THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO CHERN-SIMONS-SCHR¨ODINGER SYSTEMS

. We show the existence of nontrivial solutions to Chern-Simons-Schr¨odinger systems by using the concentration compactness principle and the argument of global compactness.

1. Introduction and main results. We are concerned with the existence of real function u ∈ H 1 (R 2 ) satisfying the following Chern-Simons-Schrödinger system (CSS system) where V (x) is external potential, f (u) is the appropriate nonlinearity. This system arises in the study of the standing wave of Chern-Simons-Schrödinger system, which describes the dynamics of large number of particles in a electromagnetic field. The system proposed in [11], [12] and [8] consists of the Schrödinger equation augmented by the gauge field. This feature of the model is important for the study of the high-temperature superconductor, fractional quantum Hall effect and Aharovnov-Bohm scattering.
Then the Euler-Lagrange equations of this Lagrangain are given by (1.3) Blowing up time-dependent solutions were investigated by Berge, De Bouard, Saut [3] and local wellposedness was studied by Liu, Smith, Tataru [15]. The (CSS) system (1.3) is invariant under the following gauge transformation φ → φe iχ , A µ → A µ − ∂ µ χ where χ : R 1+2 → R is an arbitrary C ∞ function. The standing waves of(1.3) have been investigated by Byeon, Huh and Seok in [4]. They were seeking the solutions to (1.3) of type φ(t, x) = u(|x|)e iωt , A 0 (t, x) = h 1 (|x|), where ω > 0 is a given frequency and u, h 1 , h 2 are real value functions depending only on |x|. The existence and non-existence standing wave solutions have been shown under the assumptions that f (u) = λ|u| p−1 u, λ > 0 and p > 2 by variational methods in [4], see also [9] and [10], [5]. A series of their existence results of solitary waves has been established in [6], [13], [16], [17] and [23]. We studied the existence, non-existence, and multiplicity of standing waves to the nonlinear CSS systems with an external potential V (x) without the Ambrosetti-Rabinowitz condition in [20], and the concentration of solutions in [21].
We suppose that the gauge field satisfies the Coulomb gauge condition ∂ 0 A 0 + ∂ 1 A 1 + ∂ 2 A 2 = 0, and A µ (x, t) = A µ (x), µ = 0, 1, 2. Then, we deduce that A 1 ∂ 1 u + A 2 ∂ 2 u = 0. Moreover, we see that the standing wave ψ(x, t) = e iωt u satisfies We find the weak solution of (1.4) by variational methods joined with concentration principle. To prove it, one can obtain the components A j of the gauge field represented by which come from the constrained condition ∂ 1 A 2 −∂ 2 A 1 = − 1 2 |u| 2 and the Coulomb gauge condition ∂ 1 A 1 + ∂ 2 A 2 = 0. Similarly, the representation of the component A 0 follows by solving the identity ∆A 0 = ∂ 1 (A 2 |u(y)| 2 ) − ∂ 2 (A 1 |u(y)| 2 ). We need establish the existence of critical points of the following functional in We assume that the function V (x) is positive and differenciable in R 2 satisfies By combining the variational method and the concentration compactness principle [14], we can obtain the following result. Theorem 1.2. Let f (u) = |u| p−2 u, p > 4 and suppose that V satisfies the condition (V). Then Problem (1.1) has a nontrivial solution, which solution has the asymptotic behavior lim |x|→∞ u(x)e θ|x| = 0 for some θ ∈ (0, 1).
We observe that J ∞ as in (1.5) plays the limit functional of problem (1.1). Theorem 1.2 is proven by using the mountain pass theorem [1] and the global compactness argument from [2], [7], [19].
The paper is organized as follows. In Section 2 we introduce the framework and prove some technical lemmas. In Section 3 we show the existence in Theorem 1.1. In Section 4 we study the expansion of the Palais-Smale sequences and demonstrate Theorem 1.2.
2. Mathematical framework. In the section, we outline the variational workframe for the future study.
Let H 1 (R 2 ) denote the usual Sobolev space with This implies that 2 j=1 A j ∂ j u = 0. Letting ω = 1, we can consider the following system Define the functional

YOUYAN WAN AND JINGGANG TAN
Note that We have the derivative of J in H 1 (R 2 ) as follow: for all η ∈ C ∞ 0 (R 2 ). Especially, from (2.3), we obtain that The components A j of the gauge field can be represented by solving the elliptic equations which provide where K j = −xj 2π|x| 2 , for j = 1, 2 and * denotes the convolution. The identity ∆A 0 = ∂ 1 (A 2 |u| 2 ) − ∂ 2 (A 1 |u| 2 ), gives the following representation of the component A 0 : (2.8) We know that J is well defined in H 1 (R 2 ), J ∈ C 1 (H 1 (R 2 )), and the weak solution of (2.1) is the critical point of the functional J from the following properties.
Proposition 2.1. Let 1 < s < 2 and 1 s − 1 q = 1 2 . (i) Then there is a constant C depending only on s and q such that where the integral operator T is given by (ii) If u ∈ H 1 (R 2 ), then we have that for j = 1, 2, and Proof. (i) This is the Hardy-Lilltewood-Sobolev inequality.
(iii) The Hölder inequality gives We will need the following properties of the convergence for A j , whose proof comes from the idea of Brezis-Lieb lemma.
Proposition 2.2. Suppose that u n converges to u a.e. in R 2 and u n converges weakly to u in H 1 (R n ). Let A j,n := A j (u n (x)), j = 1, 2. Then A j,n converges to A j (u(x)) a.e. in R 2 ; if u n converges weakly to u in H 1 (R n ) and u n converges to u a.e. in R 2 then, Proof. We see that dy. Taking n → ∞ and R → ∞, we obtain that Then the weak convergence implies that Hence we can deduce that which gives the desired result.

Let us denote
where we can get the right hand by differentiating both sides of (2.9) with respect to t at 1. Consider the functional J on the manifold M where 2 p−2 < α < 2 6−p for p ∈ (4, 6) and α > 1 arbitrary for p ≥ 6. We are going to establish the existence of the minimizer of the functional on this manifold, that is, (2.12) whose critical points are nontrivial solutions of (1.4).
Proof. From the Sobolev inequality From this, one can deduce that G is strictly positive if u is small.
Proof. We observe that for all u ∈ M, By Lemma 2.5, we have u > 0, then it follows that the functional J on the manifold M is strictly positive.
It is known in [10] that the stationary solutions of the CSS system satisfy the Derrick-Pohozaev type identity. For the radial case, we can find the Pohozaev identity in [4]. For no sake of completion, we provide the proof for the system (1.1) in Section 4.
Lemma 2.8. Let u be the minimizer of inf | u∈M J(u). Then G (u) = 0.
Proof. Let us denote Since u is the minimizers of J | M and G(u) = 0, we see that 14) Rewrite the Pohozaev equality Suppose G (u) = 0 by contradiction. We have G (u), u = 0, that is
Step 2. We shall prove µ = 0. Since 2αµ = 1, from Proposition 2.7, we have From G(u) = 0, we get According to (2.20) and (2.23), we get 3. Concentration compactness principle. In this section, we complete the proof of Theorem 1.1 by applying the concentration compactness principle [14], [18] to the constrained minimization problem.
Define critical values for the functional on M The following properties follows from [22].
Proof. First, we prove c = c * * . In fact, this will follow if we can show that for any u ∈ H 1 (R 2 ) \ {0} there exists unique t 0 > 0 such that t α 0 u(t 0 x) is on M as well as J t α 0 u(t 0 x) achieves the maximum of J(u). On one hand, by Lemma 3.1 in [4], there exists an unique t 0 > 0 such that J t α 0 u(t 0 x) achieves the maximum of J(u). On the other hand, Since t 0 > 0, we have G t α 0 u(t 0 x) = 0, that is, t α 0 u(t 0 x) is on M.

YOUYAN WAN AND JINGGANG TAN
Next, we prove c * = c * * . It is clear that c * * ≥ c * . Let us show c * * ≤ c * . For u ∈ H 1 (R 2 ) \ {0} fixed, let t 0 be the unique point such that t α 0 u(t 0 x) ∈ M. Then, we can write If for all γ ∈ Γ, γ ∩ K = ∅, then the inequality is proved. If there exists γ ∈ Γ such that γ(t) ∈ K for all t ∈ [0, 1], then we have which contradicts the mountain pass characterization of c * . Consequently, Proof. We will use the concentration compactness principle given in [14]. Define Because J(u n ) n → c and u n ∈ M, for n large where A 1,n := A 1 (u n ) and A 2,n := A 2 (u n ). It follows that {u n } is bounded in H 1 (R 2 ). For any n ∈ N we consider the measure By the concentration compactness lemma in [18], there exists a subsequence of {µ n }, which we will always denote by {µ n }, satisfying one of the three following possibilities: Vanishing. Suppose that there exists a subsequence of {µ n }, such that for all ρ > 0 lim n→∞ sup y∈R 2 Bρ(y) dµ n = 0.
Compactness. From the proof above, we obtain that there is a subsequence of {µ n } such that it is compact, that is, there is a sequence {ξ n } ⊂ R N such that for any δ > 0 there exists a radius ρ > 0 such that Bρ(ξn) dµ n ≥ c − δ, for all n. (3.14) Strong convergence. We define the new sequence of functions v n (·) = u n (·−ξ n ) ∈ H 1 (R 2 ). It is easy to see that A j (v n (·)) = A j (u n (· − ξ n )), j = 1, 2 and hence v n ∈ M. Moreover, by (3.14), we have that for any δ > 0 there exists a radius ρ > 0 such that v n H 1 (B c ρ ) < δ, uniformly for n ≥ 1.
where C > 0 is the constant of the embedding H 1 (B c ρ ) ⊂ L s (B c ρ ). We deduce that Note that lim n→∞ G(v n ) = 0.
Proof of Theorem 1.1. By Proposition 3.2, u ∈ M and J(u) = c. Therefore, u is a solution of (1.4). Since u has been obtained as a minimize of J restricted to M, |u| is also a minimizer. Using |u| instead ofv in Lemma 2.8 and Proposition 2.9, we obtain that |u| is a solution.
To prove that the solution u ∈ H 1 (R 2 ) does not change sign. By using Proposition 2.7, we know that We observe that for u + = max{u, 0} and u − = u + − u, which implies u − ≡ 0 or u + ≡ 0. Hence we assume that u ≥ 0, up to a change of sign. Now combining the Sobolev theorem and the Moser iteration to weak solution u ∈ H 1 (R 2 ) to (1.4). One can obtain that u is bounded in L ∞ (R 2 ). Thus, for each q ∈ [2, ∞), there exists C 1 such that u W 1,q (R 2 ) ≤ C 1 . Moreover, we have that u ∈ C γ (R 2 ) for some γ ∈ (0, 1). The standard bootstrap argument shows that u ∈ ∞ q=2 W 2,q (R 2 ). The classical elliptic estimate implies u ∈ C 1,γ (R 2 ) for some γ ∈ (0, 1). By the maximum principle, we know that u ≥ 0.
4. Global compactness. In this section we establish Theorem 1.2 by the mountain pass theorem. In order to have a better understanding of the Palais-Smale sequences of the energy functional, we need to investigate more closely the compactness question at the level of critical values.
Let us denote the function space with the equivalent norm Define the functional associated to (1.1) in the space H by We see that J V possesses the mountain pass geometry as follows. Proof. (i) Let γ(t) = t α u(tx), for t > 0.
We see that Hence, taking u 1 = γ(t), t sufficiently large, we obtain that J V (u 1 ) < 0.
(ii) We observe that where p > 4. Then we can choose u ∈ H with u H = ρ such that J V (u) > 0.
We observe that J V ∈ C 1 (H, R) satisfies the condition for some 0 < β, ρ > 0 and v 1 ∈ H with v 1 > ρ. Let c V ≥ β be characterized by  To overcome the difficulty of proving the compactness of Palais-Smale sequences for J V , one can compare it with the energy of the corresponding functional at infinity. For this, we consider the functional related to the limit problem given by Consider the functional J ∞ on the manifold M ∞ We observe that the minimizer of the minimizing problem Define where Γ := {γ ∈ C([0, 1], H) | γ(0) = 0, J ∞ (γ(1)) < 0}. Similar to Proposition 3.1, we have Moreover, we know Proof. Suppose that w 0 is the minimizer of c ∞ . Then, we obtain that Before we go to demonstrate the compact property, let us state the following the Pohozaev formula, whose proof follows from [10]. For the sake of compactness, we also sketch its demonstration. Proposition 4.3. Suppose u ∈ H be a weak solution of (1.1). Then, we have is a solution of (1.1). Multiplying the first equation of (1.1) by 2 k=1 x k (∂ k u + iA k u) and integrating on B R , we have where dσ is the arc length differential of the circle ∂B R , and Re{ By using the Coulomb gauge condition, (1.1) and (4.14) Hence, Since u ∈ H 1 (R 2 ), one can establish the desired identity by taking R → ∞.
The following proposition provides a precise description of a behavior of Palais-Smale sequence for J V , which provides the the compactness of any Palais-Smale sequence. The proof follows from [19] and [2]. Proposition 4.4. Let {u n } be a bounded Palais-Smale sequence of J V with the critical value c V . Then there exists a u 0 ∈ H such that J V (u 0 ) = 0 and either u n converges to u 0 in H or there are integer l 0 ∈ N and ξ l,n ∈ R 2 with |ξ l,n | n → ∞ for each 1 ≤ l ≤ l 0 , such that w l,n = u l,n (· + ξ l,n ) weakly converges to nonzero critical point w l of J ∞ in H. Moreover u n − u 0 + l0 l=1 w l,n (· − ξ l,n ) n → 0 strongly in H, J ∞ (w l,n (· − ξ l,n )) + o n (1).