Infinitely Many Standing Waves for the Nonlinear Chern-Simons-Schrödinger Equations

∂ 2 = ∂/∂x 2 for (t, x 1 , x 2 ) ∈ R, φ : R → C is a complex scalar field, A μ : R → R is a component of gauge potential and D μ = ∂ μ + iA μ is a covariant derivative for μ running over 0, 1, 2, and λ > 0 is a parameter. The Chern-Simons gauge theory appears in the 1980s to explain electromagnetic phenomena of anyon physics such as the high temperature superconductivity or the fractional quantum Hall effect. In this paper, we are interested in standing wave solutions of (1). In [3], the authors introduce a standing wave ansatz of the following form:


Introduction
In [1,2], Jackiw and Pi introduce a nonrelativistic model that the nonlinear Schrödinger dynamics is coupled with the Chern-Simons gauge terms as follows: Here,  denotes the imaginary unit,  0 = /,  1 = / 1 ,  2 = / 2 for (,  1 ,  2 ) ∈ R 1+2 ,  : R 1+2 → C is a complex scalar field,   : R 1+2 → R is a component of gauge potential and   =   +   is a covariant derivative for  running over 0, 1, 2, and  > 0 is a parameter.The Chern-Simons gauge theory appears in the 1980s to explain electromagnetic phenomena of anyon physics such as the high temperature superconductivity or the fractional quantum Hall effect.In this paper, we are interested in standing wave solutions of (1).In [3], the authors introduce a standing wave ansatz of the following form: where  > 0 is a phase frequency and , , ℎ are real valued functions on [0, ∞) such that ℎ(0) = 0. Inserting (2) into (1), one may check from direct computation that (1) is reduced to the following nonlinear nonlocal elliptic equation: where ℎ  () = ∫  0 (1/2) 2 ().See [3] for its derivation.It is shown in [3] that (3) is an Euler-Lagrange equation of a  1 functional, where  1  (R 2 ) denotes the set of radially symmetric functions in standard Sobolev space  1 (R 2 ).Investigating the structure 2 Advances in Mathematical Physics of , the authors of [3] obtain several existence and nonexistence results for (3), depending on the range of  > 2 and  > 0. Recently, Pomponio and Ruiz [4] improve the results in [3] for the case  ∈ (2,4).They find a threshold for the behavior of , depending on  > 0. They also study (3) on bounded domain in [5].
In this paper, we are concerned with the existence of infinitely many solutions of (3).It is proved in [6] that if  > 6,  enjoys the symmetric mountain pass geometry and satisfies the (PS) condition so that the well-known symmetric mountain pass lemma (see [7]) applies to show there exist infinitely many critical points of .For  ∈ (4,6], it turns out that  still enjoys the symmetric mountain pass geometry although checking the (PS) condition is not easy job.One aim of this paper is to show nevertheless  still admits infinitely many critical points for  > 4.Moreover, we will replace the power-type nonlinearity || −2  of ( 3) with more general one as follows: The structure conditions for  are given by the following: (V3) There exists some  > 1 such that   ()/ 2−1/ − ()/ 3−1/ is monotonically increasing to ∞ as  → ∞.
We refer to the work of Cunha et al. [8] that if we insert a sufficiently small parameter  > 0 into (5) as in then much more general assumptions for , the so-called Berestycki-Lions conditions [9], are sufficient for guaranteeing the existence and multiplicity of solutions of (6).
In our work, we assume further than the Berestycki-Lions conditions but we do not need small parameter  > 0. See also [10] in which Tan and Wan consider asymptotically linear nonlinearities.
To prove Theorem 1, we will apply the method employed in author's former paper [11] in which the Schrödinger-Poisson equation, another nonlocal field equation similar to (5), is dealt with.Instead of generating (PS) sequences, we will show the existence and compactness of the socalled approximate solution sequences of  which may be considered as more refined version of (PS) sequences.In Section 2, we give a definition of the approximate solution sequences of .Some auxiliary lemmas are also prepared in Section 2. In Section 3, we prove the compactness of approximate solution sequences.In Section 4, we construct infinitely many approximate solution sequences whose energy levels go to infinity and complete the proof of Theorem 1.

Mathematical Settings and Preliminaries
The dual space of  1  (R 2 ) is denoted by  1  (R 2 ) * .Arguing similarly to [3], it is easy to show ( 5) is an Euler-Lagrange equation of the  1 functional In this paper, we search for infinitely many critical points of  to prove Theorem 1.To do this, we insert parameter  into  as follows: Here  ranges over which converges to 1 as  → ∞, we say {  ,   } is an approximate solution sequence of  if     (  ) = 0 for all .In the following subsection, we state a variant of the famous Struwe's monotonicity trick [12], which plays a crucial role in constructing approximate solution sequences.

A Variant of Struwe's Monotonicity Trick.
Let  be Banach space.We say a subset  ⊂  is symmetric if −V ∈  for every V ∈ .Let  be a compact subset of  and  a closed subset of .We denote by Γ the set of every continuous odd function  :  →  such that (V) = V on .Let  be a closed interval in R and   one parameter family of even  1 functional on .
We define a minimax level by The following theorem is a variant of so-called Struwe's monotonicity trick [12].A more general version of it is given in [11].The property () below is first proposed by Jeanjean and Toland in [13].

Proof. By the change of variable, one can compute
We differentiate it with respect to  to get
Proof.To show the well-definedness of , we have to show that there exists unique  > 0 satisfying ‖ − ( −1 ⋅)‖ = 1 for given nonzero  ∈  1  (R 2 ).We note that this is equivalent to prove there is a unique solution  > 0 of the equation Arguing similarly to the proof of Lemma 5, we are able to see that () is monotonically decreasing on (0,  0 ) for some  0 > 0, attains its unique local minimum at  0 , and is monotonically increasing to infinity on ( 0 , ∞).Therefore there is a unique positive zero of () since (0) = 0. Also, the implicit function theorem says that () is continuous on [0, ∞) because   (()) ̸ = 0.The evenness of  follows from the fact that each coefficient of (20) is even.This completes the proof.

Compactness of Approximate Solution Sequences
In this section, we prove the compactness of an approximate solution sequence {  } of    when its energy {   (  )} is bounded above.
Proof.We divide the proof into two steps.
Step 1 (boundedness of   ).We first prove that {  } is bounded in  1  (R 2 ).Arguing indirectly, suppose that {  } is unbounded.Let   := (  ), where the function  is defined in Section 2. Equation (20) says   is unbounded.Let V  () :=  −    ( −1  ) so that ‖V  ‖ = 1.Then, up to a subsequence, {V  } converges weakly in  1  (R 2 ) and strongly in   (R 2 ) for all  > 2 to some Combining this with the Pohozaev identity (15), we obtain Then, from the change of variable and dividing by  6−4  , (22) transforms to Since ‖V  ‖ = 1 and   is unbounded,   is bounded for  but the structure assumption (V3) implies that   tends to infinity as  → ∞ provided V 0 is not identically zero.We claim that V 0 is nonzero.Suppose V 0 is identically zero.From ( 19) and ( 22), we see that Then, Lemma 5 implies that    ,  (1) is the global maximum of    ,  () on (0, ∞).Thus we see that, for each  > 1, The last equality follows from ‖V  ‖ = 1, the convergence of {V  } to 0 in   (R 2 ) for all  > 2, and the structure conditions (V1)-(V2).However, taking large  > 1, this makes a contradiction and shows V 0 is not identically zero.This proves the boundedness of {  } in  1  (R 2 ).
Step 2 (compactness of   ).Compactness of {  } follows from a standard procedure.Since {  } is bounded, there exists  0 ∈  1  (R 2 ) such that {  } converges, up to a subsequence, to  0 weakly in  1  (R 2 ) and strongly in   (R 2 ) for all  > 2. Then it follows from Lemma 3 that  0 is a critical point of .Also, it is easy to see from the boundedness of {  } that   (  ) → 0 in  1  (R which shows ‖  ‖ → ‖ 0 ‖ as  → ∞.Therefore we have
−1 ) so that ‖V‖ = 1.By a change of variable, we get from (29) that, for each  ∈   , , = {V ∈  1  (R 2 ) | ‖V‖ = 1, V  =   V (⋅) ∈   } .We claim that, for each  > 0,  , → 0 as  → ∞.To see this, suppose that  , →   > 0 as  → ∞.Choose V , ∈  , satisfyingCompletion of the Proof of Theorem 1.Now we complete the proof of Theorem 1.We first choose and fix arbitrary  ∈ N.Let {  } be an approximate solution sequence of    , given by Proposition 10.Take  ∈ Γ  satisfying max It follows from the compactness of   that   (  ) ≤ max Then Proposition 7 applies to see   converges, up to a subsequence, to some  0 which is a critical point of .Recall that    (  ) =   (  ) ≥ .By taking a limit  → ∞, we deduce ( 0 ) ≥ .Since  is arbitrary, this shows the existence of infinitely many critical points of .This completes the proof.