Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 77, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu STANDING WAVES TO CHERN-SIMONS-SCHRÖDINGER SYSTEMS WITH CRITICAL EXPONENTIAL GROWTH YOUYAN WAN, JINGGANG TAN Abstract. In this article we study the existence of standing waves to nonlin- ear Chern-Simons-Schrödinger systems with critical exponential growth. 1. Introduction and main result We study the existence of ground state to the Chern-Simons-Schrödinger system (CSS system) involving a nonlinearity f(u) in the case of critical exponential growth −∆u+ u+A0u+ 2∑ j=1 A2 ju = f(u), ∂1A0 = A2|u|2, ∂2A0 = −A1|u|2, ∂1A2 − ∂2A1 = −1 2 u2, ∂1A1 + ∂2A2 = 0, (1.1) where Aµ ∈ R, µ = 0, 1, 2, is vector potential of the gauge fields, ∂0 = ∂ ∂t , ∂1 = ∂ ∂x1 , ∂2 = ∂ ∂x2 . This system arises in the study of the standing wave of Chern-Simons- Schrödinger system that describes the dynamics of large number of particles in a electromagnetic field. Chern-Simons terms in CSS system are necessary ingredi- ents in various anyon models describing many fermimion systems such as electron paring in the high-temperature superconductor, fractional quantum Hall effect and Aharovnov-Bohm scattering, see [28, 29] and references therein. Since the gauge field Aµ is coupled to complex field φ ∈ C, the Euler-Lagrange equations of the energy which are given by iD0φ+ (D1D1 +D2D2)φ = f(φ), ∂0A1 − ∂1A0 = − Im(φ̄D2φ), ∂0A2 − ∂2A0 = Im(φ̄D1φ), ∂1A2 − ∂2A1 = −1 2 |φ|2. (1.2) Here Dµφ = (∂µ + iAµ)φ, µ = 0, 1, 2. The CSS system (1.2) is invariant under the following gauge transformation φ → φeiχ, Aµ → Aµ − ∂µχ where χ : R1+2 → R is an arbitrary C∞ function. We assume that the gauge field satisfies the Coulomb 2010 Mathematics Subject Classification. 35J50, 35J10. Key words and phrases. Chern-Simons gauge field; Schrödinger equation; Moser-Trudinger inequality. c©2021. This work is licensed under a CC BY 4.0 license. Submitted July 9, 2021. Published September 15, 2021. 1 2 Y. WAN, J. TAN EJDE-2021/77 gauge condition ∂0A0 + ∂1A1 + ∂2A2 = 0. Then the standing wave ψ(x, t) = eiωt u satisfies −∆u+ ωu+A0u+A2 1u+A2 2u = f(u), ∂1A0 = A2u 2, ∂2A0 = −A1u 2, ∂1A2 − ∂2A1 = −1 2 |u|2, ∂1A1 + ∂2A2 = 0. (1.3) We say that f(s) has subcritical growth at +∞ if for all α > 0, lim s→+∞ f(s) eαs2 = 0 (1.4) and f(s) has critical growth at +∞ if there exists α0 > 0 such that lim s→+∞ f(s) eαs2 = { 0, if α > α0, +∞, if α < α0. (1.5) We assume f(u) satisfies the following conditions: (A1) f ∈ C(R,R) and f(0) = 0, lims→0 F (s)/s2 = 0; (A2) There exist θ > 6 and s1 > 0 such that for all |s| ≥ s1 0 < θF (s) := θ ∫ s 0 f(t) dt ≤ sf(s); (A3) There exists β0 > 0 such that lim s→+∞ sf(s)e−α0s 2 ≥ β0. We remark that the condition (A2) can be replaced by 0 < F (s) ≤M0f(s), if |s| ≥ R0, for some constants R0, M0 > 0. The standing waves of (1.2) have been investigated by Byeon, Huh and Seok [2]. They were seeking the radial solutions when f(u) = λ|u|p−1u, λ > 0 and p > 2 by variational methods, see also [11, 12]. A series of existence and nonexistence results of solitary waves has been established in [4, 5, 17, 24, 25, 26, 30]. 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 [27]. Sign-changing solutions and Nodal standing waves to a gauged nonlinear Schrödinger equation have been established by [7, 18, 19, 20]. Sign- changing multi-bump solutions were found in [3]. Moreover, we have shown the existence of nontrivial solutions to Chern-Simons- Schrödinger systems (1.1) by using the concentration compactness principle with V (x) is a constant and the argument of global compactness with V ∈ C(R2) and 0 < V0 < V (x) < V∞ under the condition p > 4 in [28]. We also have obtained the concentration behavior of the solutions to system (1.1) with p > 6 in [29]. The main characteristic of system (1.1) is that the non-local term Aµ, µ = 0, 1, 2 depends on u and there is a lack of compactness in R2. By using the variational method we can obtain the following result. Theorem 1.1. If f(s) is critical growth and (A1)–(A3) hold, then Problem (1.1) has a solution. EJDE-2021/77 CSS SYSTEMS WITH CRITICAL EXPONENTIAL GROWTH 3 We mention that Zhang and Wan also proved that if f(s) is subcritical growth then Problem (1.1) has a solution in [33]. On the other hand, radial solutions for the Chern-Simons-Schrödinger equation with exponential growth can be found in [16]. To demonstrate the desired result, we employ the approach which was developed by do Ó, Medeiros and Severo [8]. Here we mention that Pan, Li, Tang [23] studied CSS system with critical growth; see also [6, 21]. Sign-changing solutions have been found for the nonlinear Chern-Simons-Schrödinger equations in [31] Normalized solutions of Chern-Simons-Schrödinger system are studied by [10, 22, 32]. This article is organized as follows. In Section 2 we introduce the framework and prove some technical lemmas. In Section 3 we prove Theorem 1.1. 2. Mathematical framework In this section, we outline the variational framework for a future study. We consider the functions which belong to the usual Sobolev space H1(R2) with ‖u‖ = (∫ R2 |∇u|2 + |u|2 dx )1/2 . Define the functional J(u) = 1 2 ∫ R2 ( |∇u|2 + |u|2 +A2 1|u|2 +A2 2|u|2 ) dx− ∫ R2 F (u) dx, (2.1) where F (u) = ∫ u 0 f(s) ds. We have the derivative of J in H1(R2) as follows 〈J ′(u), η〉 = ∫ R2 ( ∇u∇η + uη − f(u)η + (A2 1(u) +A2 2(u))uη +A0uη ) dx + 2 ∫ R2 A1u 2 ∫ R2 K2(x, y)u(y)η(y) dy dx + 2 ∫ R2 A2u 2 ∫ R2 −K1(x, y)u(y)η(y) dy dx, (2.2) for all η ∈ C∞0 (R2). Especially, from (2.4), we have 〈J ′(u), u〉 = ∫ R2 ( |∇u|2 + |u|2 + 3 ( A2 1(u) +A2 2(u) ) |u|2 − f(u)u ) dx. (2.3) Substituting ∂1A0 = A2u 2, ∂2A0 = −A1u 2 in the Coulomb gauge condition ∂1A1 + ∂2A2 = 0, we obtain 0 = ∂2∂1A0 − ∂1∂2A0 = ∂2(A2u 2) + ∂1(A1u 2) = 2u(A1∂1u+A2∂2u) + u2(∂1A1 + ∂2A2). This implies 2∑ j=1 Aj∂ju = 0. This also implies the imaginary part of the CSS system vanishes. 4 Y. WAN, J. TAN EJDE-2021/77 Again we can derive from ∂1A2 − ∂2A1 = − 1 2u 2 that∫ R2 A0|u|2 dx = −2 ∫ R2 A0(∂1A2 − ∂2A1) dx = 2 ∫ R2 (A2∂1A0 −A1∂2A0) dx = 2 ∫ R2 (A2 1 +A2 2)|u|2 dx. (2.4) Combining the equation ∂1A2 − ∂2A1 = −u2/2 and the Coulomb gauge condition ∂1A1 + ∂2A2 = 0 provides that the components Aj can be determined from u by solving elliptic system ∆A1 = ∂2( |u|2 2 ), ∆A2 = −∂1( |u|2 2 ). That are equivalent to F(A1) = −ξ2 |ξ|2 F( |u|2 2 ), F(A2) = ξ1 |ξ|2 F( |u|2 2 ) where F denotes the Fourier transform of an integrable function. Then we have the following representation of (A1, A2), A1 = A1(u) = K2 ∗ ( |u|2 2 ) = − 1 2π ∫ R2 x2 − y2 |x− y|2 |u|2(y) 2 dy, (2.5) A2 = A2(u) = −K1 ∗ ( |u|2 2 ) = 1 2π ∫ R2 x1 − y1 |x− y|2 |u|2(y) 2 dy, (2.6) where Kj = −xj 2π|x|2 for j = 1, 2 and ∗ denotes the convolution. Moreover, the system ∂1A0 = A2u 2, ∂2A0 = −A1u 2 implies that ∆A0 = ∂1(A2|u|2)− ∂2(A1|u|2), which yields the following representation A0 = A0(u) = K1 ∗ (A1|u|2)−K2 ∗ (A2|u|2) = K1 ∗ ( |u|2K2 ∗ |u|2 2 ) +K2 ∗ ( |u|2K1 ∗ |u|2 2 ) . (2.7) We know that J is well defined in H1(R2), J ∈ C1(H1(R2)), and the weak solution of (1.1) is the critical point of the functional J from the following properties, which we refer to [28, 29]. Proposition 2.1. Let 1 < s < 2 and 1 s − 1 q = 1 2 . (i) There is a constant C depending only on s and q such that(∫ R2 ∣∣Tu(x) ∣∣q dx)1/q ≤ C (∫ R2 |u(x)|s dx )1/s , where the integral operator T is defined as Tu(x) := ∫ R2 u(y) |x− y| dy. EJDE-2021/77 CSS SYSTEMS WITH CRITICAL EXPONENTIAL GROWTH 5 (ii) If u ∈ H1(R2), then we for j = 1, 2, ‖A2 j (u)‖Lq(R2) ≤ C‖u‖2L2s(R2), ‖A0(u)‖Lq(R2) ≤ C‖u‖2L2s(R2)‖u‖ 2 L4(R2). (iii) For q′ = q q−1 and j = 1, 2, we have ‖Aj(u)u‖L2(R2) ≤ ‖|Aj(u)|2‖Lq(R2)‖u‖2L2q′ (R2) . We will need the following properties of the convergence for Aj , see [29]. Proposition 2.2. Suppose that un converges to u a.e. in R2 and un converges weakly to u in H1(R2). Let Aµ,n := Aµ(un(x)), µ = 0, 1, 2. Then (i) Aµ,n converges to Aµ(u(x)) a.e. in R2. (ii) ∫ R2 A 2 j,nunu dx, ∫ R2 A 2 j,n|u|2 dx, and ∫ R2 A 2 j,n|un|2 dx converge to ∫ R2 A 2 j |u|2 dx, for j = 1, 2; ∫ R2 A0,nunu dx and ∫ R2 A0,n|un|2 dx converge to ∫ R2 A0|u|2 dx. (iii) ∫ R2 |Aj(un − u)|2|un − u|2 dx = ∫ R2 |Aj(un)|2|un|2 dx − ∫ R2 |Aj(u)|2|u|2 dx + on(1), for j = 1, 2. To prove the mountain pass construction, we need the following results from [8]. Proposition 2.3. (i) If α > 0 and u ∈ H1(R2), then∫ R2 ( eαu 2 − 1 ) dx <∞. Moreover, if ‖∇u‖22 ≤ 1, ‖u‖2 ≤ M < ∞ and α < 4π then there exists a constant C = C(M, α), which depends only on M and α, such that∫ R2 ( eαu 2 − 1 ) dx < C(M, α). (ii) Let {wn} in H1(R2) satisfy ‖wn‖ = 1. Suppose that wn weakly converges to w0 in H1(R2) with ‖w0‖ < 1. Then for all 0 < β < 4π 1−‖w0‖2 , sup n ∫ R2 ( eβ|wn|2 − 1 ) dx <∞. (iii) Let β > 0 and r > 1. Then for each α > r there exists a positive constant C = C(α) such that for all s ∈ R,( eβs 2 − 1 )r ≤ C(eαβs2 − 1 ) . In particular, if u ∈ H1(R2) then (eβu 2 − 1)r belongs to L1(R2). (iv) If v ∈ H1(R2), β > 0, q > 0 and ‖v‖ ≤ M with βM2 < 4π, then there exists C = C(β,M, q) > 0 such that∫ R2 ( eβv 2 − 1 ) |v|q dx ≤ C‖v‖q. (2.8) Next, we prove that the energy functional J has the mountain pass structure. Lemma 2.4. Assume (A1), (A2), and (1.5) hold. Then there exists ρ > 0 such that J(u) > 0 if ‖u‖ = ρ. Proof. From (A1), (A2), and (1.5), there exists ε < λ/2, where λ is the best constant of L2(R2) ↪→ H1(R2), such that |F (s)| ≤ ε|s|2 + C1|s|q(eαs 2 − 1), (2.9) 6 Y. WAN, J. TAN EJDE-2021/77 for all s ∈ R and q > 2. By (iv) of Proposition 2.3 and the Sobolev embeddings, we obtain J(u) ≥ ( 1 2 − ε λ )‖u‖2 − C1‖u‖q. (2.10) Consequently, by using ε < 1 2λ and q > 2, we can choose ρ > 0 such that for ‖u‖ = ρ J(u) ≥ ‖u‖[(1 2 − ε λ )‖u‖ − c‖u‖q−1] > 0. � Lemma 2.5. Assume that f satisfies (A2). Then there exists e ∈ E with ‖e‖ > ρ such that I(e) < inf‖u‖=ρ I(u). Proof. Let u ∈ H1(R2) such that u ≡ s1 in B1, u ≡ 0 in Bc2 and u ≥ 0. Denoting k = supp(u). From (A2), for all s ∈ R we have F (s) ≥ C1|s|θ − C2. (2.11) Then, for t > 1 we have I(tu) ≤ t2 2 ‖u‖2 + ct6‖u‖6 − ctθ ∫ {x: t|u(x)|≥s1} uθ dx+ C1|k|. Since θ > 6, we obtain I(tu) → −∞ as t → +∞. Setting e = tu with t large enough, the proof is complete. � We need the following result to prove the (PS) condition. Lemma 2.6. Assume (A2) and (1.5). Let (un) in E such that J(un) → c and J ′(un)→ 0. Then, ‖un‖ ≤ c0, ∫ R2 f(un)un dx ≤ c0, and ∫ R2 F (un) dx ≤ c0. Proof. First, we prove that ‖un‖ ≤ c0. We have 1 2 ‖un‖2 + 1 2 ∫ R2 ( A2 1,n|un|2 +A2 2,n|un|2 ) dx− ∫ R2 F (un) dx = c+ on(1) and for any ϕ ∈ E,∫ R2 ( ∇un∇ϕ+ unϕ ) dx+ ∫ R2 ( A2 1,n +A2 2,n +A0,n ) unϕdx− ∫ R2 f(un)ϕdx = on(‖ϕ‖). From (A2) and θ > 6, we obtain θc+ εn‖un‖ ≥ ( θ 2 − 1)‖un‖2 + ( θ 2 − 3) ∫ R2 ( A2 1,n|un|2 +A2 2,n|un|2 ) dx − ∫ R2 ( θF (un)− f(un)un ) dx ≥ ( θ 2 − 1)‖un‖2 − ∫ {x:|un(x)| r0. Notice that ψ̃n ∈ H1(R2), supp ψ̃n ⊂ Br0 , for a fixed r0. By using the fact∫ {a0<|x|<1} ∇ log |x| dx = 2π ∫ 1 a0 |∇ log r|2r dr = 2π ∫ 1 a0 1 r dr = −2π ln a0, we can prove that ∫ R2 |∇ψ̃n|2 dx = 1. Moreover,∫ R2 |ψ̃n|2 dx = O( 1 log n ), as n→∞. Thus, we can conclude that ‖ψ̃n‖ → 1 as n→∞. Considering ψn = ψ̃n/‖ψ̃n‖, we can rewrite ψ2 n(x) = (2π)−1 log n+ dn, for all |x| ≤ r0 n , where dn = (2π)−1(‖ψ̃n‖−1 − 1) log n. Consequently dn log n → 0 as n→∞. (3.1) On the other hand, we know that lim n→∞ ∫ R2 |ψn|2 dx = 0. By the Hölder inequality, for 2θ + q1(1− θ) = 4 we have ‖ψn‖4L4(R2) ≤ ‖ψn‖ 2θ L2(R2)‖ψn‖ (1−θ)q1 Lq1 (R2). 8 Y. WAN, J. TAN EJDE-2021/77 Then we can deduce that lim n→∞ ∫ R2 |ψn|q1 dx = 0 for q1 ≥ 2, lim n→∞ ∫ R2 Aj(ψn)2ψ2 n dx = 0. Proposition 3.1. Assume that (A2)–(A4), hold. Then there exists n ∈ N such that max t≥0 [ t2 2 + t6 2 ∫ R2 ( A2 1(ψn)|ψn|2 +A2 2(ψn)|ψn|2 ) dx− ∫ R2 F (tψn) dx ] < 2π α0 . Proof. Let us choose r0 > 0 such that β0 > 2 r2 0α0 , (3.2) where β0 has been fixed in (A3). Suppose by contradiction that for all n t2 2 + t6 2 ∫ R2 ( A2 1(ψn)|+A2 2(ψn) ) |ψn|2 dx− ∫ R2 F (tψn) dx ≥ 2π α0 . (3.3) From (A2), there exist positive constants C1, C1 such that F (s) ≥ C1e |s| M0 − C2. Consequently, if t > 0 is sufficiently large and m > 2, we have∫ R2 F (tψn) dx ≥ −C1 + ∫ {tψn≥s1} etψn/M0 dx ≥ −C1 + C3 ∫ {tψn≥s1} (ψn)m dx ≥ −C1 + C3t m ∫ {ψn≥s1} (ψn)m dx. Hence, for each n there exists unique maximum point tn such that t2n 2 + t6n 2 ∫ R2 ( A2 1(ψn) +A2 2(ψn) ) |ψn|2 dx− ∫ R2 F (tnψn) dx = max t>0 J(tψn) and d dt J(tψn) ∣∣ t=tn = 0. From which it follows that t2n + 3t6n ∫ R2 ( A2 1(ψn) +A2 2(ψn) ) |ψn|2 dx− ∫ R2 tnψnf(tnψn) dx = 0. (3.4) By (A3) for each ε > 0 there exists Rε > 0 such that ψnf(ψn) ≥ (β0 − ε) exp(α0ψ 2 n) (3.5) for all ψn ≥ Rε and |x| ≤ r0. From (3.4) and (3.5), we have t2n + 3t6n ∫ R2 ( A2 1(ψn) +A2 2(ψn) ) |ψn|2 dx ≥ (β0 − ε)π (r0 n )2 exp( α0 2π t2n log n+ 2α0t 2 ndn). (3.6) That is, 1 + 3t4n ∫ R2 ( A2 1(ψn) +A2 2(ψn) ) |ψn|2 dx EJDE-2021/77 CSS SYSTEMS WITH CRITICAL EXPONENTIAL GROWTH 9 ≥ (β0 − ε)πr2 0 exp( α0 2π t2n log n+ 2α0t 2 ndn − 2 log tn − 2 log n). Since ∫ R2 ( A2 1(ψn) + A2 2(ψn) ) |ψn|2 dx → 0, as n → ∞, we obtain that {tn} is bounded. We claim that t2n → 4π α0 , as n→∞. (3.7) In fact, by (3.3), (3.4), and (A2), we have t2n 2 + t6n 2 ∫ R2 ( A2 1(ψn) +A2 2(ψn) ) |ψn|2 dx ≥ 2π α0 + ∫ {tnψn≤s1} F (tnψn) dx Since {tn} is bounded, by (2.9) and ‖ψ̃n‖2 → 1 as n→∞, we obtain∣∣∣ ∫ {tnψn≤s1} F (tnψn) dx ∣∣∣ ≤ C ∫ R2 ψ2 n dx = C 1 ‖ψ̃n‖2 ∫ R2 ψ̃2 n dx→ 0. Note that ∫ R2 ( A2 1(ψn) +A2 2(ψn) ) |ψn|2 dx→ 0 as n→∞. Consequently, t2n ≥ 4π α0 + on(1), as n→∞. Suppose by contradiction that limn→∞ t2n > 4π α0 . From (3.6), we have t2n + 3t6n ∫ R2 ( A2 1(ψn) +A2 2(ψn) ) |ψn|2 dx ≥ (β0 − ε)πr2 0 exp ( ( α0 4π t2n − 1)2 log n+ 2α0t 2 ndn ) . Since (3.1), the last inequality contradicts the boundedness of {tn} and the claim holds. Let us denote Ω1,n := {x ∈ Br0 : tnψn ≥ Rε}, and Ω2,n := Br0 \ Ω1,n. By (3.4) and (3.5), we obtain t2n + 3t6n ∫ R2 ( A2 1(ψn) +A2 2(ψn) ) |ψn|2 dx ≥ (β0 − ε) ∫ Br0 eα0t 2 nψ 2 n + ∫ Ω2,n tnψnf(tnψn)− (β0 − ε) ∫ Ω2,n eα0t 2 nψ 2 n . (3.8) Since ψn(x) → 0 as n → ∞ and the characteristic functions χΩ2,n → 1 for almost every x such that |x| ≤ r. By the Lebesgue dominated convergence theorem, we have ∫ Ω2,n tnψnf(tnψn) dx→ 0 and ∫ Ω2,n eα0t 2 nψ 2 n dx→ πr2 0 as n→∞. By t2n ≥ 4π α0 , we obtain∫ {|x|≤r0} eα0t 2 nψ 2 n dx ≥ ∫ {|x|≤r0} e4πψ2 n dx = ∫ {|x|≤ r0 n } e4πψ2 n dx+ ∫ { r0 n ≤|x|≤r0} e4πψ2 n dx. (3.9) 10 Y. WAN, J. TAN EJDE-2021/77 A direct computation gives lim n→∞ ∫ {|x|≤ r0 n } e4πψ2 n dx = lim n→∞ ∫ {|x|≤ r0 n } e2 logn+4πdn dx = lim n→∞ π r2 0 n2 n2+4π(logn)−1dn = πr2 0. Set t = log( r0|x| )/(ξn log n), where ξn = ‖ψ̃n‖ > 1. We have∫ { r0 n ≤|x|≤r0} e4πψ2 n dx = 2πr2 0ξn log n ∫ ξ−1 n 0 e2 logn(t2−ξnt) dt. Since t2 − ξnt ≥ { −ξnt if 0 ≤ t ≤ ξ−1 n 2 , (2ξ−1 n − ξn)(t− ξ−1 n ) + (ξ−2 n − 1) if ξ−1 n 2 ≤ t ≤ ξ −1 n , we obtain lim n→∞ ∫ {r0/n≤|x|≤r0} e4πψ2 n dx ≥ 2πr2 0. Taking n→∞ in (3.8) and using (3.7), we obtain 4π α0 ≥ (β0 − ε)2πr2 0, which gives β0 ≤ 2 α0r20 . This contradicts (3.2). The proof is complete. � Assuming that lim inf n→∞ ‖un‖2 < 4π α0 , then there exists a subsequence of {un} which converges to u0 in H1(R2). Proposition 3.2. J(u0) = c. Proof. Since {un} is bounded in H1(R2), there exists a subsequence denoted again by {un} such that un ⇀ u0 in H1(R2), un → u0 in Lqloc(R2), q ≥ 1, un(x)→ u0(x) a.e. in R2. Moreover, for any R > 0, lim n→∞ ∫ BR ( F (un)− F (u0) ) dx = 0. It is known that for u ∈ L2(R2), the Schwartz symmetrization of u satisfies |u∗| ≤ ‖u∗‖L2(R2)√ π|x| . Since ∫ Bc R F (un) ≤ C1 ∫ Bc R |un|2 + C2 ∫ Bc R ( |un|eα|un|2 − 1 ) dx, EJDE-2021/77 CSS SYSTEMS WITH CRITICAL EXPONENTIAL GROWTH 11 where α ≥ α0, and∫ Bc R ( |un|eα|un|2 − 1 ) dx ≤ ∫ Bc R ∞∑ l=1 |u∗n|2l+1 l! ≤ C R , for each δ > 0, there exists R > 0 such that max{ ∫ Bc R F (un) dx, ∫ Bc R F (u0) dx, ∫ BR ( F (un)− F (u0) ) dx} ≤ δ 3 , from which it follows that ∫ R2 ( F (un)− F (u0) ) dx < δ. Hence by using J(un)→ c we conclude that 1 2 ‖un‖2 + 1 2 ∫ R2 ( A2 1(un) +A2 2(un) ) |un|2 dx = c+ ∫ R2 F (u0) dx+ on(1). We observe that limn→∞ ‖un‖ ≥ ‖u0‖ > 0, so that we define wn = un ‖un‖ and w0 = u0 limn→∞ ‖un‖ . Then ‖wn‖ = 1 and wn ⇀ w0 in H1(R2). Suppose that ‖w0‖ < 1. By Proposition 3.1, we see that α0 < 2π c−J(u0) . Let us choose β > 1 sufficiently close to 1 and δ > 0 such that βα0‖un‖2 ≤ 2π‖un‖2 c− J(u0) − δ ≤4π c+ ∫ R2 F (u0) dx− 1 2 ∫ R2 ( A2 1(u0) +A2 2(u0) ) |u0|2 dx+ on(1) c− J(u0) − δ. On the other hand, by using the formula for J(u0) and J(u0) < c we deduce that( 1− ‖w0‖2 )( c+ ∫ R2 F (u0) dx− 1 2 ∫ R2 ( A2 1(u0) +A2 2(u0) ) |u0|2 dx+ on(1) ) ≤ c+ ∫ R2 F (u0) dx− 1 2 ∫ R2 ( A2 1(u0) +A2 2(u0) ) |u0|2 dx − ‖w0‖2 ( ∫ R2 F (u0) dx− 1 2 ∫ R2 ( A2 1(u0) +A2 2(u0) ) |u0|2 dx+ on(1) ) = c+ (−J(u0) + 1 2 ‖u0‖2) − ‖w0‖2 ( c+ ∫ R2 F (u0) dx− 1 2 ∫ R2 ( A2 1(u0) +A2 2(u0) ) |u0|2 dx+ on(1) ) ≤ c− J(u0). Therefore, there exists δ > 0 such that βα0‖un‖2 ≤ 4π 1− ‖w0‖2 − δ. Thus, (β + ε)α0‖un‖2 ≤ 4π 1−‖w0‖2 , which implies by (ii) of Proposition 2.3 that∫ R2 ( e(β+ε)α0‖un‖2w2 0 − 1 ) dx ≤ C. 12 Y. WAN, J. TAN EJDE-2021/77 We observe that | ∫ R2 f(un)(un − u0) dx| ≤ | ∫ R2 |(un − u0)|eα0‖un‖2w2 n dx| ≤ C ∫ R2 |un − u0| q q−1 dx. Thus, lim n→∞ ∫ R2 ∇u0∇(un − u0) + u0(un − u0) dx = 0. Hence, {un} converges to u0 in H1(R2). � Proof of Theorem 1.1. Let {un} satisfying J(un)→ c0 and J ′(un)→ 0 as n→∞. By Lemma 2.6, {un} is bounded, up to a subsequence, we may assume that un ⇀ u0 in H1(R2), un → u0 in Lqloc(R2) for all q ≥ 2 and un → u0 almost everywhere in R2, as n → ∞. Then, if f(s) satisfies (1.5), we have for each α > α0 there exist b1, b2 > 0 such that for all s ∈ R, for all α > 0, |f(s)| ≤ b1|s|+ b2 ( eαs 2 − 1 ) . (3.10) If the vanishing case occurs, then lim n→∞ ∫ R2 |un|2 dx = 0. (3.11) Consequently, by (3.10), (3.11), Hölder’ inequality, and Proposition 2.3, we have lim n→∞ ∫ R2 |f(un)un| dx ≤ lim n→∞ ∫ R2 ( b1|un|2 + b2|un| ( eα|un|2 − 1 )) dx ≤ b1 lim n→∞ ∫ R2 |un|2 dx + b2 ( lim n→∞ ∫ R2 |un|2 dx )1/2( lim n→∞ ∫ R2 ( eα|un|2 − 1 )2 dx )1/2 = 0. (3.12) By Proposition 2.2, we have lim n→∞ ∫ R2 ( A2 1(un) +A2 2(un) ) u2 n dx = 0. (3.13) From (2.3), (3.12), (3.13), and that {un} is bounded, we have ‖un‖2 = 〈J ′(un), un〉 − 3 ∫ R2 (( A2 1(un) +A2 2(un) ) u2 n dx+ ∫ R2 f(un)un dx→ 0, (3.14) as n→∞. By (2.9), (3.12), (3.14), and Hölder’s inequality, we have lim n→∞ | ∫ R2 F (un) dx| ≤ lim n→∞ ( ε ∫ R2 |un|2 dx+ C1 ∫ R2 |un|q(eαu 2 n − 1) dx ) = 0. (3.15) This implies that 0 < J(un) → 0 as n → ∞, which means that vanishing is impossible. Hence only the nonvanishing case happens. Since∫ R2 u2(x)〈A′0(u), η〉 dx EJDE-2021/77 CSS SYSTEMS WITH CRITICAL EXPONENTIAL GROWTH 13 = ∫ R2 u2(x) (∫ R2 x1 − y1 2π|x− y|2 u(y)η(y)(y)A2(u(y)) dy − ∫ R2 x2 − y2 2π|x− y|2 u(y)η(y)A1(u(y)) dy ) dx = ∫ R2 A2(u(y))u(y)η(y) ∫ R2 x1 − y1 2π|x− y|2 u(x)2 dxdy − ∫ R2 A1(u(y))u(y)η(y) ∫ R2 x2 − y2 2π|x− y|2 u2(x) dx dy = ∫ R2 |A2(u(y))|2u(y)η(y) + |A1(u(y))|2u(y)η(y) dy, For each η ∈ C∞0 (R2), we have 0 = lim n→∞ 〈J ′(un), η〉 = lim n→∞ ∫ R2 ( ∇un∇η + unη + (A2 1(un) +A2 2(un))unη +A0(un)unη − f(un)η ) dx = 〈J ′(u0), η〉. Hence u0 is a week solution of Problem (1.1). � Acknowledgments. Both authors were supported by Jianghan University. Y. 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Xia; Existence, nonexistence and multiplicity results of a Chern-Simons-Schrödinger sys- tem, Acta Appl. Math., 166 (2020), 147–159. [31] W. Xie, C. Chen; Sign-changing solutions for the nonlinear Chern-Simons-Schrödinger equa- tions, Appl. Anal., 99 (2020), 880–898. [32] J. Yuan; Multiple normalized solutions of Chern-Simons-Schrödinger system, Nonl. Diff. Equ. Appl., 22 (2015), 1801–1816. [33] C. Zhang, Y. Wan; The existence of solutions to Chern-Simons-Schrödinger systems with exponential nonlinearities, Journal of Math., Vol. 38, 5 (2018), 804-812. Youyan Wan The Department of Mathematics, Jianghan University, Wuhan, Hubei 430056, China Email address: wanyouyan@jhun.edu.cn Jinggang Tan Department of Mathematics, Jiangxi Normal University, Nanchang 330022, China. Departamento de Matemática, Universidad Técnica Federico Santa Maŕıa, Avda. España 1680, Valparáıso, Chile Email address: jinggang.tan@usm.cl 1. Introduction and main result 2. Mathematical framework 3. Proof of main results Acknowledgments References