Infinitely many solutions for a nonlinear Schr\"{o}dinger equation with non-symmetric electromagnetic fields

In this paper, we study the nonlinear Schr\"{o}dinger equation with non-symmetric electromagnetic fields $$\Big(\frac{\nabla}{i}-A_{\epsilon} x)\Big)^2 u+V_{\epsilon}(x)u=f(u),\ u\in H^1 (\mathbb{R}^N,\mathbb{C}), $$ where $A_{\epsilon}(x)=(A_{\epsilon,1}(x),A_{\epsilon,2}(x),\cdots,A_{\epsilon,N}(x))$ is a magnetic field satisfying that $A_{\epsilon,j}(x)(j=1,\ldots,N)$ is a real $C^{1}$ bounded function on $\mathbb{R}^{N}$ and $V_{\epsilon}(x)$ is an electric potential. Both of them satisfy some decay conditions and $f(u)$ is a nonlinearity satisfying some nondegeneracy condition. Applying localized energy method, we prove that there exists some $\epsilon_{0 }>0$ such that for $0<\epsilon<\epsilon_{0 }$, the above problem has infinitely many complex-valued solutions.


Introduction and main result
In this paper, we investigate the existence of standing waves ψ(x, t) = e − iEt u(x), E ∈ R, u : R N → C to the time-dependent nonlinear Schrödinger equation with an external electromagnetic field which arises in various physical contexts such as nonlinear optics or plasma physics where one simulates the interaction effect among many particles by introducing a nonlinear term (see [28]).The function ψ(x, t) takes on complex values, is the Planck constant, i is the imaginary unit.
Here A denotes a magnetic potential and the Schrödinger operator is defined by Actually, in general dimension, the magnetic field B is a 2-form where B k,j = ∂ j A k − ∂ k A j ; in the case N = 3, B = curlA.The function G represents an electric potential.
Assuming f (x, e iθ u) = e iθ f (x, u), θ ∈ R 1 and substituting this ansatz ψ(x, t) = e − iEt u(x) into (1.1),one is led to solve the complex semilinear elliptic equation For simplicity, let V (x) = (G(x)−E) and assume that V is strictly positive on the whole space R N .The transition from quantum mechanics to classical mechanics can be formally described by letting → 0, and thus the existence of solutions for small has physical interest.Standing waves for small are usually referred as semi-classical bound states (see [16]).
When A(x) ≡ 0, problem (1.2) arises in various applications, such as chemotaxis, population genetics, chemical reactor theory, and the study of standing waves of certain nonlinear Schrödinger equations.In recent years, a considerable amount of work has been devoted to study wave solutions of (1.2) with A(x) ≡ 0. Among of them, we refer to [5,6,10,11,13,15,21,22,25,27,29].Recently, in [1], Ao and Wei applying localized energy method obtained infinitely many positive solutions for (1.2) with non-symmetric potential.
On the contrary, there are still relatively few papers which deal with the case A(x) ≡ 0, namely when a magnetic field is present.The first result on magnetic nonlinear Schrödinger equation is due to Esteban and Lions in [14].They obtained the existence of standing waves to (1.2) for fixed and for special classes of magnetic fields by solving an appropriate minimization problem for the corresponding energy functional in the cases of N = 2, 3.In [9], Cao and Tang constructed semiclassical multi-peak solutions for (1.2) with bounded vector potentials.In [8], using a penalization procedure, Cingolani and Secchi extended the result in [7] to the case of a vector potential A, possibly unbounded.The penalization approach was also used in [3] by Bartsch, Dancer and Peng to obtain multi-bump semiclassical bound for problem (1.2) with more general nonlinear term f (x, u).In [19], Kurata proved the existence of least energy solution of (1.2) for > 0 under a condition relating V (x) and A(x).In [16,17], Helffer studied asymptotic behavior of the eigenfunctions of the Schrödinger operators with magnetic fields in the semiclassical limit.See also [2] for generalization of the results and in [18] for potentials which degenerate at infinity.In [20], Li, Peng and Wang applied the finite reduction method to obtain infinitely many nonradial complex valued solutions for (1.2) with radial electromagnetic fields satisfying some algebraic decaying conditions.Liu and Wang in [23] extends the result to some weaker symmetric conditions.In [26], Pi and Wang obtained multi-bump solutions for (1.2) with = 1, f (x, u) = |u| p−2 u and an electrical potential satisfying a condition by applying the finite reduction method.
In this paper, inspired by [1,30], our main idea is to use the Lyapunov-Schmidt reduction method.We want to point out that the only assumption we need is the non-degeneracy of the bump.We have no requirements on the structure of the nonlinearity.
For simplicity of notations, in the sequel, we denote Then we are concerned with the following problem In order to state our main result, we give the conditions imposed on Ã(x), Ṽ (x) and f : has a non-degenerate solution w, i.e., Particularly, f (u) = |u| p−1 u satisfies (f 2 ).
Under the above assumptions, the spectrum of the linearized operator admits the following decompositions where each of the eigenfunction corresponding to the positive eigenvalue λ j decays exponentially.These eigenfunctions will play an important role in our secondary Lyapunov-Schmidt reduction(see Section 3 below).
Remark 1.1.It is easy to find that w is a solution of (1.4) if and only if e iA0•x w is a solution of the following problem from which and (f 3 ) we can deduce that (1.5) has a non-degenerate solution e iσ+iA0•x w, i.e.
In the sequel, the Sobolev space H 1 (R N ) is endowed with the standard norm which is induced by the inner product ∇u, ∇v = ˆ(∇u∇v + uv).
Our main result of this paper is as follows: Then there exists ǫ 0 > 0 such that 0 < ǫ < ǫ 0 , problem (1.3) has infinitely many complex-valued solutions.
In the following, we sketch the main idea in the proof of Theorem 1.2.We introduce some notations first.Let µ > 0 be a real number such that w(x) ≤ ce −|x| for |x| > µ and some constant c independent of µ large.Now we define the configuration space Let w be the non-degenerate solution of (1.4) and m ≥ 1 be an integer.Define the sum of m spikes as Let the operator be × Ω m , we define the following functions as the approximate kernels: At the maximum point of M(σ, Q m ), we show that c j,k = 0 for all j, k.Therefore we prove that the corresponding z Q m + ϕ σ,Q m is a solution of (1.3).By the arguments before, we know that there exists µ 0 large such that µ ≥ µ 0 and ǫ ≤ c µ and for any m, there exists a spike solution to (1.3) with m spikes in Ω m .Considering that m is arbitrary, then there exists infinitely many spikes solutions for ǫ < c µ0 independent of m.
There are three main difficulties in the maximization process.Firstly, we need to show that the maximum points will not go to infinity.Secondly, we have to detect the difference in the energy when the spikes move to the boundary of the configuration space.In the second step, we use the induction method and detect the difference of the m-th spikes energy and the (m+1)-th spikes energy.A crucial estimate is Lemma 3.2, where we prove that the accumulated error can be controlled from step m to step m + 1.To this end, we make a secondary Lyapunov-Schmidt reduction.This is done in Section 3. Compared with [1], since there is a magnetic filed in our problem, we have to overcome some new difficulties which involves many technical estimates.
Our paper is organized as follows.In section 2, we carry out Lyapunov-Schmidt reduction.Then we perform a second Liapunov-Schmidt reduction in section 3. Finally, we prove our main result in section 4. Notations: 1. We simply write ´f to mean the Lebesgue integral of f (x) in R N .
2. The complex conjugate of any number z ∈ C will be denoted by z.
3. The real part of a number z ∈ C will be denoted by Rez.
Lemma 2.1.( [12], Lemma 3.4) There exists a constant C N = 6 N such that for any m ∈ N + and any for all x ∈ R N and all l ∈ N. Particularly, we have Lemma 2.2.Let h with h * bounded and assume that (ϕ σ,Q m , c j,k ) is a solution to (2.3).Then there exist positive numbers µ 0 and C, such that for all 0 < ǫ < e −2µ , µ > µ 0 and (σ, where C is a positive constant independent of µ, m and Q m ∈ Ω m . Proof.We prove it by contradiction.Assume that there exists a solution ϕ σ,Q m to (2.3) and Multiplying the equation in (2.3) by Dj,k and integrating in R N , we get Considering the exponential decay at infinity of ∂w(x) ∂x k and the definition of D j,k (k = 1, . . ., N + 1), we have (2.8) +O(e −µ ), as µ → +∞, k = 1, 2, . . ., N and (2.9) On the other hand, by Lemma A.1 we have (2.10) (2.11) Here and in what follows, C stands for a positive constant independent of ǫ and µ, as ǫ → 0.
Observing that Similarly, we can get Let now θ ∈ (0, 1).It is easy to check that the function E(x) in (2.1) satisfies provided μ is large enough and μ ≤ µ 2 .Indeed, by Lemma 2.1 we have From (2.24) and direct computation, we have which yields that (2.23) is true.
Hence the function E(x) can be used as a barrier to prove the pointwise estimate . Now we prove it by contradiction.We assume that there exist a sequence of ǫ tending to 0, µ tending to ∞ and a sequence of solutions of (2.3) for which the inequality is not true.The problem being linear, we can reduce to the case where we have a sequence ǫ (n) tending to 0, µ (n) tending to ∞ and sequences h Then (2.25) implies that there exists Applying elliptic estimates together with Ascoli-Arzela's theorem, we can find a sequence Q and we can extract, from the sequence which is bounded by a constant times e −γ|x| , with γ > 0.Moreover, recall that ϕ Q m satisfies the orthogonality conditions in (2.3).Therefore, the limit function ϕ ∞ also satisfies Re ˆϕ∞ ∂z ∂x j = 0, j = 1, . . ., N, and Re ˆϕ∞ ∂z ∂σ = 0, where z = e iσ+iA0•x w(x).
From Lemma 2.2, we can obtain the following result Proposition 2.3.Then there exist positive numbers γ ∈ (0, 1), µ 0 > 0 and C > 0, such that for all 0 < ǫ < e −2µ , µ > µ 0 and for any given h with h * norm bounded, there is a unique solution Proof.Here we consider the space 3) can be rewritten as (2.31) We come to the main result in this section.
Moreover, by the assumption of ǫ, we can prove that for some β > 0. In fact, on one hand, fix j ∈ {1, 2, . . ., m} and consider the region |x − Q j | ≤ µ 2 .In this region, we have On the other hand, considering the region |x − Q j | > µ 2 for all j, we have By the same arguments with (2.39), we can prove It follows from (2.37) to (2.42) that for some β > 0 independent of µ, m and Q m .
Lemma 2.6.For any Proof.By direct computation and applying the mean-value theorem, we have (2.45)

and
(2.46) From (2.45) and (2.46), we can have Now, we are ready to prove Proposition 2.4.
Proof of Proposition 2.4.We will use the contraction theorem to prove it.Observe that ϕ σ,Q m solves (2.31) if and only if where A is the operator introduced in (2.29).In other words, ϕ σ,Q m solves (2.31) if and only if ϕ σ,Q m is a fixed point for the operator where τ > 0 small enough.We will prove that T is a contraction mapping from B to itself.On one hand, for any ϕ σ,Q m ∈ B, it follows from Lemmas 2.5 and 2.6 that On the other hand, taking ϕ 1 σ,Q m and ϕ 2 σ,Q m in B, by Lemma 2.6 we have Hence by the contraction mapping theorem, for any Now we need to prove that ϕ σ,Q m is 2π-periodic with respect to σ. Replacing σ by σ + 2π in the above reduction process, we get Combining (2.22), (2.36), (2.43) and (2.44) we have

A secondary Lyapunov-Schmidt reduction
In this section, we present a key estimate on the difference between the solutions in the m-th step and (m+1)-th step.This second Lyapunov-Schmidt reduction has been used in the paper [1,23,29].For (σ, Q m ) ∈ [0, 2π] × Ω m , we denote u Q m as z Q m + ϕ σ,Q m , where ϕ σ,Q m is the unique solution given by Proposition 2.4.The main estimate below states that the difference between u Q m+1 and u Q m + z Qm+1 is small globally in H 1 (R N , C) norm.
For this purpose, we now write By Proposition 2.4, we can easily obtain that However the estimate (3.2) is not sufficient.We need a crucial estimate for φ m+1 which will be given later.(In the following we will always assume that γ > 1 2 ) In order to obtain the crucial estimate, we will need the following lemma.
As we mentioned before, the following eigenvalue problem admits the following set of eigenvalues We denote the eigenfunctions corresponding to the positive eigenvalues λ j as ϕ j , j = 1, . . ., n.Now, we have the eigenvalue λ k (k = 1, . . ., n) with eigenfunction φ0,k = e iσ+iA0•x ϕ k of the following linearized operator , where η j is the cut-off function introduced in section 1.By the equations satisfied by φ m+1 , we have for some constants c j,k , where where Now we proceed the proof into a few steps.First we estimate the L 2 -norm of S. By the estimate in Proposition 2.4, we have the following estimate (3.9) ˆ|f We also have It follows from (3.8) to (3.10) that (3.11) By the estimate (3.2), we have the following estimate we have for j = 1, . . ., m, In order to estimate the coefficients g j,l , we use the equation (3.19).First, multiplying (3.19) by φj,l and integrating over R N , we have (3.20) By the definition of φj,l , we have thus one has Recall the definition of ϕ, we have Combining (3.18), (3.20), (3.21) and (3.22), and the orthogonal conditions satisfied by ψ We claim that for some constant c 0 > 0.
Since the approximate solution is exponentially decay away from the points Q j , we have Now we only need to prove the above estimates in the domain ∪ j B µ 2 (Qj ) .We prove it by contradiction.Otherwise, there exists a sequence µ n → ∞, and Then we can extract from the sequence a subsequence which will converge weakly in H 1 (R N ) to ψ ∞ , and µ n → ∞, we have It follows from (3.27) and (3.28) that ψ ∞ = 0. Therefore (3.29) Hence, we have By (3.23) and (3.31), we have From (3.11), (3.18) and (3.32), recalling that γ > 1 2 , we get (3.33) Since we choose γ > 1 2 , by the definition of the configuration space, we have It follows from (3.33) and (3.34) that (3.35) Hence (3.5) holds.
Moreover, from the estimates (3.18) and (3.23), and taking into consideration that η j is supposed in B µ 2 (Q j ), using the H ölder inequality, we can get a more accurate estimate on φ m+1 , (3.36)

Proof of the main result
In this section, first we study a maximization problem.Then we prove our main result.
Since both z Q m and ϕ σ,Q m are both 2π-periodic respect to σ, we only need to consider the maximum problem of We will show below that the maximization problem has a solution.Let M( Qm ) be the maximum where Qm = ( Q1 , . . ., Qm ) ∈ Ωm that is and we denote the solution by u Qm .First we prove that the maximum can be attained at finite points for each C m .
Lemma 4.1.Let assumptions (A1) − (A4), (V 1) − (V 2) and the assumptions in Proposition 2.4 be satisfied.Then, for all m: (i)There exists (ii) There holds where I(z) is the energy of the solution z of (1.5): (4.6) Proof.We divide the proof into the following two steps.
Step 1: C 1 > I(z), and C 1 can be attained at a finite point.First applying standard Liapnunov-Schmidt reduction, we have Assuming that |Q| → ∞, then we have (4.8) where we use the fact that (4.9) By the slow decay assumption on the potential Ṽ (x) and Ã(x), we have Now we will prove that C 1 can be attained at a finite point.Let Q j be a sequence such that lim j→∞ M(Q j ) = C 1 , and assume that |Q j | → +∞, (4.11) which contradicts to (4.10).Thus C 1 can be attained at a finite point.
Step 2: Assume that there exists Qm = ( Q1 , . . ., Qm ) ∈ Ω m such that C m = M(Q m ) and we denote the solution by u Qm .Next we prove that there exists Q m+1 ∈ Ω m+1 such that C m+1 can be attained.Let Q (n) m+1 be a sequence such that (4.12) We claim that Q (n) m+1 is bounded.We prove it by contradiction.In the following we omit index n for simplicity.By direct computation, we have (4.13) Moreover, we have (4.14) By estimate (2.32) in Proposition 2.4, and that the definition of D j,k , we have

By the equation satisfied by ϕ
Moreover, we can choose γ that γ + δ > 1, (1 + δ)γ > 1.Then we can easily get By the assumption that |Q On the other hand, since by the assumption, C m can be attained at ( Q1 , . . ., Qm ), so there exists other point Q m+1 which is far away from the m points which be determined later.Next let's consider the solution concentrated at the points ( Q1 , . . ., Qm , Q m+1 ), and we denote the solution by u Qm ,Qm+1 , then similar with the above argument, applying the estimate (3.36) of φ m+1 instead of (3.5), we have the following estimate: Similarly, we can prove Moreover, we have where we can choose δ > 0 small enough.From all the estimates above, then (4.9) holds.

4 .
The ordinary inner product between two vectors a, b ∈ R N will be denoted by a • b.Acknowledgements: This paper was partially supported by NSFC (No.11301204; No.11371159), self-determined research funds of CCNU from the colleges' basic research and operation of MOE (CCNU14A05036).
In the sequel, if ϕ σ,Q m is the unique solution given by Proposition 2.3, we denote(2.29)ϕσ,Q m = A(h).
H, where h is defined by duality and K : H → H is a linear compact operator.By Fredholm's alternative, we know that the equation (2.28) has a unique solution for h = 0 which in turn follows from Lemma 2.2.The estimate (2.27) follows from directly from (2.6) in Lemma 2.2.The proof is complete.* ≤ C h * .Now, we consider