Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation

1 Introduction

In this study, we are interested in the following gauged nonlinear Schrödinger equation:

where p > 6 and h u ( s ) = 1 2 ∫ 0 s r u 2 ( r ) d r . Equation (1.1) arises from finding standing wave solutions for the following Chern-Simons-Schrödinger system:

where t ∈ R , x = ( x 1 , x 2 ) ∈ R 2 , ϕ : R × R 2 → C is a scalar field, A μ : R × R 2 → R are the components of the gauge potential and D μ = ∂ μ + i A μ is the covariant derivative ( μ = 0 , 1 , 2 ) .

System (1.2) has been studied extensively in the last three decades. For p = 4 , system (1.2) corresponds to the model proposed first by Jackiw and Pi [16] to describe the dynamics of a nonrelativistic solitary wave that behaves like a particle in the three-dimensional gauge Chern-Simons theory. And since then plenty of results on (1.2) have been obtained, including the initial value problem, well-posedness, global existence and blow-up, scattering, etc., see [3,12,13,20,21,25] and references therein.

In particular, to find standing wave solutions of (1.2), Byeon et al. [5] introduced the following ansatz:

where ω ∈ R is a constant and u is a radial real valued function. Then, u solves the following equation:

where h u ( r ) = 1 2 ∫ 0 r s u 2 ( s ) d s and ξ ∈ R is an integration constant of A 0 , which takes the form

In this way, one obtains a standing wave solution for (1.2) of the form

As usual in Chern-Simons theory, system (1.2) is invariant under the gauge transformation

for arbitrary C ∞ function χ : R × R 2 → R . By taking χ = c t in the gauge invariance (1.4), we obtain a family of standing wave solutions for (1.2)

which implies that constant ω + ξ is a gauge invariant. In what follows, we can take ξ = 0 , which is also necessary for the following decaying condition:

Therefore, equation (1.3) becomes

As shown in [5] for ω > 0 , (1.5) is actually the Euler-Lagrange equation of the following C 1 energy functional:

where H r 1 ( R 2 ) ≔ { u ∈ H 1 ( R 2 ) : u is radially symmetric } . By variational methods, Byeon et al. [5] proved the existence of positive solutions to (1.5) for p > 2 and ω > 0 . We point out that if p > 4 , the functional I ω owns the mountain pass structure, but it seems hard to check the Palais-Smale condition for p ∈ ( 4 , 6 ) , so Beyon et al. [5] considered a minimization problem on a manifold of Pohozaev-Nehari type in H r 1 ( R 2 ) . While for p ∈ ( 2 , 4 ) , the minimization arguments for the case p > 4 do not work anymore and they find solutions as minimizers on a L 2 -sphere, in this case, ω occurs as a Lagrange multiplier.

Later, the results for p ∈ ( 2 , 4 ) were extended by Pomponio and Ruiz [26] by studying the global behavior of the energy functional I ω . The technique of L 2 -constraint also was extended to the case p ≥ 4 by Li and Luo in [18]. Infinitely many solutions have been found by Huh [14] for p > 6 . The existence of nodal solutions has been studied by Deng et al. [10], see also [7,19,22]. For more results on (1.5), see [1,5,8,14,26–29] and references therein. We remark that the existence of standing wave solution for (1.2) with a vortex point was also studied in [4,17]. In addition, Huh et al. [15] considered a generalized Jackiw-Pi model and obtained the existence of standing wave solutions by using variational method.

Recently, Azzollini and Pomponio [1] investigated equation (1.1), which is usually called zero-mass equation. The zero-mass equation causes some mathematical difficulties, since the corresponding functional is not well-defined in the usual Sobolev space, and the classical critical point theory and variational methods cannot be applied directly.

As in [1], we define

which is equipped with the norm ‖ u ‖ 2 , 4 ≔ ( ‖ ∇ u ‖ L 2 ( R 2 ) 2 + ‖ u ‖ L 4 ( R 2 ) 2 ) 1 2 . Naturally, ℋ r 2 , 4 ( R 2 ) is defined as

We also define

Formally, equation (1.1) is associated with the energy functional I defined by

where h u ( r ) = 1 2 ∫ 0 r s u 2 ( s ) d s . Although the functional I is well-defined on H r 1 ( R 2 ) , we cannot prove that a minimizing sequence or a (PS)-sequence is bounded in H r 1 ( R 2 ) , because of the absence of ‖ u ‖ L 2 ( R 2 ) 2 in I ( u ) . On the other hand, I ( u ) < + ∞ for each u ∈ A , but we do not know whether A is a Banach space or not.

Now, we give the definition of weak solution to (1.1), see Definition 1.4 in [1].

In [1], by perturbation method, Azzollini and Pomponio proved that equation (1.1) has a positive solution if p > 4 , and this positive solution belongs to H r 1 ( R 2 ) if p > 10 . More precisely, the authors first regarded I ω as a perturbed functional, by [5], it is easy to see that there exists a critical point u ω of I ω , for any ω > 0 . Then, they studied the behavior of the family { u ω } , as ω → 0 + . By concentration-compactness arguments, they showed that, up to a subsequence, there exists u 0 ∈ A such that the family converges to such u 0 in ℋ r 2 , 4 ( R 2 ) as ω → 0 + . Finally, they proved that, actually, u 0 is the desired solution.

In the present study, we use a different method to study equation (1.1), and we obtain infinitely many nodal solutions for (1.1), our result in this aspect can be stated as follows.

To prove Theorem 1.1, we adopt the so-called Nehari manifold technique, which was first introduced by Bartsch and Willem [2] and Cao and Zhu [6], and it has been improved by Deng et al. [9,10] to deal with nonlocal problem. However, Deng et al. [10] dealt with equation (1.5) in the case ω ≠ 0 , so the functional framework can be set in H r 1 ( R 2 ) . But, in our case, ω = 0 , we need the following definition, i.e., the direction derivative of I , which has been used in [23].

By Definition 1.2, we know that u is a solution of (1.1) if and only if ⟨ I ′ ( u ) , ϕ ⟩ = 0 for each ϕ ∈ H r 1 ( R 2 ) .

Note that ℋ 2 , 4 ( R 2 ) cannot be embedded in L 2 ( R 2 ) , it remains unclear whether the solutions obtained by Theorem 1.1 belong to L 2 ( R 2 ) . However, we can prove the following.

Inspired by Theorem 1.8 in [1], we prove Theorem 1.2 based on a comparison argument. However, Azzollini and Pomponio [1] used perturbation arguments and they only discussed positive solutions. In this study, we prove the L 2 integrability of nodal solutions directly.

The study is organized as follows. In Section 2, we first state some preliminary results for equation (1.1). Then, we modify the original functional I to a new one, which is associated with a system consisting of ( k + 1 ) equations. In Section 3, we study a minimization problem to find a solution of the system consisting of ( k + 1 ) equations for any fixed integer k ≥ 0 . In Section 4, we glue all components of the solution obtained in Section 3 to form a nodal solution for equation (1.1) and complete the proof of Theorems 1.1 and 1.2.

2 Preliminaries

In this section, we first state some preliminary results for studying equation (1.1). Based on [1], we list the following properties of ℋ 2 , 4 ( R 2 ) and ℋ r 2 , 4 ( R 2 ) .

Now, we give some notations. For any integer k ≥ 0 , we define

For any given r k ∈ Γ k , we denote

and