GROUND STATES OF LINEARLY COUPLED SCHRÖDINGER SYSTEMS

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1 Electronic Journal of Differential Equations, Vol (2017), o. 05, ISS: URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu GROUD STATES OF LIEARLY COUPLED SCHRÖDIGER SYSTEMS HAIDOG LIU Abstract. This article concerns the standing waves of a linearly couled Schrödinger system which arises from nonlinear otics and condensed matter hysics. The coefficients of the system are satially deendent and have a mixed behavior: they are eriodic in some directions and tend to ositive constants in other directions. Under suitable assumtions, we rove that the system has a ositive ground state. In addition, when the L -norm of the couling coefficient tends to zero, the asymtotic behavior of the ground states is also obtained. 1. Introduction and statement of main results onlinear Schrödinger systems of the form i t Ψ 1 = Ψ 1 V 1 (x)ψ 1 + µ 1 Ψ 1 2 Ψ 1 + β Ψ 2 2 Ψ 1 + γψ 2 i t Ψ 2 = Ψ 2 V 2 (x)ψ 2 + µ 2 Ψ 2 2 Ψ 2 + β Ψ 1 2 Ψ 2 + γψ 1 x, t > 0, Ψ j = Ψ j (x, t) C, t > 0, j = 1, 2 (1.1) model several interesting henomena in hysics. Physically, Ψ j are two comonents of a quantum system, µ j and β are the intrasecies and intersecies scattering lengths, γ is the Rabi frequency related to the external electric field. The sign of the scattering length β determines whether the interaction is reulsive or attractive. We refer to [1, 16, 17, 18, 29] and references therein for more information on the hysical background of (1.1). To study standing waves of the system (1.1), we set Ψ j (x, t) = e iλt u j (x) for j = 1, 2. Then (1.1) is reduced to the following ellitic system u 1 + (V 1 (x) + λ)u 1 = µ 1 u βu 1 u γu 2 in, u 2 + (V 2 (x) + λ)u 2 = µ 2 u βu 2 1u 2 + γu 1 in, u j (x) 0 as x, j = 1, 2. (1.2) In the resence of only nonlinearly couling terms (i.e., γ = 0), (1.2) has been studied extensively in recent years for the existence, multilicity and asymtotic 2010 Mathematics Subject Classification. 35B40, 35J47, 35J50. Key words and hrases. Linearly couled Schrodinger system; ground states; asymtotic behavior. c 2017 Texas State University. Submitted January 11, Published January 5,

2 2 H. LIU EJDE-2017/05 behavior of nontrivial solutions. We make no attemt here to give a comlete survey of all related results and only refer the reader to [4, 7, 8, 21, 23, 24, 27, 28, 30] and references therein. However, In the resence of only linearly couling terms (i.e., β = 0), (1.2) has not been much studied and we are only aware of a few aers in this direction ([2, 3, 5, 11, 12, 20]). On the other hand, the single ellitic equation u + V (x)u = µ(x) u 2 u, (1.3) has been deely studied in the literature. Among other solutions, ground states are hysically and mathematically of articular interest. We refer the reader to [6, 9, 10, 13, 14, 15, 19, 22, 25, 26, 31, 32] for related results. Here we only mention that (1.3) has a ositive ground state if 0 < V (x) lim x V (x) <, 0 < lim x µ(x) µ(x) ([14, 19, 22, 31]) or if V, µ are ositive and eriodic in each variable ([13, 19]). The otentials considered in [10] have a mixed behavior. More recisely, the otentials are eriodic in some directions and tend to ositive constants in other directions. Partially motivated by [3, 10], we deal with in this aer ground states of linearly couled Schrödinger equations in which all of the hysical arameters are satially deendent, i.e., we will consider the system u 1 + V 1 (x)u 1 = µ 1 (x) u 1 2 u 1 + γ(x)u 2 in, u 2 + V 2 (x)u 2 = µ 2 (x) u 2 2 u 2 + γ(x)u 1 in, u j H 1 ( ), j = 1, 2, (1.4) where 2, 2 < < 2, 2 = 2/( 2) for > 2 and 2 = + for = 2, and the coefficients V j, µ j, γ are continuous functions on. For system (1.4), a nontrivial solution is a solution (u 1, u 2 ) with (u 1, u 2 ) (0, 0). A nontrivial solution of (1.4) will be called a ground state if it has the least energy among all nontrivial solutions. A ositive ground state means a ground state with each comonent being ositive. We remark that each comonent of a nontrivial solution of (1.4) must be nonzero. Write = k + l with 1 k 1 and x = (x, x ) R k R l =. We study (1.4) under the following assumtions. (H1) The five functions V j, µ j, γ are τ i -eriodic in x i, τ i > 0, i = 1,..., k. (H2) 0 < V j (x) V j := lim x V j (x) < for all x, j = 1, 2; 0 < µ j := lim x µ j (x) µ j (x) for all x, j = 1, 2; 0 < γ := lim x γ(x) γ(x) for all x. (H3) γ(v 1 V 2 ) 1/2 < 1, where is the usual norm in L ( ). The first main result in our aer is as follows and is for the existence of ground states of (1.4). Theorem 1.1. If (H1) (H3) hold, then (1.4) has a ositive ground state. We assume (H3) to guarantee that the ehari manifold is bounded away from zero, see Lemma 2.1 for details. The second aim of our aer is to describe the asymtotic behavior of ground states when L -norm of the linearly couling coefficient of (1.4) tends to zero. For this urose, we relace (H1), (H2) and (H3) with (H1 ) The functions V j, µ j, γ n are τ i -eriodic in x i, τ i > 0, i = 1,..., k.

3 EJDE-2017/05 LIEARLY COUPLED SCHRÖDIGER SYSTEMS 3 (H2 ) 0 < V j (x) V j := lim x V j (x) < for all x, j = 1, 2; 0 < µ j := lim x µ j (x) µ j (x) for all x, j = 1, 2; 0 < γ n := lim x γ n (x) γ n (x) for all x, n = 1, 2,.... (H3 ) γ n 0 as n. Under the assumtions (H1 ), (H2 ) and (H3 ), we see from Theorem 1.1 that, for n sufficiently large, (1.4) with γ = γ n has a ositive ground state (u n1, u n2 ). ext we show that one comonent of (u n1, u n2 ) converges to zero in H 1 ( ). Theorem 1.2. Assume (H1 ) (H3 ) are satisfied. Let (u n1, u n2 ) be the ositive ground state of (1.4) with γ = γ n, then we have either u n1 0 in H 1 ( ) or u n2 0 in H 1 ( ). This article is organized as follows. In Section 2, we rove some reliminary results including basic roerties of the ehari manifold. Section 3 is devoted to the existence of ground states for (1.4), while Section 4 concerns the asymtotic behavior of ground states when L -norm of the couling coefficient tends to zero. 2. Preliminaries By (H1) and (H2), ( ( u j = u 2 + V j u 2)) 1/2, j = 1, 2 are equivalent norms in H 1 ( ). Set H = H 1 ( ) H 1 ( ) and, for u = (u 1, u 2 ) H, denote ( 1/2. u = u u 2 2) 2 Then is equivalent to the standard norm in H. Throughout this aer, the notation will always refer to this norm. It is well known that solutions of (1.4) corresond to critical oints of the energy functional I : H R defined by I( u) = 1 2 u 2 1 (µ 1 u 1 + µ 2 u 2 ) γu 1 u 2. Denote by the so-called ehari manifold associated with I, namely, = { u H \ {(0, 0)} : J( u) := I ( u), u = 0 }. Some useful roerties of the ehari manifold are given next. Lemma 2.1. There exists δ > 0 such that, for u, u δ. Proof. For u, we use the Hölder inequality and Sobolev inequality to deduce (1 γ(v1 V 2 ) 1/2 ) u 2 u 2 2 γu 1 u 2 R = (µ 1 u 1 + µ 2 u 2 ) C u, which imlies the desired result.

4 4 H. LIU EJDE-2017/05 ote that for u, I( u) = ( ) ( ) u 2 2 γu 1 u 2 ( 1 R 2 1 ) ( 1 γ(v1 V 2 ) 1/2 ) u 2. Therefore, as a consequence of Lemma 2.1, we have Lemma 2.2. c = inf u I( u) > 0. Lemma 2.3. If c is achieved, then (1.4) has a ositive ground state. Proof. We first claim that any minimizer of c is a ground state of (1.4). Indeed, if u = (u 1, u 2 ) is a minimizer of c, then there exists λ R such that I ( u) = λj ( u). From I ( u), u = 0 and ( ) J ( u), u = 2 u 2 2 γu 1 u 2 (µ 1 u 1 + µ 2 u 2 ) R = ( 2)( u 2 2 γu 1 u 2 ) ( 2) (1 γ(v1 V 2 ) 1/2 ) u 2 ( 2) (1 γ(v1 V 2 ) 1/2 ) δ 2 < 0 it follows that λ = 0. Then I ( u) = 0 and so u is a ground state of (1.4). ext we rove that (1.4) has a ositive ground state. Assume u = (u 1, u 2 ) is a minimizer of c and let t > 0 be such that (t u 1, t u 2 ). Then which imlies t 2 = u 2 2 γ u 1 u 2 (µ 1 u 1 + µ 2 u 2 ) u 2 2 γu 1 u 2 (µ 1 u 1 + µ 2 u 2 ) = 1, c I(t u 1, t u 2 ) = ( ) t (µ 1 u 1 + µ 2 u 2 ) ( 1 2 ) 1 (µ 1 u 1 + µ 2 u 2 ) = I( u) = c. This means t = 1 and ( u 1, u 2 ) is also a minimizer of c. From the above claim, ( u 1, u 2 ) is a ground state of (1.4). ote that none of u 1 and u 2 can be identically zero. Using the strong maximum rincile, we have u 1 > 0 and u 2 > 0. Therefore ( u 1, u 2 ) is a ositive ground state of (1.4). The roof is comlete. 3. Proof of Theorem 1.1 In this section we rove the existence of ground states of (1.4). From Lemma 2.3, it suffices to rove that c is achieved. For this urose, we need to comare the value of c with c = inf I ( u), u where the functional I : H R is defined by I ( u) = 1 ( u1 2 + V 1 u u V 2 u 2 ) 2 2

5 EJDE-2017/05 LIEARLY COUPLED SCHRÖDIGER SYSTEMS 5 and 1 (µ 1 u 1 + µ 2 u 2 ) γ u 1 u 2, = { u H \ {(0, 0)} : I ( u), u = 0} is the ehari manifold associated with I. Lemma 3.1. Assume that (H1) (H3) are satisfied and suose in addition that at least one of the five functions V j, µ j, γ is not a constant, then c < c. Proof. By [3, Lemma 3.2], c is achieved at some u = (u 1, u 2 ) with each comonent being ositive. Let t > 0 be such that t u. Then c = I ( u ) I (t u ) = I(t u ) + t2 [ (V1 V 1 ) u (V 2 V 2 ) u 2 ] 2 2 R + t [(µ1 µ1) u1 + (µ2 µ2) u2 ] R + t 2 (γ γ ) u 1 u 2 > I(t u ) c. The roof is comlete. ow we are in a osition to rove the first main result. Proof of Theorem 1.1. By Lemma 2.3, it suffices to rove that the infimum c = inf u I( u) is achieved. If the five functions V j, µ j, γ are all constants, then the result has been roved in [3, Lemma 3.2]. In what follows we always assume that at least one of the five functions V j, µ j, γ is not a constant. By Ekeland s variational rincile, there exists { u m } with u m = (u m1, u m2 ) such that I( u m ) c and (I ) ( u m ) 0 as m. Then there exists a sequence {λ m } of real numbers such that I ( u m ) λ m J ( u m ) 0 as m. From (H3) and I( u m ) = ( ) ( ) u m 2 2 γu m1 u m2 = c + o(1), we see that { u m } is bounded in H. Then o(1) = I ( u m ) λ m J ( u m ), u m ( ) = λ m ( 2) u m 2 2 γu m1 u m2 = λ m (2c + o(1)), which imlies that λ m = o(1) and so I ( u m ) 0 as m. Since { u m } H is bounded, u to a subsequence, it can be assumed that (u m1, u m2 ) (u 1, u 2 ) in H,

6 6 H. LIU EJDE-2017/05 (u m1, u m2 ) (u 1, u 2 ) in L loc (R ) L loc (R ), (u m1, u m2 ) (u 1, u 2 ) a.e. in. Then u = (u 1, u 2 ) is a critical oint of I. We have the following two cases. Case 1. u (0, 0). In this case, u and c I( u) = ( 1 2 ) 1 (µ 1 u 1 + µ 2 u 2 ) (1 lim m 2 ) 1 (µ 1 u m1 + µ 2 u m2 ) = lim I( u m) = c. m Then c is achieved by u. Case 2. u = (0, 0). In this case, we decomose into -dimensional intervals {Q j } j {0} with each of them having sides of size (τ 1,..., τ k, 1,..., 1) and chosen in such a way that 0 is the center of Q 0. For each m, we set d m = su j {0} [ Then there exists η > 0 such that Q j (µ 1 u m1 + µ 2 u m2 ) ] 1/. d m η > 0 (3.1) for all m. Indeed, using the Sobolev embeddings H 1 (Q j ) L (Q j ) and the boundedness of { u m } H leads to 2 (c + o(1)) = (µ 1 u m1 + µ 2 u m2 ) 2 = (µ 1 u m1 + µ 2 u m2 ) Q j which imlies (3.1). From j=0 j=0 d 2 m [ j=0 Cd 2 m u m 2 Cd 2 m, Q j (µ 1 u m1 + µ 2 u m2 ) Q j (µ 1 u m1 + µ 2 u m2 ) C, we see that d m is achieved. Let y m be the center of the interval Q m satisfying [ ] 1/. d m = (µ 1 u m1 + µ 2 u m2 ) Q m It follows from (3.1) and (u m1, u m2 ) (0, 0) in H that {y m } is unbounded. Without loss of generality, we assume y m as m. Set ũ mj = u mj ( + y m ) for j = 1, 2 and assume that (ũ m1, ũ m2 ) (ũ 1, ũ 2 ) in H, (ũ m1, ũ m2 ) (ũ 1, ũ 2 ) in L loc (R ) L loc (R ), ] 2/

7 EJDE-2017/05 LIEARLY COUPLED SCHRÖDIGER SYSTEMS 7 (ũ m1, ũ m2 ) (ũ 1, ũ 2 ) a.e. in. Then (ũ 1, ũ 2 ) (0, 0) as showed by [ 0 < η lim m Q m [ C lim m Q m [ = C lim ( ũ m1 + ũ m2 ) m Q 0 [ ] 1/. = C ( ũ 1 + ũ 2 ) Q 0 ] 1/ (µ 1 u m1 + µ 2 u m2 ) ] 1/ ( u m1 + u m2 ) ext we claim that { y m } is bounded. Suose by contradiction that y m as m. Then, using (H2), we deduce from I (u m1, u m2 ) 0 that I (ũ 1, ũ 2 ) = 0. ] 1/ Therefore, since (ũ 1, ũ 2 ) (0, 0), (ũ 1, ũ 2 ) and we have c I (ũ 1, ũ 2 ) = ( 1 2 ) 1 (µ 1 ũ 1 + µ 2 ũ 2 ) (1 lim m 2 ) 1 (µ 1 ũ m1 + µ 2 ũ m2 ) lim I( u m) = c, m a contradiction to Lemma 3.1. Decomose into bigger -dimensional intervals { ˆQ j } j {0} and assume that Q m ˆQ m with (y m, 0) being the center of ˆQ m. Setting û mj = u mj ( +(y m, 0)) for j = 1, 2, we see from the assumtion (H1) that I(û m1, û m2 ) c and I (û m1, û m2 ) 0 as m. Since {(û m1, û m2 )} is bounded in H, we may assume that (û m1, û m2 ) (û 1, û 2 ) in H, (û m1, û m2 ) (û 1, û 2 ) in L loc (R ) L loc (R ), (û m1, û m2 ) (û 1, û 2 ) a.e. in. Then (û 1, û 2 ) is a critical oint of I. Furthermore, [ 0 < η lim (µ 1 u m1 + µ 2 u m2 ) m Q m [ C lim ( u m1 + u m2 ) m ˆQ m [ = C lim ( û m1 + û m2 ) m ˆQ0 [ ] 1/, = C ( û 1 + û 2 ) ˆQ0 which imlies (û 1, û 2 ) (0, 0). Using the same arguments as in Case 1, we see that c is achieved by (û 1, û 2 ). ] 1/ ] 1/ ] 1/

8 8 H. LIU EJDE-2017/05 In both cases we reach the conclusion. The roof is comlete. 4. Proof of Theorem 1.2 In this section, we describe the asymtotic behavior of the ground states of (1.4) when the L -norm of the couling coefficient tends to zero. To underline the deendence on the couling coefficient, for (1.4) with γ = γ n, the corresonding functional and the ehari manifold will be denoted by I γn and γn resectively. Then we see from Theorem 1.1 that, for n sufficiently large, the infimum c γn = inf u γn I γn ( u) is achieved at (u n1, u n2 ). We also use the functional I 0 : H R defined by I 0 ( u) = 1 2 u 2 1 (µ 1 u 1 + µ 2 u 2 ) and its corresonding ehari manifold Define c 0 = inf u 0 I 0 ( u), then we have 0 = { u H \ {(0, 0)} : I 0( u), u = 0}. c 0 = min Φ j(w j ), j {1,2} where Φ j (u) = 1 2 u 2 j 1 µ j u and w j is the ositive ground state of the single ellitic equation Lemma 4.1. c γn c 0 as n. u + V j u = µ j u 2 u in. Proof. Since (w 1, 0) and (0, w 2 ) are contained in γn, we have c γn min{i γn (w 1, 0), I γn (0, w 2 )} = min j {1,2} Φ j(w j ) = c 0. (4.1) Then it is easy to see that {(u n1, u n2 )} is bounded in H. We define a sequence {t n } of ositive numbers by t 2 (u n1, u n2 ) 2 n = (µ 1 u n1 + µ 2 u n2 ). Then (t n u n1, t n u n2 ) 0. We claim that t n = 1 + o(1) as n. Indeed, since (u n1, u n2 ) is a ositive ground state of (1.4) with γ = γ n, we have (1 γn (V 1 V 2 ) 1/2 ) (u n1, u n2 ) 2 (u n1, u n2 ) 2 2 γ n u n1 u n2 R = (µ 1 u n1 + µ 2 u n2 ) C (u n1, u n2 ). Then, since γ n 0, we see that (u n1, u n2 ) 2 and (µ 1 u n1 + µ 2 u n2 ) have a ositive lower bound. From this, it can be seen that t n = 1 + o(1) as n. Therefore, c 0 I 0 (t n u n1, t n u n2 ) = ( ) t n (µ 1 u n1 + µ 2 u n2 )

9 EJDE-2017/05 LIEARLY COUPLED SCHRÖDIGER SYSTEMS 9 which combined with (4.1) comletes the roof. = ( 1 2 ) 1 (µ 1 u n1 + µ 2 u n2 ) + o(1) = c γn + o(1), Proof of Theorem 1.2. We use an argument of contradiction and, u to a subsequence, suose that u nj j α > 0 (4.2) for j = 1, 2. Define two sequences {s n } and {t n } of ositive numbers by s 2 n = u n1 2 1 µ 1 u n1, t 2 n = u n2 2 2 µ 2 u n2. Since (u n1, u n2 ) is a ositive ground state of (1.4) with γ = γ n and γ n 0, we have u n1 2 1 = µ 1 u n1 + γ n u n1 u n2 = µ 1 u n1 + o(1), R u n2 2 2 = µ 2 u n2 + γ n u n1 u n2 = µ 2 u n2 + o(1), which combined with (4.2) imlies that s n = 1 + o(1) and t n = 1 + o(1) as n. Then we have 2c 0 Φ 1 (s n u n1 ) + Φ 2 (t n u n2 ) = ( ) R s n µ 1 u n1 + ( ) R t n µ 2 u n2 = ( 1 2 ) 1 (µ 1 u n1 + µ 2 u n2 ) + o(1) = c γn + o(1), which contradicts the result in Lemma 4.1. This contradiction imlies that either u n1 0 in H 1 ( ) or u n2 0 in H 1 ( ). Acknowledgments. This research was suorted by the ZJSF (LQ15A010011). References [1]. Akhmediev, A. Ankiewicz; Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), [2] A. Ambrosetti; Remarks on some systems of nonlinear Schrödinger equations, J. Fixed Point Theory Al., 4 (2008), [3] A. Ambrosetti, G. Cerami, D. Ruiz; Solitons of linearly couled system of semilinear nonautonomous equations on R n, J. Funct. Anal., 254(2008), [4] A. Ambrosetti, E. Colorado; Standing waves of some couled nonlinear Schrödinger equations, J. London Math. Soc., 75 (2007), [5] A. Ambrosetti, E. Colorado, D. Ruiz; Multi-bum solutions to linearly couled systems of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 30 (2007), [6] A. Ambrosetti, V. Felli, A. Malchiodi; Ground states of nonlinear Schrödinger equations with otentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), [7] T. Bartsch, E.. Dancer, Z.-Q. Wang; A Liouville theorem, a-riori bounds, and bifurcating branches of ositive solutions for a nonlinear ellitic system, Calc. Var. Partial Differential Equations, 37 (2010), [8] S. Bhattarai; Stability of solitary-wave solutions of couled LS equations with ower-tye nonlinearities, Adv. onlinear Anal., 4 (2015),

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