Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity

1 Introduction and main results

In this paper, we study the existence of nontrivial solutions for the following nonlinear Choquard equation:

where 0<μ<N, N≥3, 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 [9] and plays an important role in the theory of Bose–Einstein condensation (see [15]). It is clear that Ψ⁢(x,t)=u⁢(x)⁢e-i⁢E⁢t solves the evolution equation (1.1) if and only if u solves

with V⁢(x)=W⁢(x)-E.

When the response function is the Dirac function, i.e., Q⁢(x)=δ⁢(x), the nonlinear response is local and we are lead to the Schrödinger equation

which has been studied extensively under various hypotheses on the potentials and the nonlinearities, see [3, 2, 4, 5, 10, 13, 8, 17, 16, 19, 18, 20, 21, 25].

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 1|x|μ, then we arrive at the Choquard–Pekar equation,

The case q=2 and μ=1 goes back to the description of the quantum theory of a polaron at rest by Pekar in 1954 [32] 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 [22]. 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 [22], Lieb proved the existence and uniqueness, up to translations, of the ground state to equation (1.4). Later, in [24], 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 [26] considered the generalized Choquard equation (1.4) for q≥2, 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 N,α and q, is nonempty. Under the same assumption, Cingolani, Clapp and Secchi [14] 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 [27], 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 infℝN⁡V⁢(x)>0, 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 [1], for example. For a periodic potential V that changes sign and 0 lies in the gap of the spectrum of the Schrödinger operator -Δ+V, the problem is strongly indefinite, and in [12], the authors proved the existence of nontrivial solution with μ=1 and F⁢(u)=u2 by reduction methods. For a general class of response function Q and nonlinearity f, Ackermann [1] 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 ε2⁢Δ, Secchi [33] obtained the existence result for small ε when V>0 and lim inf|x|→∞⁡V⁢(x)⁢|x|γ>0 for some γ∈[0,1) by a Lyapunov–Schmidt type reduction. In [29], the authors prove the existence and concentration behavior of the semiclassical states for the problem with F⁢(u)=up 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 ε=1 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 [2], in the present paper we intend to study the Choquard equation (SNE) with potentials vanishing at infinity, that is, V⁢(∞)=0. This class of problems goes back to the work by Berestycki and Lions in [8], where the authors studied the Schrödinger equation with zero mass and showed that the problem

has no ground state solution if f⁢(t)=|t|p-2⁢t. However, they also proved that if f behaves like |t|q-2⁢t for small t, and like |t|p-2⁢t for large t when p<2*<q, then the problem has a ground state solution. For example, these conditions are verified by the function

where h is selected so that f is a C1⁢(ℝ) function. Later Benci, Grisanti and Micheletti [7, 6] considered problems of the type

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:

q≥2*=2⁢NN-2. Assume that 0<μ<N+22 and

for some p∈(1,2⁢(N-μ)N-2). Note that interval (1,2⁢(N-μ)N-2) is not empty because of the above condition on μ. Note that by (f1), and since 2⁢(N-μ)N-2<2⁢N-μN-2<2*, we have that

Thus, by (f1)–(f2) and (1.6), there exists c0>0 such that

The nonlinearity f is also supposed to verify the Ambrosetti–Rabinowitz type superlinear condition for the nonlocal problem, that is, there exists θ>2 such that

where F⁢(t)=∫0tf⁢(s)⁢𝑑s. Without loss of generality, we will assume that θ<4 in the rest of the paper.

Related to the potential V⁢(x), we assume that it is a nonnegative continuous function, and we fix the following notations:

and we define the function 𝒱:(1,+∞)→[0,∞) by

One of the main results of this paper is the following existence result.

The constant 𝒱0 is a technical constant, which will appear in the proof of the main result, see Section 3. Next, we will introduce an example of potential V for Theorem 1.1.

If the potential V is a radial function, that is,

then we define the function 𝒲:(1,+∞)→[0,∞) by

and we can apply the arguments in Theorem 1.1 to treat the homogeneous case with nonlinearity

where 0<μ<min⁡{N+22,4}.

Involving this situation, we have the following existence result.

As an application, we can use Theorem 1.3 to study

where V is a potential as below.

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 [16]. 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.

2 The penalized problem

In this section, adapting some arguments introduced by del Pino and Felmer in [16], we are able to obtain the existence of nontrivial solution for (SNE) by studying an auxiliary 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.

The above inequality guarantees that (2.1) holds, because by (f3) and (1.7),

Thereby, by the Hardy–Littlewood–Sobolev inequality,

if F⁢(u)∈Lt⁢(ℝN) for t>1 and

that is,

However, by (2.2),

shows that F⁢(u)∈Lt⁢(ℝN) for all u∈E.

From the above commentaries, the Euler–Lagrange functional I:E→ℝ, associated to (SNE) and given by

is well defined and belongs to 𝒞1⁢(E,ℝ) 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 (PS) 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 [16]. For ℓ>1 and R>1 to be determined later, we set the functions

where 𝒳BR denotes the characteristic function of the ball BR. Using the previous notations, let us introduce the auxiliary problem

where G⁢(x,s):=∫0sg⁢(x,τ)⁢𝑑τ. A direct computation shows that, for all s∈ℝ, the following hold:

The Euler–Lagrange functional associated to (APE) is given by

where

From (1.7) and (2.4), we know that Φ is well defined and belongs to 𝒞1⁢(E,ℝ) with its derivative given by

Therefore, it is easy to see that the solutions of (APE) correspond to the critical points of the energy functional Φ.

Next, we check that Φ verifies the mountain pass geometry.

Applying the mountain pass theorem without the (PS) condition [35], we know that there exists a (PS)cV sequence (un)⊂E such that

where cV is the mountain pass level characterized by

with

Hence, from the proof of Lemma 2.4,

where

and m is given in (1.8). Here, it is important to observe that d is independent of the choice of ℓ and R.

Before proving the next lemma, we need to fix some notations. In what follows,

and

With the above notations, we are able to show the ensuing estimate.

From now on, we take ℓ≥ℓ0 and consider the penalized problem with the nonlinearity defined in (2.3).