Multiple complex-valued solutions for the nonlinear Schrödinger equations involving magnetic potentials

This paper is concerned with the following nonlinear Schrodinger equations with magnetic potentials \begin{document}$\begin{equation}\label{0} \Bigl(\frac{\nabla}{i}-α A(|x|)\Bigl)^{2}u+(1+α V(|x|))u=|u|^{p-2}u,\,\,u∈ H^{1}(\mathbb{R}^{N},\mathbb{C}),\ \ \ \ \ \ \ \ \ \ \left( 0.1 \right) \end{equation}$\end{document} where \begin{document} $2 if \begin{document} $N≥q 3$ \end{document} and \begin{document} $2 if \begin{document} $N=2$ \end{document} . \begin{document} $α$ \end{document} can be regarded as a parameter. \begin{document} $A(|x|)=(A_{1}(|x|),A_{2}(|x|),···,A_{N}(|x|))$ \end{document} is a magnetic field satisfying that \begin{document} $A_{j}(|x|)>0(j=1,...,N)$ \end{document} is a real \begin{document} $C^{1}$ \end{document} bounded function on \begin{document} $\mathbb{R}^{N}$ \end{document} and \begin{document} $V(|x|)>0$ \end{document} is a real continuous electric potential. Under some decaying conditions of both electric and magnetic potentials which are given in section 1, we prove that the equation has multiple complex-valued solutions by applying the finite reduction method.

1. Introduction and main result. In this paper, we investigate the existence of a standing wave solution ψ(x, t) = e − iEt u(x), E ∈ R, u : R N → C to the timedependent nonlinear Schrödinger equation with an external electromagnetic field which arises in many fields of physics, in particular condensed matter physics and nonlinear optics(see [30]). 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.
Denote w yj (x) = w(x − y j ), u yj (x) = e iσ w yj (x) and In order to prove Theorem 1.1, we will show the following result.
Our paper is organized as follows. In section 2, we will carry out the reduction. Then, we will study the reduced finite dimensional problem in section 3. Some technical estimates are left in the appendix. Notations: 1. The complex conjugate of any number u ∈ C will be denoted byū.
2. The real part of a number u ∈ C will be denoted by Reu.
3. The ordinary inner product between two vectors a, b ∈ R N will be denoted by a · b.
4. C,C, c i denote generic constants, which may vary inside a chain of inequalities.
2. The reduction. We assume where τ > 0 is a small constant. Define By the assumptions of A(|x|) and V (|x|), the norm of H 1 (R N , C) can be defined as follows: Define It is easy to check that is a bounded bi-linear functional in H. Hence, by the Lax-Milgram Theorem there is a bounded linear operator L from H to H, such that The following result implies that L is invertible in H.
Proof. We only prove the lemma for the case 4sin π k > b > 2 sin π k , since the other case is similar. Here we prove it by a contradiction argument. Suppose to the contrary that there exist n → +∞, r n ∈ Λ k and ϕ n ∈ E with Lϕ n = o(1) ϕ n .
Let us recall the following results which are used later.
Proof. Denote J(ϕ) = I(U r,σ + ϕ), ϕ ∈ H. By direct computation, we have Hence, L is the bounded linear map from H to H in Lemma 2.1, and It is not difficult to verify that f (ϕ) is a bounded linear functional in H, so there exists an f k ∈ H such that f (ϕ) = f k , ϕ . Thus, to find a critical point for J(ϕ), we only need to solve (2.13) From Lemma 2.1 we know that L is invertible. Therefore, (2.13) can be rewritten as where τ < τ 1 < 1.
When 2 < p ≤ 3, we can verify that Hence Lemma 2.6 below implies (2.14) Thus, A maps N into N when 2 < p ≤ 3. Meanwhile, when 2 < p ≤ 3, we see Thus, where ε ∈ (0, 1). Thus, we have proved that when 2 < p ≤ 3, A is a contraction map. When p > 3, noting the fact that for any a ∈ C, |Rea| ≤ |a|, then by Lemma 2.4, the Hölder inequality and the Sobolev inequality, we get Hence, by Remark 4.2, we get In order to estimate of R (ϕ) , by the Hölder inequality and the Sobolev inequality, we have

Thus, we have
and where ε ∈ (0, 1). Hence, A is also a contraction map from N to N . Now applying the contraction mapping theorem, we can find a unique ϕ such that (2.13) holds. Moreover, it follows from (2.14) and (2.15) that (2.11) holds.
Considering the symmetry of the problem and using the same argument in (4.2) and (4.3), we have (2.17) Also we have By the same argument, we can get

20)
Since it follows from (3.18) and (3.19) in [24], When 2 < p ≤ 3, by Lemma 2.2, we have where τ > 0 is any small fixed constant. When p > 3, we also have where τ > 0 is any small fixed constant. Hence, combining all the estimates above we have 3. The proof of the main result. In this section we will prove Theorem 1.2.
It is well-known that if (r, σ) is a critical point of F(r, σ), then U r,σ + ϕ(r, σ) is a solution of (1.3) (see [11]). As a consequence, in order to complete the proof of the proposition, we only need to prove that F(r, σ) has a critical point in Λ k × [0, 2π]. By Proposition 2.5 and Lemma 4.3, we have where A, B, B 1 and B 2 are defined in Lemma 4.3. We only prove the theorem for the case b < 2d, since the other case is similar. If b < 2d, then For the case b < 2sin π k , we consider the following maximum respect to r : max{F(r) : r ∈ Λ k }.
Assume that (3.1) is achieved by some r k in Λ k , we will prove that r k is an interior point of Λ k . Investigating the following smooth function in Λ k , g(r) := α 2 B 2 r a e −br − Br It is easy to check that g(r) has a maximum pointr k = (1+o(1)) ln α b−2 sin π k , satisfying By direct computation, we deduce that On the other hand, we suppose that r k = 1+τ b−2 sin π k ln α, then This is a contradiction to (3.3). Similarly Hence we can check that (3.1) is achieved by some r k , which is in the interior of Λ k . As a result, r k is a critical point of F(r). Therefore is a solution of (1.3).
For the case 4sin π k > b > 2sin π k , we consider the following minimum respect to r : min{F(r) : r ∈ Λ k }.
(3.5) Assume that (3.5) is achieved by some r k in Λ k , we will prove that r k is an interior point of Λ k .
Investigating the following smooth function in Λ k , It is easy to check that g(r) has a minimum pointr k = (1+o (1) (3.6) By direct computation, we deduce that On the other hand, we suppose that r k = 1+τ b−2 sin π k ln α, then This is a contradiction to (3.3). Similarly Hence we can check that (3.5) is achieved by some r k , which is in the interior of Λ k . As a result, r k is a critical point of F(r). Therefore is a solution of (1.3). 4. Some technical estimates. In this section, we will give some estimates of the energy expansion for the approximate solutions. Firstly, we recall y j = r cos 2(j − 1)π k , r sin 2(j − 1)π k , 0 , j = 1, ..., k, At the begining, we give the following basic estimate.    B 2 and B are positive constants. Proof. By a direct computation, we have Noting the symmetry, we have By Lemma 4.1, we have where we choose (2 − η)(1 − δ) > d.
And by Lemma 2.2, we have :  Hence, we get