Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions

1 Introduction

In this paper, we consider the following singularly perturbed nonlinear Choquard equation:

where ε > 0 is a parameter, N ≥ 3, α ∈ (0, N) and I_α : ℝ^N → ℝ is the Riesz potential defined by

F(t) = ∈ ∫0t f(s)ds, V : ℝ^N → ℝ and f : ℝ → ℝ satisfy the following basic assumptions:

1. V ∈ 𝓒(ℝ^N, [0, ∞)) and V_∞ := lim_|x|→∞V(x) > 0;

2. f ∈ 𝓒(ℝ, ℝ) and there exists a constant 𝓒₀ > 0 such that

   |f(t)t|≤C0|t|(N+α)/N+|t|(N+α)/(N−2),    ∀ t∈R;

3. F(t) = o(t^(N+α)/N) as t → 0 and F(t) = o(t^{(N+α)/(N–2)}) as |t| → ∞;

4. there exists s₀ > 0 such that F(s₀) ≠ 0.

Note that (F1)-(F3) were almost necessary and sufficient conditions and regarded as the Berestycki-Lions type conditions to Choquard equations, which were introduced by Moroz and Van Schaftingen in [22] for the study of (1.1) with ε = 1.

In recent years, semiclassical problems like (1.1), i.e. the parameter ε goes to zero, have received attention from the mathematical community. For small ε > 0, bound states are called semiclassical states, which describe a kind of transition from Quantum Mechanics to Newtonian Mechanics. There are some nice work on semiclassical states for (1.1). For example, for a special form of (1.1) with N = 3, α = 2 and F(s) = s²/2, by proving the uniqueness and non-degeneracy, of the ground states for the limit problem, Wei and Winter [37] constructed a family of solutions by a Lyapunov-Schmidt type reduction; Cingolani et al. [9] proved the existence of solutions concentrating around several minimum points of V by a global penalization method. Moroz and Van Schaftingen [24] developed a nonlocal penalization technique to show that problem (1.1) with F(s) = |s|^p/p and p ≥ 2 has a family of solutions concentrating at the local minimum of V provided V satisfies some additional assumptions at infinity. However, for (1.1) with general nonlinearity F which only satisfies (F1)-(F3), there seem to be no results in the existing literature. One of main purpose of this paper is to deal with this case.

When ε = 1, (1.1) reduces to the nonlinear Choquard equation of the form:

which has been extensively studied by using variational methods, see [1, 2, 3, 8, 17, 20, 21, 22, 23, 24, 25, 29, 36] and references therein. In view of (F1), (F2) and Hardy-Littlewood-Sobolev inequality, for some p ∈ (2, 2^*) and any ϵ > 0, one has

It is standard to check using (1.3) that under (V1), (F1) and (F2), the energy functional defined in H¹(ℝ^N) by

is continuously differentiable and its critical points correspond to the weak solutions of (1.2).

If the potential V(x) ≡ V_∞, then (1.2) reduces to the following autonomous form:

its energy functional is as follows:

Problem (1.5) is a semilinear elliptic equation with a nonlocal nonlinearity. For N = 3, α = 2, V_∞ = 1 and F(t) = t²/2, it covers in particular the Choquard-Pekar equation

introduced by Pekar [27] at least in 1954, describing the quantum mechanics of a polaron at rest. In 1976, Choquard [16] used (1.7) to describe an electron trapped in its own hole. In 1996, Penrose [19] proposed (1.7) as a model of self-gravitating matter. In this context (1.7) is usually called the nonlinear Schrödinger-Newton equation, see Moroz-Schaftingen [22].

If we let α → 0 in (1.5), then we can get the following limiting problem:

where g = Ff. In the fundamental paper [4], Berestycki-Lions proved that (1.8) has a radially symmetric positive solution provided that g satisfies the following assumptions:

1. g ∈ 𝓒(ℝ, ℝ) is odd and there exists a constant 𝓒₀ > 0 such that

   |g(t)|≤C01+|t|(N+2)/(N−2),∀ t∈R;

2. g(t) = o(t) as t → 0 and g(t) = o(t^{(N+2)/(N–2)}) as t → +∞;

3. there exists s₀ > 0 such that G(s0)>12V∞s02, where G(t) = ∫0t g(s)ds.

To prove the above result, Berestycki-Lions [4] considered the following constrained minimization problem

where

they first showed that by the Pólya-Szegö inequality for the Schwarz symmetrization, the minimum can be taken on radial and radially nonincreasing functions. Then they showed the existence of a minimum ŵ ∈ H¹(ℝ^N) by the direct method of the calculus of variations. With the Lagrange multiplier Theorem, they concluded that ū(x) := ŵ(x/t_ŵ) with tw^=N−22N∥∇w^∥2 is a least energy solution of (1.8). By noting the one-to-one correspondence between 𝓢 and

Jeanjean-Tanaka [13] proved that ū minimizes the value of the energy functional on the Pohožaev manifold for (1.8).

However, the approach of Berestycki-Lions [4] fails for nonlocal problem (1.5) due to the appearance of the nonlocal term. In [22], Moroz-Van Schaftingen proved firstly the existence of a least energy solution to (1.5) under (F1)-(F3). To do that, they employed a scaling technique introduced by Jeanjean [11] to construct a Palais-Smale sequence ((PS)-sequence in short) that satisfies asymptotically the Pohožaev identity (a Pohožaev-Palais-Smale sequence in short), where the information related to the Pohožaev identity helps to ensure the boundedness of (PS)-sequences, and then used a concentration compactness argument to overcome the difficulty caused by lack of Sobolev embeddings. Such an approach could be useful for the study of other problems where radial symmetry of solutions either fails or is not readily available. For more related results on nonlocal problems, we refer to [6, 17, 18, 26, 31, 38].

We would like to point out that the approach used in [22] is only valid for autonomous equations, it does not work any more for nonautonomous equation (1.2) with V ≠ constant, since one could not construct a Pohožaev-Palais-Smale sequence as Moroz-Van Schaftingen did in [22]. Thus new techniques are required for the study of the nonautonomous equation (1.2) with f satisfying (F1)-(F3), which is another focus of this paper.

In view of [22, Theorem 3], every solution u of (1.5) satisfies the following Pohožaev type identity:

Therefore, the following set

is a natural constraint for the functional 𝓘^∞. Moreover, the least energy solution u₀ obtained in [22] satisfies 𝓘^∞(u₀) ≥ inf_𝓜^∞𝓘^∞. A natural question is whether there exists a solution ū ∈ 𝓜^∞ such that

In the first part of this paper, motivated by [4, 7, 13, 22, 33, 35], we shall develop a more direct approach to obtain a ground state solution for (1.2) which has minimal “energy” 𝓘 in the set of all nontrivial solutions, moreover, this solution also minimizes the value of 𝓘 on the Pohožaev manifold associated with (1.2), under (F1)-(F3), (V1) and the following two additional conditions on V:

1. V(x) ≤ V_∞ for all x ∈ ℝ^N;

2. V ∈ 𝓒¹(ℝ^N, ℝ) and there exists θ ∈ [0, 1) such that t↦NV(tx)+∇V(tx)⋅(tx)tα+(N−2)3θ4tα+2|x|2 is nonincreasing on (0, ∞) for all x ∈ ℝ^N ∖ {0}.

To state our first result, we define a functional on H¹(ℝ^N) as follows:

which is associated with the Pohožaev identity 𝓟(u) = 0 of (1.2), see Lemma 3.2. Let

Our first result is as follows.

With the help of the Pohožaev type identity (1.11) established in [22], we easily prove that the solution ū obtained in Corollary 1.2 is also the least energy solution for (1.5). More precisely, we have the following theorem:

In the second part of this paper, we are interested in the existence of the least energy solutions for (1.2) under (F1)-(F3). In this case, we can replace (V3) by the following weaker decay assumption on ∇V:

(V4) V ∈ 𝓒¹(ℝ^N, ℝ) and there exist θ′ ∈ (0, 1) and R̄ ≥ 0 such that

In this direction, we have the following theorem.

Applying Theorem 1.6 to the following perturbed problem:

where V_∞ is a positive constant and the function h ∈ 𝓒¹(ℝ^N, ℝ) verifies:

1. h(x) ≥ 0 for all x ∈ ℝ^N and lim_|x|→∞ h(x) = 0;

2. sup_x∈ℝ^N[–∇h(x) ⋅ x] < ∞.

   Then we have the following corollary.

In the last part of the present paper, we consider the singularly perturbed nonlinear Choquard equation (1.1), and prove the existence of semiclassical ground state solutions for (1.1) under weaker assumptions on V:

1. 0 < V(x₀) := min_x∈ℝ^N V(x) < V_∞ for some x₀ ∈ ℝ^N;

2. V ∈ 𝓒¹(ℝ^N, ℝ) and there exists θ″ ∈ (0, 1) such that

   ∇V(x)⋅x≤θ″αV(x),∀ x∈RN.

Condition (V5) was introduced by Rabinowitz in [28]. Our last result is as follows.

To prove Theorem 1.1, we shall divide our arguments into three steps: i). Choosing a minimizing sequence {u_n} of 𝓘 on 𝓜, which satisfies

Then showing that {u_n} is bounded in H¹(ℝ^N). ii). With a concentration-compactness argument and “the least energy squeeze approach”, showing that {u_n} converges weakly to some ū ∈ H¹(ℝ^N) ∖ {0}. And then showing that ū ∈ 𝓜 and 𝓘(ū) = inf_𝓜𝓘. iii). Showing that ū is a critical point of 𝓘. Of them, Step ii) is the most difficult due to lack of global compactness and adequate information on 𝓘′(u_n). To avoid relying radial compactness, we establish a crucial inequality related to 𝓘(u), 𝓘(u_t) and 𝓟(u) (Lemma 2.2), it plays a crucial role in our arguments, see Lemmas 2.7, 2.11, 2.13, 3.5, 4.2. With the help of this inequality, we then can complete Step ii) by using Lions’ concentration compactness, the least energy squeeze approach and some subtle analysis. Moreover, such an approach could be useful for the study of other problems where radial symmetry of bounded sequence either fails or is not readily available.

Classically, in order to show the existence of solutions for (1.2), one compares the critical level with the one of (1.5) (i.e. the problem at infinity). To this end, it is necessary to establish a strict inequality similar to

for some path y₀ ∈ 𝓒([0, 1], H¹(ℝ^N)). Clearly, y₀(t) > 0 is a natural requirement under (V1), which usually involves an additional assumption on f besides (F1)-(F3), such as f(t) is odd and f(t)t ≥ 0, see [22, Theorem 1.4]. We would like to point out that the above strict inequality is not used in our arguments, see Section 2. Our approach could be useful for the study of other problems where paths or the ground state solutions of the problem at infinity are not sign definite.

To prove Theorem 1.6, as in Jeanjean-Tanaka [13], for λ ∈ [1/2, 1] we consider the family of functionals 𝓘_λ : H¹(ℝ^N) → ℝ defined by

These functionals have a Mountain Pass geometry, and denoting c_λ the corresponding Mountain Pass levels. Corresponding to (1.17), we also let

By Corollary 1.2, for every λ ∈ [1/2, 1], there exists a minimizer uλ∞ of  Jλ∞ on Mλ∞, where

and

Let

Then 𝓘_λ(u) = A(u) – λB(u). Since B(u) is not sign definite, it prevents us from employing Jeanjean’s monotonicity trick [12]. More trouble, it is difficult to show the following key inequality

due to the minimizer uλ∞ being not positive definite.

Thanks to the work of Jeanjean-Toland [15], 𝓘_λ still has a bounded (PS)-sequence {u_n(λ)} ⊂ H¹(ℝ^N) at level c_λ for almost every λ ∈ [1/2, 1]. Different from the arguments in the existing literature, by means of u1∞ and the key inequality established in Lemma 2.2, we can find a constant λ̄ ∈ [1/2, 1) and then prove directly the following inequality

see Lemma 3.5. In particular, it is not require any information on sign of u1∞ in our arguments. Applying (1.22) and a precise decomposition of bounded (PS)-sequences in [13], we can get a nontrivial critical point u_λ of 𝓘_λ which possesses energy c_λ for almost every λ ∈ [λ̄, 1]. Finally, with a Pohožaev identity we proved that (1.2) admits a least energy solution under (V1), (V2), (V4) and (F1)-(F3).

Throughout the paper we make use of the following notations:

1. H¹(ℝ^N) denotes the usual Sobolev space equipped with the inner product and norm

   (u,v)=∫RN(∇u⋅∇v+uv)dx,|u∥=(u,u)1/2,  ∀ u,v∈H1(RN);

2. L^s(ℝ^N) (1 ≤ s < ∞) denotes the Lebesgue space with the norm |u|_s = (∫_ℝ^N|u|^s dx)^1/s;

3. For any u ∈ H¹(ℝ^N) ∖ {0}, u_t(x) := u(t^–1x) for t > 0;

4. For any x ∈ ℝ^N and r > 0, B_r(x) := {y ∈ ℝ^N : |y – x| < r};

5. C₁, C₂, ⋯ denote positive constants possibly different in different places.

The rest of the paper is organized as follows. In Section 2, we give some preliminaries, and give the proofs of Theorems 1.1 and 1.3. Section 3 is devoted to finding a least energy solution for (1.2) and Theorem 1.6 will be proved in this section. In the last section, we show the existence of semiclassical ground state solutions for (1.1) and prove Theorem 1.9.

2 Ground state solutions for (1.2)

In this section, we give the proofs of Theorems 1.1 and 1.3. To this end, we give some useful lemmas. Since V(x) ≡ V_∞ satisfies (V1)-(V3), thus all conclusions on 𝓘 are also true for 𝓘^∞. For (1.5), we always assume that V_∞ > 0. First, by a simple calculation, we can verify Lemma 2.1.

From Lemma 2.2, we have the following two corollaries.

To show 𝓜 ≠ ∅, we define a set Λ as follows:

From Corollary 2.4, Lemmas 2.6, 2.7 and Corollary 2.8, we have 𝓜 ≠ ∅, 𝓜^∞ ≠ ∅ and the following lemma.

The following lemma is a known result which can be proved by a standard argument(see [32]).