–Lagrange functional associated to (APE) is given by
Therefore, it is easy to see that the solutions of (APE) correspond to the critical points of the energy functional Φ.
Assume that condition (f3) holds. Then,
For all , a direct computation gives
Since , the conclusion follows. ∎
Next, we check that Φ verifies the mountain pass geometry.
There exist such that for all .
There exist and with such that .
(1) By (1.7),
from which it follows that
and so, by the Hardy–Littlewood–Sobolev inequality,
Since , assertion (1) follows if we choose ρ small enough.
(2) Fixing with , we set
By direct calculus,
Hence, integrating the above inequality over with , we find
and assertion (2) follows for with s large enough. ∎
Applying the mountain pass theorem without the condition , we know that there exists a sequence such that
where is the mountain pass level characterized by
Hence, from the proof of Lemma 2.4,
and m is given in (1.8). Here, it is important to observe that d is independent of the choice of and R.
By Lemma 2.3,
which means is bounded in E. Moreover, we may assume that
which shows the lemma, because d is independent of the choice of and R. ∎
Before proving the next lemma, we need to fix some notations. In what follows,
With the above notations, we are able to show the ensuing estimate.
From the definition of G,
Choosing , it follows from Hölder’s inequality that
Once , there exists such that
From this, there exists verifying
From now on, we take and consider the penalized problem with the nonlinearity defined in (2.3).
From Lemma 2.5,
where d is independent of the choice of and R. Therefore, we can assume that there exists such that in E. Thus, for each , there exists such that
where is the volume of the unitary ball in .
Let be such that if with and . Note that
Since is bounded in E, it follows that . Moreover, recalling that in , we obtain
From Lemma 2.6, we have
By Hölder’s inequality,
Since in and is bounded in E, it follows that
Using Hölder’s inequality again, we get
Since in E, and , it follows that
Now, our goal is to show that the following limit holds:
We begin with recalling that by Lemma 2.6, there exists such that
Recall (1.7) and notice that with for each . Then, the Sobolev compact embedding theorem implies
From (1.7), it also holds that , and so
Combining Lemma 2.7 with the Sobolev embedding theorem we have that given , there exists such that
Similarly, applying Hölder’s inequality, we can also prove that
Let u be the solution obtained in Theorem 2.9. Then, there exists a constant , which depends only on , such that
By hypothesis, u is a solution of
where and . Since for all and
it follows that
where . By a Trudinger–Brézis–Kato iteration argument, see Struwe’s book [34, Lemma B3, p. 273], we deduce that for all . Moreover, once does not depend on , the norms also do not depend on . Now, fixing s large enough, the bootstrap arguments implies that there exists , which is independent of R, such that . ∎
3 Proof of Theorem 1.1
Let v be the harmonic function
By Lemma 2.10, we have the inequality
which implies that the function
belongs to . Since in , on and , it follows that
showing that , i.e., in , and thus
Using the inequality , we have
Now, fix such that . Then, the last inequality combined with definition (1.9) gives
Thus, setting the number
and if there is such that
then we obtain the desired result for the above fixed, that is,
4 Proof of Theorem 1.3
In the proof of Theorem 1.3, we replace the space by and consider
In this case, due to the Hardy–Littlewood–Sobolev inequality, the energy function
is well defined and belongs to . Now, repeating the same ideas of the previous sections, we can consider again problem (APE). Here, the reader is invited to check that the functional Φ still verifies, with natural modifications, Lemmas 2.3, 2.4 and 2.5.
Proof of Theorem 1.3.
From Lemmas 2.3, 2.4, 2.5, and Theorem 2.9, (APE) has a positive solution . Thereby, in order to prove that it is indeed a solution of problem (SNE), we must show that there exists such that satisfies the inequality
We begin by recalling that
from where it follows that
where . Then,
Since , we have
Using the definition of , it follows that
and assuming that there exists such that
then for the above , we can ensure that
implying that in . Now, using the principle of symmetric criticality due to Palais , we can conclude that in E, thus finishing the proof. ∎
Funding source: Conselho Nacional de Desenvolvimento Científico e Tecnológico
Award Identifier / Grant number: 301807/2013-2
Award Identifier / Grant number: 301292/2011-9
Award Identifier / Grant number: 552101/2011-7
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11571317
Award Identifier / Grant number: 11101374
Award Identifier / Grant number: 11271331
Funding statement: C. O. Alves was partially supported by CNPq/Brazil 301807/2013-2. G. M. Figueiredo was partially supported by CNPq/Brazil 301292/2011-9 and 552101/2011-7. M. Yang was supported by NSFC (11571317, 11101374, 11271331) and ZJNSF (LY15A010010)
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