Concentrations for nonlinear Schrödinger equations with magnetic potentials and constant electric potentials

1 Introduction

The magnetic Schrödinger equation in R N is given by

where ε > 0 is a small parameter, 1 < p < N + 2 N − 2 if N ≥ 3 , p > 1 if N = 2 and i is the imaginary unit. Here the function ψ is complex-valued. The vector A = ( A 1 , A 2 , … , A N ) : R N → R N denotes the magnetic potential and Q : R N → R represents the electric potential. The magnetic Laplacian ( i ε ∇ + A ) 2 is defined by

The vector A models the presence in some quantum model of the magnetic field B , which is given by

For the discussion of this operator, one may refer to [16].

Equation (1) arises in various physical contexts such as Bose-Einstein condensates and nonlinear optics [17]; or plasma physics where one can simulate the interaction effect among many particles by introducing some nonlinear terms, see [19].

From now on we consider standing wave solutions to problem (1), namely ψ ( t , x ) = e i λ ε − 1 t u ( x ) for some complex-valued function u ( x ) . Substituting this ansatz into problem (1), u ( x ) should satisfy the following nonlinear magnetic Schrödinger equation:

where V ( x ) = Q ( x ) + λ . The potential V ( x ) is then usually assumed to be smooth and satisfy

Concerning nonlinear Schrödinger equations with the magnetic field, the pioneer work is by Esteban-Lions [11], in which they prove the existence of standing waves to (1) by a constrained minimization approach, in the case V ( x ) = 1 and for special classes of magnetic fields. For more results, one can refer to [1,13,14] and references therein. On the other hand, problem (2) seems to be a very interesting problem since the correspondence’s principle establishes that classical mechanics is, roughly speaking, contained in quantum mechanics. The mathematical transition from quantum mechanics to classical mechanics can be formally described by letting the Planck constant ε → 0 , and thus the existence of solutions for ε small has physical interest. Standing waves for ε small are usually referred to as semi-classical bound states, see e.g., [13]. In the linear case, Helffer et al. [14] studied the asymptotic behavior of the eigenfunctions of the Schrödinger operators with magnetic fields in the semiclassical limit.

Concentrations play an important role in the study of problem (2). When A ≡ 0 , the pioneer work of concentrations to problem (2) in one dimension was carried out by Floer and Weistein [12]. They proved that if the electric potential V ( x ) has a non-degenerate critical point, then u ( x ) concentrates near this critical point as ε → 0 . This kind of solution with point concentrations is sometimes called peak solutions. After that, there are too many peak solution results for nonlinear Schrödinger problems to give a full reference list here.

The first concentration result for nonlinear magnetic Schrödinger equation (2) is given by Kurata [15]. Under some assumptions linking the magnetic field and electric potential, he showed that least energy solutions exist for every small ε and they exhibit peak concentrations at a global minimum point of V . Moreover, Kurata [15] mentioned that the magnetic potential A only contributes the phase factor of least energy solutions. Next, previous studies [4,7] proved that peak concentrations may also occur at any non-degenerate critical point, not necessarily a minimum of V , as ε → 0 by a reduction method. Cingolani and Secchi [8] again show that there exist peak solutions concentrating at topologically nontrivial critical points of V by a penalization procedure. In the aforementioned papers, the presence of a magnetic field produce a phase in the complex wave, but does not influence the location of peaks. Later Cingolani et al. [6] used a variational approach and proved the existence of a multi-peak solution without any non-degenerate condition. But the locations of peaks are still near local minimum points of V . For more related results see [5,20] and references therein.

It is interesting to ask whether the magnetic field affects the locations of concentrations. Secchi and Squassina in [18] show that the magnetic potential A might perhaps affect the locations of concentration points for the three-dimensional magnetic Schrödinger equation. Nevertheless, in the particular but important case of power-type nonlinearities just as our case, they emphasize that the locations of peaks are independent of A . DiCosmo and Van Schaftingen in [10] then consider a strong magnetic field case A = O ( ε − 2 ) , where the interaction between the magnetic field and the electric field is thus comparable. In this case, they obtain solutions concentrating around global or local minima of a limit energy that depends on the electric potential and the magnetic field. However, their limit energy does not show the effect of magnetic fields explicitly. Later Bonheure et al. [2] proved the existence of semiclassical cylindrically symmetric solutions to three-dimensional problem (2) whose moduli concentrates around a circle, which is driven by the magnetic and electric potentials. Their result shows that the magnetic field really influences the locations of concentrations if it occurs around a locus. Actually, this is a high-dimensional concentration phenomenon, not a peak concentration. See also [21] for a curve concentration phenomenon under a weak magnetic potential without symmetric conditions.

Thus, a nature question arises that whether a common, neither weak nor strong, magnetic potential A influences the locations of peak concentrations to the nonlinear magnetic Schrödinger problem. What role does A play in this kind of concentration? To this aim, we assume the electric potential V to be a constant, otherwise A has no effect on the locations of peaks due to the aforementioned works [4,6,7,15].

Recently, Bonheure et al. [3] considers the ground state solution of magnetic Schrödinger equations with a constant electric potential and no Planck constant ε . Based on the assumption of a weak constant magnetic field, they show that the ground state solution is unique up to magnetic translations and rotations. Furthermore, the energy expansion of the ground state solution is also given in [3] as the constant magnetic field B is small enough.

In this article, we study the following nonlinear magnetic Schrödinger equation:

Here p > 1 is Sobolev subcritical, and A = ( A 1 , … , A N ) : R N → R N is assumed in W 1 , ∞ ( R N , R N ) and smooth everywhere for simplicity. To state our main result, we introduce the Frobenius norm of a matrix ℳ = ( m i j ) I × J

And denote by w ( y ) = w ( ∣ y ∣ ) the unique radial real-valued solution of

Now our main results are the following.

For general critical points, i.e., not local extremum points, we also have the following result.

From the point view of physics, the magnetic field B is essential, not the particular choice of magnetic potential A . At the same time, there is a gauge invariance for the magnetic Laplacian correspondingly, that is, the magnetic field B is invariant under the transform of the potential A → A + ∇ f . Also, it is easy to see that the energy of problem (3) is unchanged under the gauge invariance, with which our result coincides (see Proposition 4.3).

The proofs of main theorems are based on the reduction method. When the electric potential V ( x ) has some non-degeneracy, we just need to make an approximation from the limit equation, which is enough to deduce the roles of function V ( x ) . But in order to see the roles of magnetic vector potential A in our case, we need an approximation up to order ε of problem (3), i.e., a second approximation have to be done. Finally, we mention that the solutions given in both Theorems 1.1 and 1.2 show simple peak concentrations. In the forthcoming article, we will deal with multi-bump solutions for constant electric potential.

The article is organized as follows. In Section 2, the ansatz and the estimate of the error are given. The corresponding nonlinear problem is solved in Section 3. In Section 4, the original problem is reduced to the finite dimension problem using variational reduction process and the expansion of the energy functional is shown. Finally, Theorems 1.1 and 1.2 are proved in Section 5.

Notations.

1. Constant β = min { p − 1 , 1 } .

2. The real part of z ∈ C will be denoted by Re z .

3. The complex conjugate of z ∈ C will be denoted by z ¯ .

4. C denotes a generic positive constant, which may be different from lines to lines.

5. Landau symbol O ( ε ) is a generic function such that ∣ O ( ε ) ∣ ≤ C ε and o ( ε ) means that lim ε → 0 + ∣ o ( ε ) ∣ ε = 0 .

2 Ansatz

In this section, we present the approximation of the problem and give the corresponding error estimate.

Recall in [15] that the problem

possesses a unique ground state solution w ˜ ( y ) = w ( y ) e i σ , ∀ σ ∈ [ 0 , 2 π ] where w ( y ) is the radial solution of problem (4). Thus, by the gauge invariance,

is also the ground state solution to the constant magnetic potential problem

In the frame of large variable y = x / ε , the original problem (3) is equivalent to

Therefore, the function

formally approximates the solution at least near y = ζ ′ . Now it is time to give the first approximation

where ζ m ∈ Ω m , m = 1 , 2 , … , K . Denote

and take a positive number δ such that δ ≤ 1 4 ρ . Let R ^ ( y ) be the error caused by the first approximation W , which is

It is checked that for ∣ y − ζ 1 ′ ∣ ≤ δ ε , ∣ U m ∣ ≤ ∣ U 1 ∣ e − ∣ ζ m ′ − ζ 1 ′ ∣ 3 , m = 2 , 3 , … , K and

where

As usual one writes ζ m ′ = ( ζ m , 1 ′ , ζ m , 2 ′ , … , ζ m , N ′ ) . Direct calculation shows that

Similarly, it is verified that

As for the term ∇ x ⋅ A ( ε y ) w ( ∣ y − ζ 1 ′ ∣ ) , we expand it similarly to

Therefore, in the region ∣ y − ζ 1 ′ ∣ < δ ε , we obtain that

Note that the imaginary part of R ˜ ( y ) e − i σ 1 − i A ( ζ 1 ) ⋅ y is O ( ε ) . It is of less accuracy for later application. Complying with the guideline that the better approximation, the more possibility to obtain a solution, we should improve the accuracy of the approximation. To this purpose, we find that the real-valued function Ψ 1 , i j ( y ) = 1 2 ( y i − ζ 1 , i ′ ) ( y j − ζ 1 , j ′ ) w ( ∣ y − ζ 1 ′ ∣ ) ( i ≠ j ) is the solution to

Besides, Ψ 1 , i i ( y ) = 1 2 ( y i − ζ 1 , i ′ ) 2 w ( ∣ y − ζ 1 ′ ∣ ) satisfies the equation

Then obviously the function

satisfies

since Re ( U ¯ 1 Ψ 1 ) = 0 . Moreover, the same computation may also be carried out in the region ∣ y − ζ m ′ ∣ < δ ε for m = 2 , … , K . The ultimate approximation is then selected as

where Ψ ( y ) = ∑ m = 1 K Ψ m ( y ) . Here

and

Now the approximation W is good enough to help us find a solution. Precisely, our aim is to find a solution with the form W ( y ) + ϕ ( y ) of problem (3), where ϕ is a small perturbation and satisfies the equation

Here R ( y ) denotes the error caused by W , which is

and the nonlinear term

Obviously, with the notation β = min { p − 1 , 1 } ,

To solve problem (10), it is important to estimate the error R ( y ) .