Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 251, pp. 1–11. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMICLASSICAL SOLUTIONS FOR LINEARLY COUPLED SCHRÖDINGER EQUATIONS SITONG CHEN, XIANHUA TANG Abstract. We consider the system of coupled nonlinear Schrödinger equations −ε2 ∆u + a(x)u = Hu (x, u, v) + µ(x)v, x ∈ RN , −ε2 ∆v + b(x)v = Hv (x, u, v) + µ(x)u, x ∈ RN , 1 N u, v ∈ H (R ), C(RN ) where N ≥ 3, a, b, µ ∈ and Hu , Hv ∈ C(RN × R2 , R). Under conditions that a0 = inf a = 0 or b0 = inf b = 0 and |µ(x)|2 ≤ θa(x)b(x) with θ ∈ (0, 1) and some mild assumptions on H, we show that the system has at least one nontrivial solution provided that 0 < ε ≤ ε0 , where the bound ε0 is formulated in terms of N, a, b and H. 1. Introduction In this article, we study the existence of semiclassical solutions of the system of coupled nonlinear Schrödinger equations −ε2 ∆u + a(x)u = Hu (x, u, v) + µ(x)v, x ∈ RN , −ε2 ∆v + b(x)v = Hv (x, u, v) + µ(x)u, x ∈ RN , 1 (1.1) N u, v ∈ H (R ), where z := (u, v) ∈ R2 , N ≥ 3, a, b, µ ∈ C(RN , R) and H, Hu , Hv ∈ C(RN × R2 , R). Systems of this type arise in nonlinear optics [1]. In the past several years, there are many papers about the semiclassical solutions of the nonlinear perturbed Schrödinger equation −ε2 ∆u + V (x)u = f (x, u), u ∈ H 1 (RN ) under various hypotheses on the potential and the nonlinearity (see [2, 6, 7, 12, 13, 14, 16, 19, 22, 25]). However, by Kaminow [17], we know that single-mode optical fibers are not really “single-mode”, but actually bimodal due to the presence of birefringence. And recently, different authors focused their attention on coupled nonlinear Schrödinger 2000 Mathematics Subject Classification. 35J20, 58E50. Key words and phrases. Nonlinear Schrödinger equation; semiclassical solution; coupled system. c 2014 Texas State University - San Marcos. Submitted October 23, 2014. Published December 1, 2014. 1 2 S. CHEN, X. H. TANG EJDE-2014/251 systems (see [3, 4, 8, 9, 10, 11, 18]) which describe physical phenomena (see, e.g., [1, 5, 15]). In a recent article, [8], Chen and Zou studied the system of nonlinear Schrödinger equations −ε2 ∆u + a(x)u = f (u) + µv, x ∈ RN , −ε2 ∆v + b(x)v = g(v) + µu, u, v > 0 in RN , x ∈ RN , (1.2) u, v ∈ H 1 (RN ), where N, a and b are the same as in (1.1). Under the assumptions (i) there exists a constant a0 > 0 such that a(x), b(x) ≥ a0 and 0 ≤ µ < a0 ; g(s) (ii) f, g ∈ C(RN , R) and lims→0 f (s) s = s = 0; ∗ (iii) there exists a constant p0 ∈ (1, 2 − 1) such that lim sup s→+∞ f (s) < +∞, sp0 Rs lim sup s→+∞ g(s) < +∞; sp0 Rs f (t) dt s2 g(t) dt (iv) either lim sups→+∞ 0 = +∞ or lim sups→+∞ 0 s2 = +∞. They proved that (1.2) has a positive solution for sufficiently small ε > 0 and all µ ∈ (0, µ1 ] for some µ1 ∈ (0, a0 ). Obviously, if a0 = 0, their arguments become invalid due to the fact that 0 ≤ µ < a0 can not be satisfied. To the best of our knowledge, the existence of semiclassical solutions to system (1.1), under the assumption of a0 = inf a = 0 or b0 = inf b = 0, has not ever been studied by variational methods. In addition, as the nonlinearity is non-autonomous and dependent on u and v, the problem will become more complex. Motivated by [8, 20, 24, 26], we shall choose the case a0 = inf a = 0 or b0 = inf b = 0 as the objective of the present paper. Before presenting the main results, we introduce the following assumptions. (A0) a(x) ≥ a(0) = 0, b(x) ≥ 0 and there exist a0 , b0 > 0 such that the sets Aa0 := {x ∈ RN : a(x) < a0 } and Bb0 := {x ∈ RN : b(x) < b0 } have finite measure; (A1) there exists a constant θ ∈ (0, 1) such that |µ(x)|2 ≤ θa(x)b(x), for all x ∈ RN ; (B0) a(x) ≥ 0, b(x) ≥ b(0) = 0 and there exist a0 , b0 > 0 such that the sets Aa0 := {x ∈ RN : a(x) < a0 } and Bb0 := {x ∈ RN : b(x) < b0 } have finite measure; (H1) there exist constants p ∈ (2, 2∗ ) and C > 0 such that |H(x, z)| ≤ C(|z| + |z|p ), ∀(x, z) ∈ RN × R2 ; (H2) Hz (x, z) · z = o(|z|2 ), as |z| → 0, uniformly in x ∈ RN ; (H3) lim|z|→∞ |H(x,z)| = ∞ uniformly in x ∈ RN ; |z|2 (H4) there exist c0 > 0, T0 > 0 and q ∈ (2, 2∗ ) such that H(x, u, 0) ≥ c0 |u|q , ∀x ∈ RN , u ∈ [−T0 , T0 ] and −2 4−N u Z h |x|≤h H(λ−1/2 x, u/h, 0) dx ≥ (N 2 + 2)ωN , 2N (1 − 2−N )2 for all h ≥ 1, λ ≥ 1, u ≥ hT0 ; here and in the sequel, ωN = meas(B1 (0)) = 2π N/2 /N Γ(N/2); EJDE-2014/251 SEMICLASSICAL SOLUTIONS 3 (H4’) there exist c0 > 0, T0 > 0 and q ∈ (2, 2∗ ) such that H(x, 0, v) ≥ c0 |v|q , ∀x ∈ RN , v ∈ [−T0 , T0 ] and v −2 h4−N Z H(λ−1/2 x, 0, v/h) dx ≥ |x|≤h (N 2 + 2)ωN , 2N (1 − 2−N )2 for all h ≥ 1, λ ≥ 1, v ≥ hT0 ; (H5) H(x, z) := 12 Hz (x, z) · z − H(x, z) ≥ 0 for all (x, z) ∈ RN × R2 , and there exist c1 > 0 and κ > max{1, N/2} such that (1 − θ)m0 Hz (x, z) · z ≥ ⇒ |Hz (x, z) · z|κ ≤ c1 |z|2κ H(x, z), |z|2 3 where m0 := min{a0 , b0 }; (H6’) there exist c0 > 0 and q ∈ (2, 2∗ ) such that H(x, u, 0) ≥ c0 |u|q for all (x, u) ∈ RN × R; (H6”) there exist c0 > 0 and q ∈ (2, 2∗ ) such that H(x, 0, v) ≥ c0 |v|q for all (x, v) ∈ RN × R. Remark 1.1. It is easy to check that (H6’) and (H6”) imply (H4) and (H4’) with T0 = 1/(q−2) N2 + 2 , −N 2 2c0 (1 − 2 ) respectively, but (H4), (H4’) can not yield (H6’), (H6”). We give the following nonlinear example to illustrate it. Let H(x, u, v) = (|u|2 + |v|2 ) ln(1 + |u| + |v|). Clearly, H satisfies both (H4) and (H4’) with ln(1 + T0 ) = N2 + 2 2 1 − 2−N 2 , but neither (H6’) nor (H6”). Example 1.2. Let q ∈ (2, 2∗ ). Then it is easy to see that following two functions satisfy (H1)–(H3) and (H6’): q/2 H(x, u, v) = a1 |u|q + a2 |v|q , H(x, u, v) = ζ(x) |u|2 + |v|2 , where a1 , a2 > 0 and ζ ∈ C(RN ) with 0 < inf RN ζ ≤ supRN ζ < +∞. Since (q − 2)N − 2q < 0, we can let h0 ≥ 1 be such that n N 2 + 2(N + 2) oq/(q−2) (q − 2)ωN [(q−2)N −2q]/(q−2) h0 2N q(qc0 )2/(q−2) (N + 2)(1 − 2−N )2 (2κ−N )/2 (1 − θ)κ m0 = 3κ c1 (γ2∗ γ0 )N (1.3) , where γ0 and γ2∗ are embedding constants, see (2.1) and (2.2). If a and b satisfy (A0), we can choose λ0 > 1 such that sup λ1/2 |x|≤2h0 |a(x)| ≤ h−2 0 , ∀λ ≥ λ0 , (1.4) 4 S. CHEN, X. H. TANG EJDE-2014/251 if a and b satisfy (B0), we can choose λ0 > 1 such that |b(x)| ≤ h−2 0 , sup λ1/2 |x|≤2h ∀λ ≥ λ0 . (1.5) 0 Letting ε−2 = λ, (1.1) is rewritten as −∆u + λa(x)u = λHu (x, u, v) + λµ(x)v, x ∈ RN , −∆v + λb(x)v = λHv (x, u, v) + λµ(x)u, x ∈ RN , 1 (1.6) N u, v ∈ H (R ). Let Z 1 2 (|∇u|2 + |∇v|2 + λa(x)|u|2 + λb(x)|v|2 ) dx Z Z −λ H(x, z) dx − λ µ(x)uv dx, z = (u, v). Φλ (z) = RN RN (1.7) RN Obviously, the solutions of (1.1) are the critical points of Φε−1/2 (z); the solutions of (1.6) are the critical points of Φλ (z). We are now in a position to state the main results of this paper. Theorem 1.3. Assume that a, b, µ and H satisfy (A0), (A1), (H1)–(H5). Then −1/2 for 0 < ε ≤ λ0 , (1.1) has a solution zε = (uε , vε ) such that (2κ−N )/2 0 < Φε−1/2 (zε ) ≤ (1 − θ)κ m0 3κ c1 (γ2∗ γ0 )N εN −2 , (2κ−N )/2 Z H(x, zε ) dx ≤ RN (1 − θ)κ m0 3κ c1 (γ2∗ γ0 )N εN . Theorem 1.4. Assume that a, b, µ and H satisfy (A1), (B0), (H1)–(H3), (H4’), −1/2 (H5). Then for 0 < ε ≤ λ0 , (1.1) has a solution zε = (uε , vε ) such that (2κ−N )/2 0 < Φε−1/2 (zε ) ≤ (1 − θ)κ m0 3κ c1 (γ2∗ γ0 )N εN −2 , (2κ−N )/2 Z H(x, zε ) dx ≤ RN (1 − θ)κ m0 3κ c1 (γ2∗ γ0 )N εN . Theorem 1.5. Assume that a, b, µ and H satisfy (A0), (A1), (H1)–(H5). Then for λ ≥ λ0 , (1.6) has a solution zλ = (uλ , vλ ) such that (2κ−N )/2 0 < Φλ (zλ ) ≤ (1 − θ)κ m0 3κ c1 (γ2∗ γ0 )N λ1−N/2 , (2κ−N )/2 Z H(x, zλ ) dx ≤ RN (1 − θ)κ m0 3κ c1 (γ2∗ γ0 )N λ−N/2 . Theorem 1.6. Assume that a, b, µ and H satisfy (A1), (B0), (H1)–(H3), (H4’), (H5). Then for λ ≥ λ0 , (1.6) has a solution zλ = (uλ , vλ ) such that (2κ−N )/2 0 < Φλ (zλ ) ≤ (1 − θ)κ m0 3κ c1 (γ2∗ γ0 )N (2κ−N )/2 Z H(x, zλ ) dx ≤ RN λ1−N/2 , (1 − θ)κ m0 3κ c1 (γ2∗ γ0 )N λ−N/2 . EJDE-2014/251 SEMICLASSICAL SOLUTIONS 5 The rest of the article is organized as follows. In Section 2, we provide some preliminaries and lemmas. In Section 3, we give the proofs of Theorems 1.3–1.6. 2. Preliminaries Let E = (u, v) ∈ H 1 (RN ) × H 1 (RN ) : Z [a(x)|u|2 + b(x)|v|2 ] dx < +∞ , RN Z [|∇u|2 + λa(x)|u|2 + |∇v|2 + λb(x)|v|2 ] dx kzkλ† = 1/2 , ∀z = (u, v) ∈ E. RN Analogous to the proof of [23, Lemma 1], by using (A0) or (B0) and the Sobolev inequality, one can show that there exists a constant γ0 > 0 independent of λ such that kzkH 1 (RN ) ≤ γ0 kzkλ† , ∀z ∈ E, λ ≥ 1. (2.1) This shows that (E, k · kλ† ) is a Banach space for λ ≥ 1. Furthermore, by the Sobolev embedding theorem, we have kzks ≤ γs kzkH 1 (RN ) ≤ γs γ0 kzkλ† , ∀z ∈ E, λ ≥ 1, 2 ≤ s ≤ 2∗ , (2.2) here and in the sequel, we denote by k · ks the usual norm in space Ls (RN ). In view of the definition of the norm k · kλ† , we can re-write Φλ in the form Z Z 1 H(x, z) dx − λ µ(x)uv dx, ∀z ∈ E. (2.3) Φλ (z) = kzk2λ† − λ 2 RN RN It is easy to see that Φλ ∈ C 1 (E, R) and Z hΦ0λ (z), z̃i = [∇u · ∇ũ + ∇v · ∇ṽ + λa(x)uũ + λb(x)vṽ] dx RN Z −λ [Hu (x, z)ũ + Hv (x, z)ṽ] dx N ZR −λ µ(x)(uṽ + vũ)] dx, ∀z = (u, v), z̃ = (ũ, ṽ) ∈ E. (2.4) RN As in [20], we let ϑ(x) := 1 , h0N −1 h0 1−2−N 0, |x| ≤ h0 , [|x|−N − (2h0 )−N ], h0 < |x| ≤ 2h0 , (2.5) |x| > 2h0 . Then ϑ ∈ H 1 (RN ), moreover, Z N 2 ωN 2 k∇ϑk2 = |∇ϑ(x)|2 dx ≤ hN −4 , (N + 2)(1 − 2−N )2 0 RN Z 2ωN 2 kϑk2 = |ϑ(x)|2 dx ≤ hN −2 . −N )2 N 0 (1 − 2 N R (2.6) (2.7) Let eλ (x) = ϑ(λ1/2 x). Then we can prove the following lemma which is used for our proofs. 6 S. CHEN, X. H. TANG EJDE-2014/251 Lemma 2.1. Let H(x, z) ≥ 0, for all (x, z) ∈ RN × R2 . Suppose that (A0), (A1), (H1)–(H4) are satisfied. Then (2κ−N )/2 sup{Φλ (seλ , 0) : s ≥ 0} ≤ (1 − θ)κ m0 3κ c1 (γ2∗ γ0 )N λ1−N/2 , ∀λ ≥ λ0 . (2.8) Proof. From (H4), (1.3), (1.4) ,(1.7), (2.5), (2.6) and (2.7), we obtain Φλ (seλ , 0) Z Z s2 (|∇eλ |2 + λa(x)|eλ |2 ) dx − λ H(x, seλ , 0) dx = 2 RN RN Z h s2 Z i (|∇ϑ|2 + a(λ−1/2 x)|ϑ|2 ) dx − = λ1−N/2 H(λ−1/2 x, sϑ, 0) dx 2 RN RN h s2 k∇ϑk22 + kϑk22 sup |a(λ−1/2 x)| ≤ λ1−N/2 2 |x|≤2h0 Z i − H(λ−1/2 x, s/h0 , 0) dx (2.9) |x|≤h0 ≤λ 1−N/2 h s2 (k∇ϑk22 2 ∀s ≥ 0, λ ≥ λ0 , + 2 h−2 0 kϑk2 ) s2 2 (k∇ϑk22 + h−2 0 kϑk2 ) − 2 Z Z i H(λ−1/2 x, s/h0 , 0) dx , − |x|≤h0 H(λ−1/2 x, s/h0 , 0) dx |x|≤h0 2 s2 (N + 2)ωN N −4 2 ≤ [k∇ϑk22 + h−2 h ] ≤ 0, ∀s ≥ h0 T0 , λ ≥ λ0 0 kϑk2 − 2 N (1 − 2−N )2 0 and Z s2 −2 2 2 (k∇ϑk2 + h0 kϑk2 ) − H(λ−1/2 x, s/h0 , 0) dx 2 |x|≤h0 ≤ ≤ s2 c0 ωN q N −q 2 (k∇ϑk22 + h−2 s h0 0 kϑk2 ) − 2 N 2 q/(q−2) (q − 2)(k∇ϑk22 + h−2 0 kϑk2 ) −q 2/(q−2) 2q( qc0NωN hN ) 0 n (q − 2)ωN N 2 + 2(N + 2) oq/(q−2) [(q−2)N −2q]/(q−2) ≤ h0 2N q(qc0 )2/(q−2) (N + 2)(1 − 2−N )2 (2.10) (2.11) (2κ−N )/2 = (1 − θ)κ m0 3κ c1 (γ2∗ γ0 )N , ∀0 ≤ s ≤ h0 T0 , λ ≥ λ0 . The conclusion of Lemma 2.1 follows from (2.9), (2.10) and (2.11). We can prove the following lemma in the same way as Lemma 2.1. Lemma 2.2. Let H(x, z) ≥ 0 for all (x, z) ∈ RN × R2 . Suppose that (A1), (B0), (H1)–(H3), (H4’) are satisfied. Then (2κ−N )/2 sup{Φλ (0, seλ ) : s ≥ 0} ≤ (1 − θ)κ m0 3κ c1 (γ2∗ γ0 )N λ1−N/2 , ∀λ ≥ λ0 . (2.12) EJDE-2014/251 SEMICLASSICAL SOLUTIONS 7 Applying the mountain-pass lemma without the (PS) condition, by standard arguments, we can prove the following two lemmas. Lemma 2.3. Let H(x, z) ≥ 0 for all (x, z) ∈ RN × R2 . Suppose that (A0), (A1), (H1)–(H4) are satisfied. Then there exist a constant dλ ∈ (0, sups≥0 Φλ (seλ , 0)] and a sequence {zn } ⊂ E satisfying Φλ (zn ) → dλ , kΦ0λ (zn )kE ∗ (1 + kzn kλ† ) → 0. (2.13) Lemma 2.4. Let H(x, z) ≥ 0 for all (x, z) ∈ RN × R2 . Suppose (A1), (B0), (H1)– (H3), (H4’) are satisfied. Then there exist a constant dλ ∈ (0, sups≥0 Φλ (0, seλ )] and a sequence {zn } ⊂ E satisfying Φλ (zn ) → dλ , kΦ0λ (zn )kE ∗ (1 + kzn kλ† ) → 0. (2.14) Lemma 2.5. Suppose that (A0), (A1), (H1)–(H5) are satisfied. Then any sequence {zn } ⊂ E satisfying (2.13) is bounded in E. Proof. We argue by contradiction for proving boundedness of {zn }. Suppose that kzn kλ† → ∞. Let z̃n = zn /kzn kλ† := (ũn , ṽn ). Then kz̃n kλ† = 1. In view of (A1), we obtain Z Z p 2λ µ(x)ũn ṽn dx ≤ 2θλ a(x)b(x)|ũn ṽn | dx N RN Z R (2.15) 2 2 ≤ θλ [a(x)ũn + b(x)ṽn ] dx ≤ θ. RN If Z δ := lim sup sup n→∞ y∈RN |z̃n |2 dx = 0, B(y,1) then by Lions’ concentration compactness principle [21] or [27, Lemma 1.21], z̃n → (0, 0) in Ls (RN ) for 2 < s < 2∗ . Set zn · Hz (x, zn ) (1 − θ)m0 ≤ Ωn := x ∈ RN : , 2 |zn | 3 D := Aa0 ∪ Bb0 . Hence, from (A0) and the Hölder inequality it follows that Z |Hz (x, zn ) · zn | λ dx kzn k2λ† Ωn Z |Hz (x, zn ) · zn | =λ |z̃n |2 dx |zn |2 Ωn Z (1 − θ)λm0 ≤ |z̃n |2 dx 3 Ωn Z Z (1 − θ)λm0 (1 − θ)λm0 2 ≤ |z̃n | dx + |z̃n |2 dx 3 3 RN \D D ≤ 1−θ (1 − θ)λm0 [meas(D)]1/(N +1) kz̃n k2λ† + 3 3 Z N/(N +1) × |z̃n |2(N +1)/N dx D 1−θ = + o(1). 3 (2.16) 8 S. CHEN, X. H. TANG From (2.3), (2.4) and (2.13), there holds Z dλ + o(1) = λ EJDE-2014/251 H(x, zn ) dx. (2.17) RN Let κ0 = κ/(κ − 1), then 2 < 2κ0 < 2∗ . By (H5), (2.17) and the Hölder inequality, one obtains Z |Hz (x, zn ) · zn | dx λ kzn k2λ† N R \Ωn Z |Hz (x, zn ) · zn | =λ |z̃n |2 dx |zn |2 N R \Ωn hZ |H (x, z ) · z | κ i1/κ Z 1/κ0 z n n (2.18) 2κ0 dx ≤λ |z̃ | dx n |zn |2 RN \Ωn RN \Ωn Z 1/κ Z 1/κ0 0 ≤ λ c1 H(x, zn ) dx |z̃n |2κ dx RN \Ωn ≤λ 1−1/κ RN 1/κ [c1 dλ + o(1)] kz̃n k22κ0 = o(1). Combining (2.17) with (2.18) and using (2.4), (2.13) and (2.15), we have kzn k2λ† − hΦ0λ (zn ), zn i kzn k2λ† Z Z |Hz (x, zn ) · zn | =λ + 2λ µ(x)ũn ṽn dx kzn k2λ† RN RN Z Z |Hz (x, zn ) · zn | |Hz (x, zn ) · zn | ≤λ dx + λ dx + θ 2 kz k kzn k2λ† N n λ† Ωn R \Ωn 1 + o(1) ≤ (2.19) 1 + 2θ + o(1). 3 This contradiction shows that δ > 0. Going to a subsequence if necessary, we assume the existence of kn ∈ ZN such R that B √ (kn ) |z̃n |2 dx > 2δ . Let wn (x) = z̃n (x + kn ); then 1+ N Z δ |wn |2 dx > . (2.20) 2 √ B1+ N (0) ≤ Now we define ẑn (x) = zn (x + kn ), then ẑn /kzn kλ† = wn and kwn k2H 1 (RN ) = kz̃n k2H 1 (RN ) . Passing to a subsequence, we have wn * w in H 1 (RN ), wn → w in Lsloc (RN ), 2 ≤ s < 2∗ and wn → w a.e. on RN . Obviously, (2.20) implies that w 6= (0, 0). For a.e. x ∈ {z ∈ RN : w(z) 6= (0, 0)}, we have limn→∞ |ẑn (x)| = ∞. Hence, it follows from (H3), (2.3), (2.13), (2.15) and Fatou’s lemma that dλ + o(1) Φλ (zn ) = lim 2 n→∞ kzn k2 n→∞ kzn k λ† λ† Z Z h1 i H(x + kn , ẑn ) 2 = lim kz̃n k2λ† − λ |w | dx − λ µ(x)ũn ṽn dx n 2 n→∞ 2 |ẑn | RN RN Z 1+θ H(x + kn , ẑn ) −λ |wn |2 dx = −∞. ≤ lim inf 2 n→∞ 2 |ẑ | N n R 0 = lim This contradiction shows that {kzn kλ† } is bounded. EJDE-2014/251 SEMICLASSICAL SOLUTIONS 9 We can prove the following lemma in the same way as Lemma 2.5. Lemma 2.6. Suppose that (A1), (B0), (H1)–(H3), (H4’), (H5) are satisfied. Then any sequence {zn } ⊂ E satisfying (2.14) is bounded in E. 3. Proofs of main results In this section, we give the proofs of Theorems 1.3–1.6. Proof of Theorem 1.5. Applying Lemmas 2.1, 2.3 and 2.5, we deduce that there exists a bounded sequence {zn } ⊂ E satisfying (2.13) with (2κ−N )/2 dλ ≤ (1 − θ)κ m0 3κ c1 (γ2∗ γ0 )N λ1−N/2 , ∀λ ≥ λ0 . (3.1) Going to a subsequence, if necessary, we can assume that zn * zλ in (E, k · kλ† ) and Φ0λ (zn ) → 0. Next, we prove that zλ 6= (0, 0). Arguing by contradiction, suppose that zλ = (0, 0), i.e. zn * (0, 0) in E, and so zn → (0, 0) in Lsloc (RN ), 2 ≤ s < 2∗ and zn → (0, 0) a.e. on RN . Since D is a set of finite measure, there holds kzn k22 Z Z 2 |zn | dx + = RN \D |zn |2 dx ≤ D 1 kzn k2λ† + o(1). λm0 (3.2) For s ∈ (2, 2∗ ), it follows from (2.2), (3.2) and the Hölder inequality that 2(2∗ −s)/(2∗ −2) kzn kss ≤ kzn k2 ≤ (γ2∗ γ0 )2 ∗ 2∗ (s−2)/(2∗ −2) kzn k2∗ (s−2)/(2∗ −2) (λm0 )−(2 ∗ −s)/(2∗ −2) kzn ksλ† + o(1). (3.3) According to (3.2), one can obtain that Z Hz (x, zn ) · zn |zn |2 dx |zn |2 Ωn (1 − θ)λm0 1−θ ≤ kzn k22 ≤ kzn k2λ† + o(1). 3 3 Z Hz (x, zn ) · zn dx = λ λ Ωn (3.4) By (2.3), (2.4) and (2.13), we have 1 Φλ (zn ) − hΦ0λ (zn ), zn i = λ 2 Z H(x, zn ) dx = dλ + o(1). RN (3.5) 10 S. CHEN, X. H. TANG EJDE-2014/251 Using (H5), (3.1), (3.3) with s = 2κ/(κ − 1) and (3.5), we obtain Z λ Hz (x, zn ) · zn dx RN \Ωn |H (x, z ) · z | κ 1/κ z n n dx kzn k2s |zn |2 RN \Ωn Z 1/κ ∗ ∗ ≤ (γ2∗ γ0 )2·2 (s−2)/s(2 −2) λ c1 H(x, zn ) dx ≤λ Z RN \Ωn −2(2∗ −s)/s(2∗ −2) × (λm0 ) 1/κ kzn k2λ† + o(1) ∗ 1/κ ≤ c1 (γ2∗ γ0 )N/κ λ1−1/κ dλ (λm0 )−2(2 −s)/s(2∗ −2) kzn k2λ† + o(1) (3.6) 1/κ = c1 (γ2∗ γ0 )N/κ (N −2)/2 1/κ λ dλ kzn k2λ† + o(1) (2κ−N )/2κ m0 ≤ c1 (γ2∗ γ0 )N/κ (1 − θ)κ m0 (2κ−N )/2κ 3κ c1 (γ2∗ γ0 )N m (2κ−N )/2 1/κ 1/κ kzn k2λ† + o(1) 0 1−θ = kzn k2λ† + o(1), 3 which, together with (2.4), (2.13) and (3.4), yields o(1) = hΦ0λ (zn ), zn i Z 2 = kzn kλ† − λ Z Hz (x, zn ) · zn dx − 2λ µ(x)un vn dx RN Z Z (3.7) ≥ (1 − θ)kzn k2λ† − λ Hz (x, zn ) · zn dx − λ Hz (x, zn ) · zn dx RN RN \Ωn Ωn 1−θ kzn k2λ† + o(1). 3 Consequently, it follows from (2.3) and (2.13) that ≥ 0 < dλ = lim Φλ (zn ) ≤ n→∞ 1+θ lim kzn k2λ† = 0, 2 n→∞ since H(x, z) ≥ 0, ∀(x, z) ∈ RN × R2 . This contradiction shows zλ 6= (0, 0). By a standard argument, we easily certify that Φ0λ (zλ ) = 0 and Φλ (zλ ) ≤ dλ . Then zλ is a nontrivial solution of (1.7), moreover Z 1 dλ ≥ Φλ (zλ ) = Φλ (zλ ) − hΦ0λ (zλ ), zλ i = λ H(x, zλ ) dx. (3.8) 2 RN Proof of Theorem 1.6. Applying Lemmas 2.2, 2.4 and 2.6, we deduce that there exists a bounded sequence {zn } ⊂ E satisfying (2.14) with (2κ−N )/2 dλ ≤ (1 − θ)κ m0 3κ c1 (γ2∗ γ0 )N λ1−N/2 , ∀λ ≥ λ0 . The rest of the proof is the same as Theorem 1.5, so we omit it. Theorem 1.3 and 1.4 are direct consequences of Theorem 1.5 and 1.6, respectively. We omit their proofs. EJDE-2014/251 SEMICLASSICAL SOLUTIONS 11 Acknowledgements. This work is partially supported by the National Natural Science Foundation of China (No: 11171351). References [1] N. Akhmediev, A. Ankiewicz; Novel soliton states and bifurcation phenomena in nonlinear fiber couplers, Phys. Rev. Lett. 70 (1993) 2395-2398. [2] A. Ambrosetti, M. Badiale, S. Cingolani; Semiclassical states of nonlinear Schödinger equations, Arch. Rat. Mech. Anal. 140 (1997) 285-300. [3] A. Ambrosetti, G. Cerami, D. Ruiz; Solutions of linearly coupled systems of semilinear nonautonomous equations on RN , J. Func. Anal. 254 (2008) 2816-2845. [4] A. Ambroseti, E. Colorado, D. Ruiz; Multi-bumo solutions to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. Paratial Differential Equations. 30 (2007) 85122. [5] N. N. Akhmediev, V. M. Eleonskii, N. E. Kulagin, L. P. Shil’nikov; Stationary pulses in nonlinear double refracting optical fiber-solit on multiplication, Pis’ma Zh. Tekh. Fiz. 15, 19 (1989), Sov. Tech. Phys. Lett. 15, 587 (1989). [6] A. Ambrosetti, A. Malchiodi, W. M. Ni; Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I, Comm. Math. Phys. 235 (2003) 427-466. [7] A. Ambrosetti, A. Malchiodi, W. M. Ni; Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. II, Indiana Univ. Math. J. 53 (2004) 297-329. [8] Z. J. Chen, W. M. Zou; Standing waves for a coupled system of nonlinear Schödinger equations, Annali di Matematica. (2013) Doi:10.1007/s10231-013-0371-5. [9] Z. J. Chen, W. M. Zou; Standing waves for linearly coupled Schödinger equations with critical exponent, Ann. I. H. Poincaré CAN. 31 (2014) 429-447. [10] Z. J. Chen, W. M. Zou; On coupled systems of Schrödinger equations, Adv. Differential Equations 16 (2011) 755-800. [11] Z. J. Chen, W. M. Zou; Ground states for a system of Schrödinger equations with critical exponent, J. Func. Anal. 262 (2012) 3091-3107. [12] Y. H. Ding, Fanghua Lin; Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. 30 (2007) 231-249. [13] Y. H. Ding, A. Szulkin; Bound states for semilinear Schrödinger equations with signchanging potential, Calc. Var. PDE, 29 (2007) 397-419. [14] Y. H. Ding, Juncheng Wei; Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Func. Anal. 251 (2007) 546-572. [15] V. M. Eleonskii, V. G. Korolev, N. E. Kulagin, L. P. Shil’nikov; Bifurcations of branching of vector solitons of envelopes, Zh. Eksp. Teor. Fiz. 99, 1113 (1991), Sov. Phys. JETP. 72, 619 (1982). [16] W. N. Huang, X. H. Tang; Semi-classical solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl. 415(2014)791-802. [17] I. P. Kaminow; Polarization in optical fibers, IEEE J. Quantum Electron. 17 (1981) 15-22. [18] G. B. Li, X. H. Tang; Nehari-type ground state solutions for Schrödinger equations including critical exponent, Appl. Math. Lett. 37 (2014) 101-106. [19] Y. Y. Li; On a singularly perturbed elliptic equation, Adv. Differential Equations. 2 (1997) 955-980. [20] X. Y. Lin, X. H. Tang; Semiclassical solutions of perturbed p-Laplacian equations with critical nonlinearity, J. Math.Anal. Appl. 413 (2014) 438-449. [21] P. L. Lions; The concentration-compactness principle in the calculus of variations. The locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non Linéaire. 1 (1984) 223-283. [22] A. Pomponio; Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations. 227 (2006) 258-281. [23] B. Sirakov; Standing wave solutions of the nonlinear Schrödinger equations in RN , Annali di Matematica. 183 (2002) 73-83. [24] X. H. Tang; New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Advance Nonlinear Studies, 2014, 14: 361-373. 12 S. CHEN, X. H. TANG EJDE-2014/251 [25] X. H. Tang; Non-Nehari manifold method for superlinear Schrödinger equation, Taiwan J. Math., 2014, 18: 1957-1979. [26] X. H. Tang; New conditions on nonlinearity for a periodic Schrd̈inger equation having zero as spectrum, J. Math. Anal. Appl., 2014, 413: 392-410. [27] M. Willem; Minimax Theorems, Birkhäuser, Boston, 1996. Sitong Chen School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China E-mail address: sitongchen2041@hotmail.com Xianhua Tang School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China E-mail address: tangxh@mail.csu.edu.cn