Multiplicity of concentrating solutions for a class of magnetic Schrödinger-Poisson type equation

1 Introduction and main results

In this paper, we are concerned with multiplicity and concentration results for the following Schrödinger-Poisson type equation

where u ∈ H¹(ℝ³, ℂ), ε > 0 is a parameter, V : ℝ³ → ℝ is a continuous function, f ∈ C(ℝ, ℝ), 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.

Problem (1.1) arises in quantum mechanics, abelian gauge theories, plasma physics, and so on which can be used to simulate the mutual interactions of many particles. In fact, the linear Schrödinger equation describes the behavior of a single particle. However, the interaction among particles can be simulated by adding a nonlinear term f. Moreover, the convolution potential is a solution of Poisson equation which implies that the particles move in their own gravitational field generated by the probability density of particles via classical Newton field equation. Therefore, problem (1.1) can be regarded as the coupling of the Schrödinger equation and Poisson equation.

There is a vast literature concerning the existence and multiplicity of solutions for nonlinear equation without magnetic field. We notice that Fiscella, Pucci and Zhang [16] studied the existence of solutions for p-fractional Hardy-Schrödinger-Kirchhoff systems with critical nonlinearities, Ji and Radulescu [17] considered the multiplicity of multi-bump solutions for quasilinear elliptic equations with variable exponents and critical growth in ℝ^N, for more results, we refer to the Monograph [25]. Recently, by using the method of Nehari manifold and Ljusternik-Schnirelmann theory, He [21] proved the multiplicity and concentration of solutions of problem (1.1) for f ∈ C¹(ℝ, ℝ) and the potential satisfying a global condition introduced by Rabinowitz [26]. In [22], on the similar assumptions, He and Zou studied the existence and concentration behavior of ground state solutions for a class of Schrödinger-Poisson system with critical the nonlinearity f ∈ C¹(ℝ, ℝ). Then, under a local assumption introduced by del Pino and Felmer [14], He and Zou [23] studied the multiplicity of concentrating positive solutions for Schrödinger-Poisson equations with critical nonlinear f ∈ C¹(ℝ, ℝ). For further results about existence and nonexistence of solutions, multiplicity of solutions, ground states, semiclassical limit and concentrations of solutions for Schrödinger-Poisson system(see [1, 2, 3, 4, 11, 12, 27, 28, 31, 35] and the references therein).

On the other hand, the magnetic nonlinear Schrödinger equation (1.1) has been extensively investigated by many authors applying suitable variational and topological methods (see [5, 6, 7, 9, 10, 13, 15, 18, 19, 20, 32, 33, 34] and references therein). It is well known that the first result involving the magnetic field was obtained by Esteban and Lions [15]. They used the concentration-compactness principle and minimization arguments to obtain solutions for ε > 0 fixed. In [34], Xiang, Rădulescu and Zhang studied multiplicity and concentration of solutions for magnetic relativistic Schrödinger equations, Xia [32] studied a critical fractional Choquard-Kirchhoff problem with magnetic field. In particular, due to our scope, we want to mention [36] 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 [36] failed.

In this paper, motivated by [23, 29, 36], for the case f is only continuous, we intend to prove multiplicity and concentration of nontrivial solutions for problem (1.1). We note that, due to the appearance of magnetic field A(x), problem (1.1) will be more difficult in employing the methods and some estimates. On the other hand, due to the nonlocal term ∣x∣⁻¹ ∗ ∣u∣², some estimates are also more complicated.

Throughout the 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

Moreover, let the nonlinearity f ∈ C(ℝ, ℝ) be a function satisfying:

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

2. there exists q ∈ (4, 6) such that

   limt→+∞f(t)tq−22=0;

3. there is a positive constant θ > 4 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 the following:

2 Abstract setting and preliminary results

In this section, we present the functional spaces and some useful preliminary remarks which will be useful for our arguments.

For u : ℝ³ → ℂ, let us denote by

and

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) we will frequently use the following diamagnetic inequality (see e.g. [24, Theorem 7.21])

Moreover, making 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.

The diamagnetic inequality (2.1) implies that, 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.

By using the continuous embedding H¹(ℝ³, ℝ) ↪ L^r(ℝ³, ℝ) for 2 ≤ r ≤ 6, we can see that

For any u ∈ H_ε, we get ∣u∣ ∈ H¹(ℝ³, ℝ), and the linear functional 𝓛_∣u∣ : D^1,2(ℝ³, ℝ) → ℝ given by

is well defined and continuous in view of the Hölder inequality and (2.4). Indeed, we can see that

where

Then, by the Lax-Milgram Theorem, there exists a unique ϕ_∣u∣ ∈ D^1,2(ℝ³, ℝ) such that

Therefore we obtain the following t-Riesz formula

In the sequel, we will omit the constant for simplicity. The function ϕ_∣u∣ possesses the following properties.

3 The modified problem

To study problem (1.1), or equivalently (2.2) by variational methods, we shall 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 K f(a) = V₀, 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.

In view of (f1)–(f4), we have that 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, and there exists q ∈ (4, 6) such that

   limt→+∞g(x,t)tq−22=0uniformly inx∈R3;

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

4. 0 < θ G(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).

The functional associated to problem (3.2) is

defined in H_ε. It is standard to prove that J_ε ∈ C¹(H_ε, ℝ) and its critical points are the weak solutions of the modified problem (3.2).

We denote by 𝓝_ε the Nehari manifold of J_ε, that is

and define the number c_ε by

Let Hε+ be open subset H_ε given by

and Sε+=Sε∩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}.

Now we show that the functional J_ε satisfies the Mountain Pass Geometry.