Multiplicity and concentration results for magnetic relativistic Schrödinger equations

1 Introduction and statement of main results

In this paper, we consider the mean field limit of a quantum system with rest mass m > 0 in the presence of a magnetic vector potential A(x) and an electric potential V(x). More precise, we focus our attention on the following time-depend pseudo-relativistic magnetic Schrödinger equation

where ε > 0 is a small positive constant, i is the imaginary unit, m > 0, N ≥ 1, ψ : ℝ^N × ℝ → ℂ is a wave field, A : ℝ^N → ℝ^N is a continuous vector potential, V : ℝ^N → ℝ is an external continuous scalar potential and function f : ℝ^N → ℝ. The magnetic relativistic Schrödinger operator relate to the classical relativistic Hamiltonian symbol in Fourier variables

which is the sum of the kinetic energy term. This operator is known as a spinless particle in electromagnetic fields where we ignore quantum field theoretic effect like particles creation and annihilation but should take relativistic effect into consideration, see [1, 2]. We should remark that there are three type of relativistic Hamiltonian depending on how we quantize the kinetic energy symbol ξ−A(x)2+m2. The first two quantized operators defined by mean formulas, that is, for any function φ ∈ C0∞ (ℝ^N, ℂ),

and

We note that the Weyl pseudo-differential operator HA1 is not covariant under gauge transformations, that is, HA+∇ϕ1≠eiϕHA1e−iϕ. The operator HA2 is a modification of operator HA1 , which is gauge covariant, see [3]. The third quantized HA3 is the square of the nonnegative selfadjoint operator (–i∇ – A(x))² + m², that is,

The operator HA3 is gauge covariant and is used in the description of the stability of the matter in relativistic quantum mechanics, see for example [4, 5]. All three quantized operators are different from one another (see [1, 6]). As we know that they coincide if A(x) is linear, that is, A(x) = A ⋅ x, with A is a real symmetric constant matrix, see [1]. Particularly, this holds for constant magnetic field when N = 3, that is, B = ∇ × A is constant.

A solution of the form ψ(x, t) = e^iEt/εu(x) is called a solitary wave. Then ψ(x, t) is a solution of (1.1) if and only if the function u satisfies

where we write V instead of V + (E – m) for simplicity.

Recently, Cingolani and Secchi in [7] studied the interwining solutions of magnetic relativistic Hartree type equations, that is,

where 2 ≤ p < 2N/(N – 1) and (N – 1)p – N < α < N. Their proofs are based on the variational methods and Caffarelli and Silvestre’s type extension (see [8]) for pseudo-relativistic magnetic Schrödinger operator −i∇−A(x)2+m2+V(x) when A(x) is uniformly bounded or linear in x. If N = 3 and α = p = 2, which corresponds to the Coulomb kernel, equation (1.3) is often referred to a boson star in astrophysics, see for example [9, 10]. If also assume A ≡ 0 and V(x) = –m, equation (1.3) is reduced to the classical pseudo-relativistic Hartree equation which introduced by Lieb and Yau [11], see also [12, 13, 14] and references therein.

In the literature, the existence of standing waves solutions to nonlinear magnetic Schrödinger equation

has been first studied by Lions and Esteban [15], for ε > 0 fixed and special classes of magnetic fields. They have found existence results by solving appropriate minimization problems and concentration-compactness method for the corresponding energy functional in the cases N = 2 and 3. Lately, Kurata [16] studied the existence of a least energy solution of (1.3) under a condition relating V(x) and A(x); Cingolani [17] and Alves et al. [18] investigated the multiplicity results of (1.3) by applying the Ljusternik-Schnirelmann theory. We refer readers to [17, 19, 20, 21] and references therein for other results about nonlinear magnetic Schrödinger equation.

For the nonlocal magnetic Schrödinger equations have been investigated recently. The fractional magnetic Laplacian is defined by

which is deduced from the magnetic operator HA1 for smooth functions u. In quantum mechanics, when ε → 0, the existence and concentration of solution is of particular importance. The existence and concentration results for fractional magnetic Schrödinger equations were studied by Ambrosio and d’Avenia [22], Fiscella, Pinamonti and Vecchi [23], Zhang, Squassina and Xia [24], Mao and Xia [25]. We also refer to d’Avenia and Squassina [26] for the existence of ground states and other useful estimates. Lastly, for the existence and multiplicity results of semilinear or quasilinear Schrödinger equations, we refer readers to [27, 28, 29] and references therein.

Motivated by the about results, in this paper we deal with multiplicity and concentration results of the more general class of pseudo-relativistic magnetic Schrödinger equation (1.2). In what follows, on potentials we assume that

1. A : ℝ^N → ℝ is a continuous functions and uniformly bounded.

2. V : ℝ^N → ℝ is a continuous functions satisfies V + V₀ ≥ 0 for some V₀ ∈ (0, m) and every x ∈ ℝ^N.

3. There is a bounded open set 𝓞 ⊂ ℝ^N such that

   min∂OV>−V0,

   and M = {x ∈ 𝓞 : V(x) = –V₀} ≠ ∅.

Also, we suppose continuous function f satisfying

1. f(s) = 0, for all s ≤ 0 and f(s) = o(s) as s → 0⁺.

2. There exists constants q, σ ∈ (2, 2^♯) where 2♯:=2NN−1 if N ≥ 2 and 2^♯ := ∞ if N = 1, C₀ > 0 such that f(s) ≥ C₀s^q–2 for all s > 0 and

   lims→+∞f(s)sσ−2=0.

3. There exists a constant θ ∈ (2, 2^♯) such that

   0<θF(s):=θ∫0sf(τ)τdτ≤f(s)s2for alls>0.

4. The function f(s) is increasing in (0, +∞).

We shall establish a relation between the number of solutions of (1.2) and the topology of the set M. In order to make a precise statement let us recall that, for any closed subset Y of a topological space X, the Ljusternik-Schnirelmann category of Y in X, cat_X(Y), stands for the least number of closed and contractible sets in X which cover Y.

The main result of this article is

It should be pointed out that we only assume the potential V(x) satisfies local conditions (V1) – (V2) and no information on the behavior of the potential V(x) at infinity, so we will use the penalization method introduced by del Pino and Felmer [30] rather than minimax theorem to prove our main results. It is worthwhile to remark that in the arguments developed in [30], one of the key points is the existence of estimates involving the L^∞-bounds of the modified problem. Here we obtain the desired L^∞-bounds via Moser’s iteration method (see [31]) instead of Kato’s inequality. Moreover, we get the multiplicity results by Ljusternik-Schnirelmann theory (see [32]). As far as we known, this is the first time that penalization scheme and topological arguments are combined to get multiple solutions for magnetic pseudo-relativistic equations.

We also remark that we assume the nonlinearity term f is only continuous, so we can not use the standard arguments on the Nehari manifold. To overcome the non-differentiability for the Nehari manifold, we shall use some variants of critical point theorems from Szulkin and Weth [32]. This idea has been used extensively for nonlocal elliptic problems, see for example [33, 34].

Our proof based on the Caffarelli and Silvestre’s type extension (see [8]) for pseudo-relativistic magnetic Schrödinger operator −i∇−A(x)2+m2+V(x) when A(x) is uniformly bounded, which is prove by Cingolani and Secchi in [7]. However, some difficulties appear since the nonlinearity is on the boundary. In particular, in order to obtain the L^∞-bounds in Section 4 we will establish an inverse Hölder inequality for y(w) = u and we my iterate the inequality for y(w).

This paper is organized as follows. In section 2, we present the variational setting of the original and the extended variables problems, and we modify the original problem. We also prove the Palais-Smale condition for the modified functional and obtain some tools which are useful to establish a multiplicity result. In section 3, we study the autonomous problem associated which allow us to prove the modified problem has multiple solutions. Finally, we prove Theorem 1.1 via Morse iteration method.

2 Extension and modified problem

In this paper, we will systematically consider spaces of complex-valued functions. Precisely, the L²(ℝ^N, ℂ) space will be endowed with the real scalar product

In what follows, we will write |⋅|_p for the norm in L^p(ℝ^N) and ∥⋅∥_p for the norm in L^p( R+N+1 ). Moreover, for any w ∈ H¹( R+N+1 , ℂ), we denote

Let Ã(x, y) = (A(x), 0) : R+N+1 → ℝ^N+1 be the trivial lifting of a vector field A(x) : ℝ^N → ℝ^N for every (x, y) ∈ R+N+1 . Then, we define the magnetic Sobolev spaces on the half-space HA~1(R+N+1,C) as

which endowed with the norm

and the scalar product

where

For simplicity, we will write HA~1(R+N+1,C) and ∥w∥HA~1 as HA1(R+N+1,C) and ∥w∥HA1 respectively.

Next, we recall the following result about trace in magnetic Sobolev space operator which proved in [7].

This result allows us to generalized the well-known Dirichlet-to-Neumann extension for fractional Laplacian to the magnetic pseudo-relativistic operator. Letting ΔA~=−∇A~2=ΔA+∂2∂y2 where

then Cingonali and Secchi in [7] showed that

We remark that the key point of the proof of Proposition 2.2 is to show that magnetic Sobolev spaces HA1(R+N+1,C) and HA1/2 (ℝ^N, ℂ) are equivalent to H¹( R+N+1 , ℂ) and H^1/2(ℝ^N, ℂ) respectively when A(x) is bounded. Therefore, the existence of trace operator follows immediately from the standard theory of Sobolev traces in non-magnetic spaces. Hence, by Proposition 2.1, we deduce that the embeddings

are continuous when A is uniformly bounded.

By Proposition 2.2, we know that every function w ∈ HA1(R+N+1,C) possesses a trace y(w) ∈ H^1/2(ℝ^N, ℂ). Moreover, the following inequality holds

provides 2 ≤ p ≤ 2^♯. For the proofs of (2.5), one can find in [7].

It is easy to see that problem (1.2) is equivalent, after a change of variable, to the following one

where A_ε(x) = A(εx) and V_ε(x) = V(εx). Once we obtain a solution of (2.6), then the function ũ_ε(x) = u_ε(x/ε) is a solution of (1.2). Moreover, the maximum ζ_ε of ũ_ε is related to the maximum point z_ε of u_ε by ζ_ε = εz_ε.

By applying Proposition 2.2, we are interested to the study of the relativistic magnetic nonlocal equation

where Ã_ε = (A_ε, 0). We also observe that, for every m > 0, (2.5) implies that

Since there is no information about the infinity of V(x), we adapt the penalization method introduced by del Pino and Felmer [30] to establish the multiplicity results. Let K>2V0m−V0,a>0 such that f(a)=V0K where V₀ given in (V1). Define

and

where χ_𝓞(x) is the characteristic of set 𝓞. By the assumptions (f1) – (f4), it is easy to check that g is a Carathéodory function and satisfies

1. lim_s→0⁺ g(x, s) = 0.

2. g(x, s) ≤ f(s) for all x ∈ ℝ^N, s > 0.

3. 1. 0 < θ G(x, s) := θ ∫0s g(x, τ)τdτ ≤ g(x, s)s² for all x ∈ 𝓞, s > 0;

   2. 0 ≤ 2G(x, s) < g(x, s)s² ≤ V0K s² for all x ∈ ℝ^N ∖ 𝓞 and s > 0.

4. For each x ∈ 𝓞, the function s ↦ g(x, s) is increasing in (0, +∞), and for each x ∈ 𝓞^c, the function s ↦ g(x, s) is increasing in (0, a).

Therefore, we study the auxiliary problem

where g_ε(x, w) = g(εx, w). Note that solution of (2.10) with w(x) ≤ a for each x ∈ 𝓞_ε are also the solution of (2.7), where 𝓞_ε = {x ∈ ℝ^N : εx ∈ 𝓞}.

Consider the Euler-Lagrange functional associated to (2.10) given by

which is C¹ with Gataux derivative

where ∇_Ãε is defined as (2.2).

Next, we define the Nehari manifold 𝓝_ε (see [35]) related to 𝓘_ε. We say w ∈ 𝓝_ε means w ∈ HAε1(R+N+1,C) and satisfies

We denote by H~Aε1(R+N+1,C) the open subset of HAε1(R+N+1,C) given by

and S͠_ε = S_ε ∩ H~Aε1(R+N+1,C) , where S_ε is the unit sphere of HAε1(R+N+1,C) . Note that S͠_ε is a non-complete C^1,1-manifold of codimension 1, modeled on HAε1(R+N+1,C) and contained in the open H~Aε1(R+N+1,C) (see for example [32]). Then, HAε1(R+N+1,C) = T_wS͠_ε ⨁ ℝw for each w ∈ S͠_ε, where

We can check the functional 𝓘_ε satisfies the Mountain pass geometry [35].

Since we only assume f is continuous, in order to overcome the non-differential of 𝓝_ε and the in completeness of S͠_ε, we need the following two results.