Concentration results for a magnetic Schrödinger-Poisson system with critical growth

1 Introduction and main results

In this paper, we study multiplicity and concentration of the nontrivial solutions of the following Schrödinger-Poisson type equations with critical growth

where u ∈ H¹(ℝ³, ℂ), ϵ > 0 is a parameter, V : ℝ³ → ℝ is a continuous function, f ∈ C(ℝ, ℝ) has a subcritical growth, the magnetic potential A : ℝ³ → ℝ³ is Hölder continuous with exponent α ∈ (0, 1], and the convolution potential is defined by |x|^–1 * |u|² = ∫_ℝ³|x – y|^–1|u(y)|²dy.

In recent years a considerable amount of work has been devoted to investigating the existence and multiplicity of solutions for nonlinear Schrödinger-Poisson system without magnetic field. We notice that, by using minimax theorema and the Ljusternik-Schnirelmann theory, He [25] gave multiplicity and concentration of positive solutions of the following problem

where f ∈ C¹(ℝ) has the subcritical growth and the potential V satisfies a global condition introduced by Rabinowitz [31]. In [26], He and Zou studied the existence and concentration of ground state solutions for the following Schrödinger-Poisson system with the critical growth

where f ∈ C¹(ℝ) and the potential V satisfies a global condition. Then, He [27] studied the multiplicity of concentrating positive solutions for Schrödinger-Poisson system (1.2) with nonlinear term f ∈ C(ℝ) under a local assumption introduced by del Pino and Felmer [17]. For further results on Schrödinger-Poisson system without magnetic field, we refer to[1, 4, 5, 14, 15, 22, 32, 33, 36, 40] and the references therein(see also [21] for the fractional case).

Concerning the magnetic nonlinear Schrödinger equation (1.1), we refer to [6, 7, 8, 10, 11, 12, 13, 16, 19, 23, 24, 29, 38, 39] and references therein. It is well known that the first result involving the magnetic field was obtained by Esteban and Lions [19]. They used the concentration-compactness principle and minimization arguments to obtain solutions for ε > 0 fixed. In [39], the authors studied multiplicity and concentration of solutions for magnetic relativistic Schrödinger equations, Xia [38] studied a critical fractional Choquard-Kirchhoff problem with magnetic field. In particular, due to our scope, we want to mention [41] where the authors studied a Schrödinger-Poisson type equation with magnetic field by using the method of the Nehari manifold, the penalization method and Ljusternik-Schnirelmann category theory for subcritical nonlinearity f ∈ C¹. If f is only continuous, then the arguments in [41] failed. Recently, by variational methods, penalization technique, and Ljusternick-Schniremann theory, for the magnetic Schrödinger-Poisson system with subcritical growth nonlinearity f which is only continuous, in [29] we proved multiplicity and concentration properties of nontrivial solutions for ϵ > 0 small. For the fractional Schrödinger-Poisson type equations with magnetic field, we refer to [2, 3].

Inspried by [27, 29], we intend to prove multiplicity and concentration of nontrivial solutions for problem (1.1) with critical growth. Since the probem we deal with has the critical growth, we need more refined estimates to overcome the lack of compactness. On the other hand, due to the appearance of magnetic field A(x) and the nonlocal term |x|^–1 * |u|², problem (1.1) will be more difficult, and some estimates are also more complicated.

In this paper, we make the following assumptions on the potential V:

1. There exists V₀ > 0 such that V(x) ≥ V₀ for all x ∈ ℝ³;

2. There exists a bounded open set Λ ⊂ ℝ³ such that

   V0=minx∈ΛV(x)<minx∈∂ΛV(x).

Observe that

On the nonlinearity f ∈ C(ℝ, ℝ), we require that:

1. f(t) = 0 if t ≤ 0, and limt→0+f(t)t=0;

2. There exist σ, q ∈ (4, 6) and μ > 0 such that

   f(t)≥μtσ−22∀t>0,andlimt→+∞f(t)tq−22=0;

3. there is a positive constant θ ∈ (4, 6) such that

   0<θ2F(t)≤tf(t),∀t>0,where F(t)=∫0tf(s)ds;

4. f(t)t is strictly increasing in (0, ∞).

The main result of this paper is listed as follows:

The paper is organized as follows. In Section 2 we indicate the functional setting and give some preliminary results. In Section 3, we study the modified problem, and prove the Palais-Smale condition for the modified functional and provide some tools which are useful to establish a multiplicity result. In Section 4, we study the autonomous limit problem associated. It allows us to show the modified problem has the multiple solutions. Finally, the proof of Thereom 1.1 is derived in Section 5.

2 Abstract setting and preliminary results

For u: ℝ³ → ℂ, let us denote by

and

The space HA1 (ℝ³, ℂ) is an Hilbert space endowed with the scalar product

where Re and the bar denote the real part of a complex number and the complex conjugation, respectively. Moreover we denote by ∥u∥_A the norm induced by this inner product.

On HA1 (ℝ³, ℂ), an important tool is the following diamagnetic inequality (see e.g. [28, Theorem 7.21])

Now, by a simple change of variables, we can see that (1.1) is equivalent to

where A_ϵ(x) = A(ϵx) and V_ϵ(x) = V(εx).

Let H_ϵ be the Hilbert space obtained as the closure of Cc∞ (ℝ³, ℂ) with respect to the scalar product

and let us denote by ∥⋅∥_ϵ the norm induced by this inner product.

By the diamagnetic inequality (2.1), we have, if u ∈ HAϵ1 (ℝ³, ℂ), then |u| ∈ H¹(ℝ³, ℝ) and ∥u∥ ≤ C ∥u∥_ϵ. Therefore, the embedding H_ϵ ↪ L^r(ℝ³, ℂ) is continuous for 2 ≤ r ≤ 6 and the embedding H_ϵ ↪ Llocr (ℝ³, ℂ) is compact for 1 ≤ r < 6.

Arguing as in [29], by the Lax-Milgram Theorem there exists a unique ϕ_|u| ∈ D^1,2(ℝ³, ℝ) such that

We obtain the following t-Riesz formula

Arguing as in [14, 32, 40], the function ϕ_|u| possesses the following properties.

For compact supported functions in H¹(ℝ³, ℝ), the following result will be very useful for some estimates below.

3 The modified problem

To study (1.1), we modify suitably the nonlinearity f so that, for ϵ > 0 small enough, the solutions of such modified problem are also solutions of the original one. More precisely, we choose K > 2. By (f4) there exists a unique number a > 0 verifying f(a) + a² = V₀/K, where V₀ is given in (V1). Hence we consider the function

Now we introduce the penalized nonlinearity g : ℝ³ × ℝ → ℝ

where χ_Λ is the characteristic function on Λ and G(x,t):=∫0tg(x,s)ds.

From (f1)–(f4), g is a Carathéodory function satisfying the following properties:

1. g(x, t) = 0 for each t ≤ 0;

2. limt→0+g(x,t)t=0 uniformly in x ∈ ℝ³;

3. g(x, t) ≤ f(t) + t² for all t ≥ 0 and uniformly in x ∈ ℝ³;

4. 0 < θG(x, t) ≤ 2 g(x, t)t, for each x ∈ Λ, t > 0;

5. 0 < G(x, t) ≤ g(x, t)t ≤ V₀t/K, for each x ∈ Λ^c, t > 0;

6. for each x ∈ Λ, the function t ↦ g(x,t)t is strictly increasing in t ∈ (0, +∞) and for each x ∈ Λ^c, the function t ↦ g(x,t)t is strictly increasing in (0, a).

Then we consider the modified problem

Note that, if u is a solution of problem (3.2) with

then u is a solution of problem (2.2).

We observe that the weak solutions of the modified problem (3.2) can be found as the critical points of the C¹ functional

defined in H_ϵ. Moreover, we denote by 𝓝_ϵ the Nehari manifold of J_ϵ,

and define the number c_ϵ by

Let Hϵ+ be open subset H_ϵ given by

and Sϵ+ = S_ϵ ∩ Hϵ+ , where S_ϵ is the unit sphere of H_ϵ. Note that Sϵ+ is a non-complete C^1,1-manifold of codimension 1, modeled on H_ϵ and contained in Hϵ+ . Therefore, H_ϵ = T_u Sϵ+ ⨁ ℝu for each u ∈ T_u Sϵ+ , where T_u Sϵ+ = {v ∈ H_ϵ : 〈 u, v〉_ϵ = 0}.

Arguing as in [29, Lemma 3.1], we can show that the functional J_ϵ satisfies the Mountain Pass Geometry [37].

Due to f is only continuous, the next results are very important because they allow us to overcome the non-differentiability of 𝓝_ϵ and the incompleteness of Sϵ+ .

At this point we define the function

by Ψ̂_ϵ(u) = J_ϵ(m̂_ϵ(u)) and denote by Ψ_ϵ := (Ψ^ϵ)|Sϵ+.

From Lemma 3.2, arguing as in [35, Corollary 10] we may obtain the following lemma.

As in [35], we have the variational characterization of the infimum of J_ϵ over 𝓝_ϵ:

The following lemma provides a range of levels in which the functional J_ϵ verifies the Palais-Smale condition.