Multiplicity of standing wave solutions of nonlinear Schrödinger equations with electromagnetic fields

Transcription

1 Z. angew. Math. Phys. 59 (2008) /08/ DOI /s c 2007 Birkhäuser Verlag, Basel Zeitschrift für angewandte Mathematik und Physik ZAMP Multilicity of standing wave solutions of nonlinear Schrödinger equations with electromagnetic fields Zhongwei Tang Abstract. In this aer, we are concerned with the multilicity of standing wave solutions of nonlinear Schrödinger equations with electromagnetic fields ( + ia(x)) 2 u(x) + (λa(x) + 1)u(x) = u(x) 2 u(x), x R N (S λ ) for sufficiently large λ, where i is the imaginary unit, 2 < < 2N for N 3 and 2 < < + N 2 for N = 1,2. a(x) is a real continuous function on R N, A(x) = (A 1 (x), A 2 (x),..., A N (x)) is such that A (x) is a real local Hölder continuous function on R N for = 1, 2,..., N. We assume that a(x) is nonnegative and has a otential well Ω := int a 1 (0) consisting of k comonents Ω 1,..., Ω k. We show that for any non-emty subset 1, 2,..., k}, (S λ ) has a standing wave solution which is traed in a neighborhood of Ω for λ large. Mathematics Subect Classification (2000). 3560, 35B33. Keywords. Nonlinear Schrödinger equation, multilicity, otential well, magnetic fields, variational methods. 1. Introduction We are concerned with nonlinear Schrödinger equations with electromagnetic otential ( + ia(x)) 2 u(x) + (λa(x) + 1)u(x) = u(x) 2 u(x), x R N (S λ ), here i is the imaginary unit, 2 < < 2N N 2 for N 3 and 2 < < + for N = 1,2. L A = ( + ia(x)) 2 denotes a Schrödinger oerator with a real valued magnetic vector otential A(x) = (A 1 (x),a 2 (x),...,a N (x)), where A (x) is a real valued function on R N for = 1,2,...,N. Actually, the magnetic field B is nothing but B = curla if N = 3; in general dimension, B should be thought of as a 2-form where B,k = A k k A and a(x) is a continuous real valued electric otential function on R N. Our hyothesis on A(x) and a(x) are: (A 1 ) A (x) Cloc α (RN, R)( = 1,2,...,N) for some α > 0; Suorted by the National Science Foundation of China( ).

2 Vol. 59 (2008) Nonlinear Schrödinger equations 811 (A 2 ) a(x) C(R n, R) satisfies a(x) 0 and Ω := int a 1 (0) is non-emty and has smooth boundary and Ω = a 1 (0); (A 3 ) Ω consists of k comonents: Ω = Ω 1 Ω 2 Ω k and (A 4 ) There exists M 0 > 0 such that Ω i Ω =,for all i. µ ( x R N : a(x) M 0 } ) <. where µ denotes the Lebesgue measure on R N. We comare our roblem with the related roblem ( + ia(x)) 2 u(x) + V (x)u(x) = u(x) 2 u(x), x R N. (1.1) In recent years, much attention has been devoted to the study of the existence for one-bum or multi-bum bound states of (1.1) under the case A(x) 0, which leads to investigate the ositive solutions u : R N R to the semilinear ellitic equation 2 u(x) + V (x)u(x) = u(x) 2 u(x), x R N. (1.2) In [15], using a Lyaunov Schmidt reduction, Floer and Weinstein established the existence of a standing wave solutions of (1.2) when N = 1, = 3 and V (x) is a bounded function having a non-degenerate critical oint for sufficiently small > 0. Moreover, they showed that u concentrates near the given non-degenerate critical oint of V when tends to 0. Their method and results were later generalized by Oh [19, 20] to the higher-dimensional case with 2 < < 2N N 2 and the existence of multi-bum solutions concentrating near several non-degenerate critical oints of V as tends to 0 was obtained. We also refer to A. Ambrosetti, A. Malchiodi, S. Secchi [2], A. Ambrosetti, M. Badiale and S. Cingolani [1], S. Cingolani and M. Lazzo [8], S. Cingolani and M. Nolasco [10], M. Del. Pino, P. Felmer [11], [12] for the case that A(x) 0. There is also much work on (1.1) with A(x) 0. The existence of solutions of (1.1) has been roved by Lions Esteban [14] for > 0 fixed and for secial classes of magnetic fields. They found existence by solving a aroriate minimization roblems for the corresonding energy functional in the case of N = 2 and N = 3. More recently, K. Kurata [17] has roved the existence of least energy solution of (1.1) for > 0 under a condition relating V (x) and A(x). S. Cingolani [7] roved the multile results of solutions of (1.1) which concentrate at a single oint for small > 0 by using toological argument, and he also roved that the magnetic fields A(x) only contributes to the hase factor of the solitary solutions of (1.1) as small enough. Moreover, S. Cingolani and S. Secchi [9] roved the existence of the one-bum bound states of (1.1) which concentrates at a non-degenerate critical oint of V (x) as goes to zero. D. Cao and Z. Tang [6] have verified the existence and uniqueness of multi-bum bound states of (1.1) which concentrate simultaneously near several different non-degenerate critical oints of V (x) as goes to zero.

3 812 Z. Tang ZAMP Though equation (S λ ) and (1.1) are related, there is also a distinctive difference. Setting v(x) := u( x), equation (1.1) is transformed to ( + ia( x)) 2 v(x) + V ( x)v(x) = v(x) 2 v(x), x R N. (1.3) Similarly, the scaling v(x) := 2/( 2) u( x) transforms (S λ ) with λ = 2 to ( + i A( x)) 2 v(x) + (a( x) + 2 )v(x) = v(x) 2 v(x), x R N. (1.4) In all the above mentioned aers on (1.1), it is assumed that V (x) V 0 > 0 is bounded away from 0. This is not the case in our situation. As 0, the otential vanishes in (1.3) rovided a(0) = 0 which we assume. In addition, recall that liminf x a(x) = 0 is allowed. Thus even at infinity the otential is not bounded away from 0. Since we do not imose on any further assumtions on the behavior of a(x) for x not much can be said about the sectrum of L A = ( + ia(x)) 2. In articular, the otential may oscillate for x roducing comlicated essential sectrum. We refer to T. Bartsch, E. N. Dancer and S. Peng [4] in the case of V (x) 0, they obtained the existence of multi-bum semi-classic bound states of (1.1) which concentrate simultaneously at the local minima of V (x) under the condition that V (x) is non-negative. Moreover, they obtained the asymtotic behavior of the bound states as sufficiently small. For A(x) 0, the same roblems was considered by Y. Ding and K. Tanaka [13], we also refer to T. Barstch and Zhi Qiang Wang [5]. The main technique in our aer comes from the idea of Y. Ding and K. Tanaka [13], However, since the aearance of electro-magnetic otential A(x), we must consider our roblem for comlex valued functions and so we need more delicate estimates. Our aer is organized as follows: In Section 2, we describe our main results (Theorem 2.2). Section 3 is devoted to reliminary results. Section 4 contains the roofs of the main results. We will use the same C to denote various generic ositive constants, and we will use o(1) to denote quantities that tend to 0 as λ( or n). 2. Main results Suose A C α loc (RN, R N ), write Let A u = ( + ia)u. H 1 A(R N ) := u L 2 (R N ) : A u L 2 (R N ) }. and hence HA 1 (RN ) is the Hilbert sace under the scalar roduct ( ) (u,v) = Re ( u + ia(x)u)( v + ia(x)v) + u v, R N

4 Vol. 59 (2008) Nonlinear Schrödinger equations 813 the norm induced by the roduct (.,.) is u H 1 A (R N ) = Let with the norms ( RN ( A u 2 + u 2) ) 1 2 ( RN ( = u + ia(x)u 2 + u 2)) 1 2 ( ( = u 2 + ( A(x) 2 + 1) u 2) 2Re ia(x)ū u R N R N E := } u HA(R 1 N ) : a(x) u 2 < R N u 2 ( E = A u 2 + (a(x) + 1) u 2). R N We can easily see that (E,. E ) is a Hilbert sace and E H 1 A (RN ). We define for oen set K R N, HA(K) 1 = u L 2 (K) : A u L 2 (K) }, ( ( u H 1 A (K) = A u 2 + u 2) ) 1 2, K } E(K) = u HA(K) 1 : a(x) u 2 <, ( u E(K) = K K ( A u 2 + (a(x) + 1) u 2)) 1 2. Let H 0,1 A (K) be the Hilbert sace defined by the closure of C 0 (K, C) under the scalar roduct (.,.). Thus u 2 = ( H 0,1 A u 2 + u 2). A (K) K Moreover, we have the following diamagnetic inequality(see [14] for examle): A u(x) u(x), for u H 1 A(R N ). and this fact means that if u H 1 A (RN ), then u H 1 (R N ). Remark 2.1. The saces HA 1 (RN ) and the saces H 1 (R N ) are not comarable; more recisely, in general HA 1 (RN ) H 1 (R N ) and H 1 (R N ) HA 1 (RN ). However it is roved by G. Arioli and A. Szulkin [3] that if K is bounded domain with regular boundary, then H 1 (K) and HA 1 (K) is equivalent, where H1 A (K) = u L 2 (K) : A u L 2 (K) } with the norm u H 1 A (K) = ( K ) 1 2 ( A u 2 + u 2) ) 1 2..

5 814 Z. Tang ZAMP Let with the norms E λ := } u HA(R 1 N ) : a(x) u 2 < R N u 2 ( λ = A u 2 + (λa(x) + 1) u 2). R N The energy functional associated with (S λ ) is defined by λ (u) = 1 ( u + ia(x)u 2 + (λa(x) + 1) u 2 ) 1 u for u E λ. 2 R N R N We say that u(x) E λ is a least energy solution of (S λ ) if and only if λ (u) = c λ := inf λ (u) : u E λ \ 0} is a solution of (S λ ) }. For λ large, the otential well Ω =int(a 1 (0)) lays an imortant role and the following roblem ( + ia(x)) 2 u(x) + u(x) = u(x) 2 u(x), x Ω, u(x) H 0,1 A (Ω) (D Ω ) is some kind of limit roblem of (S λ ) and the solutions are characterized as critical oints of I Ω (u) = 1 + ia(x)u 2 Ω( u 2 + (λa(x) + 1) u 2 ) 1 u for u H 0,1 A (Ω). Ω We also say that u H 0,1 A (Ω) is a least energy solution of (D Ω) if and only if I Ω (u) = c(ω) := infi Ω (u) : u u H 0,1 A (Ω) \ 0} is a solution of (D Ω)}. In [21], for λ large and a(x) only satisfies the condition (A 1 ),(A 2 ) and (A 4 ), we roved that (S λ ) has a least energy solution u λ and it tends to a least energy solution u 0 (x) of (D Ω ) as λ. By our assumtion on a(x) in this aer, Ω := int a 1 (0) = Ω 1 Ω 2 Ω k from (A 3 ), we have I Ω (u) = k =1 I Ω (u), for all u H 0,1 A (Ω). Here I Ω (u),c(ω ) is defined similarly with I Ω (u) and c(ω) with Ω is relaced by Ω. We see that c(ω) = min =1,2,...,k c(ω ), thus there must exists 0 1,2,...,k} such that u 0 Ω0 is a least energy solution of the following roblem ( + ia(x)) 2 u(x) + u(x) = u(x) 2 u(x), x Ω 0, u(x) H 0,1 A (Ω 0 ). Moreover, u 0 (x) = 0 in R N \Ω 0. It is natural to ask that for a given 1,2,...,k}, whether (S λ ) has a family solution u λ that converges to a least energy solution in Ω and to 0 elsewhere? In

6 Vol. 59 (2008) Nonlinear Schrödinger equations 815 this aer, we aim to answer this question and the answer is affirmative. Moreover, we also can construct multi-bum tye solutions. Our main results are: Theorem 2.2. Suose (A 1 ) (A 4 ) hold. Then for any ε > 0 and any non-emty subset of 1,2,...,k}, there exists Λ = Λ(ε) > 0 such that, for λ Λ, (S λ ) has a solution u λ E satisfying ( uλ + ia(x)u λ 2 +(λa(x) + 1) u λ 2) 1 ( Ω 2 1 ) c(ω ) ε for, (2.1) ( uλ + ia(x)u λ 2 + u λ 2) ε, (2.2) R N \Ω where Ω = Ω. Moreover, for any sequence λ n, we can extract a subsequence λ ni such that u λni converges strongly in HA 1 (RN ) to a function u(x) which satisfies u(x) = 0 for x Ω, and the restriction u Ω is a least energy solution of ( + ia(x)) 2 u(x) + u(x) = u(x) 2 u(x), x Ω, for. u(x) H 0,1 A (Ω ) Corollary 2.3. Under the same assumtion of Theorem 2.1, there exists Λ > 0 such that for λ > Λ, (S λ ) has at least 2 k 1 bound states. 3. Preliminaries From the assumtion (A 3 ) on a(x), for 1,2,...,k}, we can find bounded oen subset Ω with smooth boundary such that (i) Ω Ω for all, (ii) Ω i Ω = for all i. For any set K R N, we can define a norm on E(K) by u 2 λ,k = ( A u 2 + (λa(x) + 1) u 2 ) for all λ 0. K It is easy to see that. λ,k is equivalent to. E(K). It is also easy to find two constants 0 < ν 0 < 1 and δ 0 > 0 such that for any set K and for all u E(K) δ 0 u 2 λ,k u 2 λ,k ν 0 u 2 for all λ 0. (3.1) K

7 816 Z. Tang ZAMP We define f(t) : R R by mint 2 2,ν0 } for t 0, f(t) = 0 for t < 0, ν 0 t 2 t ν 2 0 for t [ν 2 2 0, ), F(t) = f(s)ds = 2 0 t 2 for t [0,ν ], 0 for t (,0]. In what follows, we fix non-emty subset 1,2,...,k} and we set Ω = Ω,Ω = Ω 1 for x Ω, χ Ω (x) =, 0 for x Ω and let We define of g(x,ξ 2 ) = χ Ω (x)ξ 2 + (1 χ Ω (x))f(ξ 2 ), G(x,ξ 2 ) = Φ λ (u) = 1 2 ξ 2 0 g(x,t)dt = 2 χ Ω (x)ξ + (1 χ Ω (x))f(ξ 2 ). R N ( A u 2 + (λa(x) + 1) u 2 ) 1 2 R N G(x, u 2 ) : E R. It is easy to check that Φ λ (u) C 2 (E, R) and its critical oints are solutions ( + ia(x)) 2 u(x) + (λa(x) + 1)u(x) = g(x, u 2 )u(x), x R N. We remark that f(t) = t 2 2 for t [0,ν ] and a critical oint u(x) of Φ λ (u) is solution of (S λ ) if and only if u ν in R N \Ω. We have the following comactness results. Proosition 3.1. For λ 0, Φ λ (u) satisfies (PS) c condition for all c R. That is any sequence (u n ) E satisfying for c R Φ λ (u n ) c, (3.2) Φ λ(u n ) 0 strongly in E (3.3) has a strongly convergent subsequence in E, where E is the dual sace of E. For giving the roof of Proosition 3.1, we need the following the lemma firstly. Lemma 3.2. Suose that a sequence (u n ) E satisfies (3.2) and (3.3). Then there exists constants m(c) and M(c) which is indeendent of λ 0 such that m(c) liminf u n 2 λ limsu u n 2 λ M(c). n n

8 Vol. 59 (2008) Nonlinear Schrödinger equations 817 Proof. It follows from (3.2) and (3.3) that where ε n 0 as n. Thus 2 1 ) Since and we have + 1 Φ λ (u n ) 1 Φ λ(u n )u n = c + o(1) + ε n u n λ, ( A u n 2 + (λa(x) + 1) u n 2) 1 R 2 N R N g(x, u n 2 ) u n 2 = c + o(1) + ε n u n λ. R N G ( x, u n 2) G(x, u n 2 ) = 2 χ Ω (x) u n + (1 χ Ω (x))f( u n 2 ), g(x, u n 2 ) u n 2 = χ Ω (x) u n + (1 χ Ω (x))f( u n 2 ) u n 2, 1 G(x, u n 2 ) 1 g(x, u n 2 ) u n 2 2 R N R [ N 1 = 2 F( u n 2 ) 1 g(x, u n 2 ) u n 2]. R N We remark that for t [ ν 2 2 0, ), 1 2 F(t2 ) 1 g(x,t2 )t 2 = 1 ( ν 0 t ν 0 = 2 1 ) (ν0 t 2 ν and for t ν 2 2 0, F(t2 ) 1 g(x,t2 )t 2 = 0. ) 1 t2 ) 2 1 ) ν 0 t 2, Thus we obtained that 2 ) 1 ( ) u n 2 λ ν 0 u n 2 c + o(1) + ε n u n λ. R N Hence from (3.1), we have 2 1 ) δ 0 u n 2 λ c + o(1) + ε n u n λ. Thus u n λ is bounded as n and lim su u n 2 λ M(c) := n 2 1 ) 1δ 1 0 c.

9 818 Z. Tang ZAMP On the other hand, since 1 2 F(t2 ) 1 f(t2 )t 2 0 for all t R, we have that Therefore c + o(1) + ε n u n λ 2 1 ) u n 2 λ. lim inf u n 2 λ m(c) := n 2 1 ) 1c. This comletes the roof of Lemma 3.2. Now we give the roof of Proosition 3.1. Proof of Proosition 3.1. From Lemma 3.2, we know that (u n ) is bounded in E λ and thus is bounded in H 1 A (RN ), so there exists a subsequence of (u n ) still denote (u n ) such that u n u weakly in E λ (H 1 A (RN )), u n u strongly in L loc (RN ). Now we rove that u n u in E λ. First of all, it is easy to check that u is critical oint of Φ λ (u), namely for any ψ E λ ( Re A u A ψ + (λa(x) + 1)u ψ ) = Re u 2 u ψ. R N R N It follows from (3.2) and (3.3) that (Φ λ(u n ) Φ λ(u))(u n u) 0 that is ( A (u n u) 2 + (λa(x) + 1) u n u 2) Re g(x, u n 2 )u n (u n u) R N R N + Re g(x, u 2 )u(u n u) R N ( = A (u n u) 2 + (λa(x) + 1) u n u 2) Re u n 2 u n (u n u) R N Ω Re f( u n 2 )u n (u n u) + Re u 2 u(u n u) + Re R N \Ω R N \Ω f( u 2 )u(u n u), Ω

10 Vol. 59 (2008) Nonlinear Schrödinger equations 819 by the definition of f(t), we have Re ( f( un 2 )u n f( u 2 )u ) (u n u) R N \Ω = Re (f( u n 2 )u n f( u n 2 )u)(u n u) R N \Ω + Re (f( u n 2 )u f( u 2 )u)(u n u) R N \Ω ν 0 u n u 2 L + ν 2 0 Re u(u n u). R N \Ω Since u n u in E λ, we have Re u(u R N n u) 0, from u n u in L 2 (Ω ), we know that Re u(u Ω n u) 0. Thus, we have ν 0 Re u(u n u) 0. R N \Ω We also remark that u n u strongly in L (Ω ), thus by (3.1) we have δ 0 u n u 2 λ u n u 2 λ ν 0 u n u 2 L 2 Re u n 2 u n (u n u) Re Ω Ω u 2 u(u n u) 0 as n. Therefore u n u in E λ and this comletes the roof of Proosition 3.1. Proosition 3.3. Assume sequence (u n ) E and (λ n ) [0, ) satisfying λ n, (3.4) Φ λn (u n ) c, (3.5) Φ λ n (u n ) λ n 0. (3.6) Then after extracting a sequence, still denoted by n, we have u n u weakly in E and H 1 A(R N ) for some u E. Moreover (i) u 0 in R N \ Ω and u(x) is a solution of ( + ia(x)) 2 u(x) + u(x) = u(x) 2 u(x), x Ω, u(x) H 0,1 A (Ω ) for. (ii) u n converges to u(x) in a stronger sense, namely u n u λn 0, u n u strongly in E and H 1 A(R N ). (3.7)

11 820 Z. Tang ZAMP (iii) u n (x) also satisfying λ n a(x) u n 2 0, R N Φ λ (u n ) I Ω (u), u n u λn,r N \Ω 0, u n u 2 λ n,ω Ω A u 2 + u 2 for. Proof. As the similar roof with Lemma 3.2, we can rove that m(c) liminf u n 2 λ n n limsu u n 2 λ n M(c). n Thus (u n ) stays bounded as n in E and H 1 A (RN ), we may assume that for some u E u n u weakly in E and H 1 A (RN ), u n u a.e. in R N, u n u strongly in L q loc (RN ) for 2 q < 2N N 2. Now we come to show (i). Set C m := x R N : a(x) 1 m }, for n large, we have u n 2 m λ n a(x) u n 2 m (λ n a(x) + 1) u n 2 C m λ n R λ N n R N m λ n R N ( (λn a(x) + 1) u n 2 + A u n 2) = m λ n u n 2 λ n 0. Thus u(x) = 0 on m=1 = R N \ Ω. Next, for any ϕ C 0 (Ω, C), 1,2,...,k}, we have Φ λ n (u n )ϕ Φ λ n (u n ) λ n ϕ λn 0, here we use the fact that ϕ λn indeed does not deendent on λ n. Thus we have Re ( A u A ϕ + u ϕ) = Re Ω g(x, u 2 )u ϕ. Ω By the definition of g(x,t), we know that for, u(x) satisfies (3.7). For 1,2,...,k} \, setting ϕ = u(x) we have Ω A u 2 + u 2 f( u 2 ) u 2 = 0, that is u 2 1,Ω Ω f( u 2 ) u 2 = 0.

12 Vol. 59 (2008) Nonlinear Schrödinger equations 821 On the other hand, we know that 0 = u 2 1,Ω u 2 1,Ω ν 0 Ω f( u 2 ) u 2 Ω u 2 δ 0 u 2 1,Ω. Thus u = 0 in Ω for 1,2,...,k} \ and thus we get (i). For (ii), we know that Φ λ n (u n )(u n u) Φ λ n (u)(u n u) = u n u 2 λ n Re f( u n 2 )u n (u n u) + Re R N \Ω R N \Ω Re u n 2 u n (u n u) + Re u 2 u(u n u). Ω Ω Since u n u in L (Ω ), we have Re ( u n 2 u n u 2 u)(u n u) 0 as n. On the other hand Ω Φ λ n (u n )(u n u) Φ λ n (u n ) λ n u n u λn Thus we have u n u 2 λ n Re Φ λ n (u n ) λ n ( u n λn + u λn ) 0. R N \Ω f( u 2 )u(u n u) (f( u n 2 )u n f( u 2 )u)(u n u) 0. As the similar argument in the roof of Proosition 3.1, we obtain that δ 0 u n u 2 λ n u n u 2 λ n ν 0 u n u 2 L 2 (R N ) = u n u 2 λ n Re (f( u n 2 )u n f( u 2 )u)(u n u) + o(1) 0 R N \Ω and thus (ii) is obtained. Now we show (iii). Indeed 1 λ n a(x) u n 2 = 1 λ n a(x) u n 2 2 R 2 N R N \Ω = 1 λ n a(x) u n u 2 u n u 2 λ 2 n 0. R N \Ω This comletes the roof of Proosition 3.3. Proosition 3.4. There exists a constant Λ 0 > 0 such that if u λ is a critical oint of Φ λ (u) for λ Λ 0, then u λ ν In articular, u λ solves the original roblem (S λ ).

13 822 Z. Tang ZAMP Proof. We use notation B r (x) = y R N : x y < r}. Since u λ E is a critical oint of Φ λ (u), namely u λ satisfies the following equation ( + ia(x)) 2 u λ (x) + (λa(x) + 1)u λ (x) = g(x, u λ 2 )u λ (x), x R N. By Kato s inequality there holds ( ) uλ u λ Re u λ ( + ia(x))2 u λ (x), u λ (x) (λa(x) + 1) u λ (x) g(x, u λ 2 ) u λ (x) 0, x R N, since u λ 0 and a(x) 0 we have u λ (x) (1 + g(x, u λ 2 )) u λ (x) 0, x R N, we use the subsolution estimate (see Theorem 8.17 in [16]) to get that there exists a constant C(r) such that for any 1 < q < 2 u λ (x) C(r) u λ (x) q. B r(x) By Proosition 3.3, for any sequence λ n we can extract a subsequence still denote λ n such that In articular, u λn u 0 H 0,1 A (Ω ) strongly in H 1 A(R N ). u λn u 0 H 0,1 A (Ω ) strongly in L 2 A(R N \ Ω ). Since λ n is arbitrary, we have u λ u 0 H 0,1 A (Ω ) strongly in L 2 A(R N \ Ω ) as λ. Thus, choosing r (0,dist(Ω, R N \ Ω )), we have uniformly in x RN \ Ω that u λ (x) C(r) u λ (x) q B r(x) C(r)(meas B r (x)) 1 q 2 uλ q L 2 (B r(x)) C(r)(meas B r (x)) 1 q 2 uλ q L 2 (R N \Ω ) 0. This comletes the roof of Proosition Proof of main results For we consider the following two functionals I Ω (u) = 1 ( u + ia(x)u 2 + u 2) 1 u for u H 0,1 A 2 Ω (Ω ), Ω

14 Vol. 59 (2008) Nonlinear Schrödinger equations 823 and for u E(Ω ) = H1 A (Ω ), Φ λ,ω (u) = 1 u + ia(x)u 2 Ω ( 2 + (λa(x) + 1) u 2) 1 Ω u. (4.1) One can easily to see that both of I Ω (u) and Φ Ω (u) has mountain ass geometry. That is, (i) I Ω (0) = Φ λ,ω (0) = 0. (ii) There exists ρ 0 > 0 and ρ 1 > 0 indeendent of λ 0 such that u 0,Ω ρ 0 I Ω (u) 0, u 0,Ω = ρ 0 I Ω (u) ρ 1, u 0,Ω ρ 0 Φ λ,ω (u) 0, u 0,Ω = ρ 0 Φ λ,ω (u) ρ 1, Here we use the notation: u 0,Ω = ( u + ia(x)u 2 + u 2 ) for u H 0,1 A (Ω ). Ω (iii) There exists ϕ (x) C 0 (Ω, C) such that ϕ (x) λ,ω = ϕ (x) 0,Ω ρ 1, Φ λ,ω (ϕ ) = I Ω (ϕ ) < 0. We define the following minimax values(mountain ass): where c = inf γ Γ max t [0,1] I Ω (γ(t)), c λ, = inf γ Γ λ, max t [0,1] Φ λ,ω (γ(t)), Γ = γ C([0,1],H 0,1 A (Ω )) : γ(0) = 0,I Ω (γ(1)) < 0 }, Γ λ, = γ C([0,1],H 1 A(Ω )) : γ(0) = 0,Φ λ,ω (γ(1)) < 0 }. (4.2) (4.3) It is standard to verify the Palais Smale condition for I Ω (u) and Φ λ,ω (u) and c,c λ, are achieved by critical oints. We denote the corresonding critical oints by ω (x) and ω λ, (x) resectively. We have the following lemma: Lemma 4.1. (i) 0 < ρ 1 c λ, c for all λ 0. (ii) c (c λ, resectively) is a least energy level for I Ω (u) (Φ λ,ω (u) resectively), that is c = inf I Ω (u) : u H 0,1 A (Ω ) \ 0} is a critical oint of I Ω }, c λ, = inf Φ λ,ω (u) : u H 1 A(Ω ) \ 0} is a critical oint of Φ λ,ω },

15 824 Z. Tang ZAMP (iii) c = max r>0 I Ω (rω ), c λ, = max r>0 Φ λ,ω (rω λ, ). (iv) c λ, c as λ. Proof. From (4.3), it is easy to see that c λ, ρ 1. On the other hand, for any u H 0,1 A (Ω ), we may extend u to ũ HA 1 (Ω ) by u(x) in Ω, ũ = 0 in Ω \ Ω, we regard H 0,1 A (Ω ) H 1 A (Ω ). Thus we have Γ Γ λ, and c λ, = inf max Φ λ,ω (γ(t)) γ Γ λ, t [0,1] inf γ Γ max t [0,1] Φ λ,ω (γ(t)) = inf γ Γ max t [0,1] I Ω (γ(t)) = c (4.4) Thus we have (i). Using the monotonicity of the term u with resect to u, the roof of (ii) and (iii) is standard. Now we show (iv). Using Proosition 3.3, we may extract a subsequence λ n such that ω λ, u 0 strongly in H 1 A(Ω ), where u 0 H 0,1 A (Ω ) is a solution of (3.7) and By the definition of c, we have Φ λ,ω (ω λ, ) I Ω (u 0 ). lim su c λ, = lim su Φ λ,ω (ω λ, ) I Ω (u 0 ) c. λ λ Comare with (4.4), we get (iv) and this comlete the roof of this lemma. We remark that a simle rescaling argument give us the following ( c = I Ω (v) : v H 0,1 1 A (Ω ), v = Ω 2 1 ) } 1 c, c λ, = Φ λ,ω (v) : v HA(Ω 1 ), v = 2 1 ) 1 c λ,}. Ω (4.5) Now we give a minimax argument for Φ(u). We choose R 2 such that I Ω (Rω ) < 0, ) 1 c (4.6) R ω L 2 2 1

16 Vol. 59 (2008) Nonlinear Schrödinger equations 825 for all. Without loss of generality, we assume that = 1,2,...,l} (l k). We remark that the roect t trω belongs to Γ and satisfies max t [0,1] I Ω (trω ) = c for any. Now we set γ 0 (s 1,s 2,...,s l )(x) = Γ = l s Rω (x) for all (s 1,s 2,...,s l ) [0,1] k, (4.7) =1 γ C([0,1] l,e) : γ(s 1,s 2,...,s l ) = γ 0 (s 1,s 2,...,s l ) for all (s 1,s 2,...,s l ) ([0,l] l ) } and b λ, = inf max γ Γ (s 1,s 2,...,s l ) ([0,1] l ) Φ λ (γ(s 1,s 2,...,s l )). We remark that Γ since γ 0 Γ and thus b λ, is well defined. We denote c = l =1 c, we have the following lemmas. Lemma 4.2. (i) l =1 c λ, b λ, c for all λ 0. (ii) Φ λ (γ(s 1,s 2,...,s l )) c ρ 1 for all λ 0,γ Γ and (s 1,s 2,...,s l ) ([0,1] l ). Here ρ 1 is given in (4.2),(4.3) and (i) in Lemma 4.1. Proof. For any given γ Γ, we define a ma T : [0,1] l R l as follows ( T (s 1,s 2,...,s l ) = γ(s1,s 2,...,s l )(x),..., γ(s 1,s 2,...,s l )(x) ). Ω 1 We have for (s 1,s 2,...,s l ) ([0,1] l ) thus for any T (s 1,s 2,...,s l ) = (s 1 R ω 1,s 2 R ω 2,...,s l R ω l ) (ξ 1,ξ 2,...,ξ l ) ([0,R ω 1 L ] [0,R ω 2 L ] [0,R ω l L]) (4.8) Ω l We have deg(t,[0,1] l,(ξ 1,ξ 2,...,ξ l )) = 1, By the roerty of toological degree, there exists (s 1,s 2,...,s l ) [0,1] l such that γ(s 1,s 2,...,s l )(x) = ξ for all = 1,2,...,l. (4.9) Ω Now we come to show (i).

17 826 Z. Tang ZAMP Since γ 0 Γ, we have b λ, max Φ λ (γ 0 (s 1,s 2,...,s l )) (s 1,s 2,...,s l ) [0,1] l = max = (s 1,s 2,...,s l ) [0,1] l =1 l c = c. =1 On the other hand, remarking (4.6), let ( (1 (ξ 1,ξ 2,...,ξ l ) = 2 1 ) 1 c λ,1,, l I Ω (s Rω ) 2 1 ) 1 c λ,2,..., 2 1 ) 1 c λ,l), from (4.8) and (4.9) we have for any γ Γ, there exists s γ [0,1] l such that γ(s γ )(x) = 2 1 ) 1 c λ, for all = 1,2,...,l. Ω Thus for u(x) = γ(s γ )(x), we have where Φ λ,r N \Ω (u) = 1 2 l Φ λ (u) = Φ λ,rn \Ω (u) + Φ λ,ω (u), R N \Ω =1 ( u + ia(x)u 2 + (λa(x) + 1) u 2 ) 1 2 Since F( u 2 ) ν 0 u 2, we have Φ λ,rn \Ω (u) = 1 ( u + ia(x)u 2 + (λa(x) + 1) u 2 ) Thus R N \Ω 1 2 u 2 λ,r N \Ω 1 2 u 2 L 2 (λ,r N \Ω ) δ 0 2 u 2 λ,r N \Ω 0. l Φ λ (u) = Φ λ,r N \Ω (u) + Φ λ,ω (u) = =1 l Φ λ,ω (u) =1 l inf Φ λ,ω (v) : v HA(Ω 1 ), =1 l c λ,. =1 Ω v = R N \Ω R N \Ω F( u 2 ). F( u 2 ) 2 1 ) 1 c λ,}

18 Vol. 59 (2008) Nonlinear Schrödinger equations 827 Since γ Γ is arbitrary, we have b λ, c λ,. For (ii), we remark that for any γ Γ γ(s 1,s 2,...,s l ) = γ 0 (s 1,s 2,...,s l ) on ([0,1] l ), thus by the definition of γ 0, for (s 1,s 2,...,s l ) ([0,1] l ) we have Φ(γ 0 (s 1,s 2,...,s l )) = l I Ω (s Rω ) and I Ω (s Rω ) c for all = 1,2,...,l. On the other hand, for some 0, s 0 = 1 or s 0 = 0 and thus I Ω0 (s 0 Rω 0 ) 0. Therefore =1 Φ(γ 0 (s 1,s 2,...,s l )) 0 I Ω (s Rω ) c ρ 1. This comletes the roof of the whole lemma. Corollary 4.3. b λ, c as λ, moreover b λ, is a critical oint of Φ λ for large λ. Proof. From Lemma 4.1, we know that c λ, c as λ, thus from above lemma, it is clear that b λ, c as λ. Thus, we may choose λ 0 large enough such that for all λ λ 0, b λ, > c ρ 1. Since Φ λ (u) satisfies Palais Smale condition, by the standard deformation argument we can see that b λ, is a critical value of Φ λ (u) for λ λ 0. This comletes the roof of the corollary. We use the following notation: We choose Φ c λ = u E : Φ λ(u) c }. 0 < µ < 1 3 min (1 2 1 ) 1 c (4.10) and define (1 D µ λ = u E : u λ,r N \Ω µ, u λ,ω 2 1 ) 1 c µ for all. We remark that ω is the least energy solution of (3.7) and ( ω + ia(x)ω 2 + ω 2 ) = Ω 2 1 ) 1 c. Thus D µ λ Φc λ We have the following lemma: contains all the functions of the following form ω (x) x Ω, ω(x) = 0, x R N \ Ω.

19 828 Z. Tang ZAMP Lemma 4.4. There exists σ 0 > 0 and Λ 0 0 indeendent of λ such that Φ λ(u) λ σ 0 for λ Λ 0 and for all u (D 2µ λ \ Dµ λ ) Φc λ. (4.11) Proof. We rove it by contradiction. Suose that there exist λ n and u n (D 2µ λ n \ D µ λ n ) Φ c λ n such that Φ λ n (u) λ n 0. Since u n D 2µ λ n, thus u n is bounded in E (HA 1 (RN )) and it imlies Φ λn (u n ) stays bounded as n. We may assume that Φ λn (u n ) c c u to a subsequence. Alying Lemma 4.1, we can extract a subsequence of u n still denote u n such that u n u in E (H 1 A (RN )) and lim Φ λ n n (u n ) = l I Ω (u) c, (4.12) =1 lim u n 2 n λ = n,ω ( u + ia(x)u 2 + u 2 ) for all, (4.13) Ω lim ( u n + ia(x)u n 2 + (λ n a(x) + 1) u n 2 ) = 0. (4.14) n R N \Ω Since c = l =1 c and c is the least energy level for I Ω (u), thus we have two ossibilities: 1) I Ω (u Ω ) = c for all, 2) I Ω0 (u Ω0 ) = 0, that is u Ω0 = 0 for some 0. If 1) occurs, we have Ω ( u + ia(x)u 2 + u 2 ) = 2 1 ) 1 c for all and it follows from (4.13) and (4.14) that u n D 2µ λ n for large n which is a contradiction to u n (D 2µ λ n \ D µ λ n ). If 2) occurs, from (4.13) that (1 u n λn,ω ) 1 (1 c ) 1 c 0 3µ. This is also a contradiction to u n (D 2µ λ n \ D µ λ n ) and we comlete the roof. The following roosition is the key of the roof of our main result. Proosition 4.5. Let µ satisfy (4.10) and let Λ 0 be the constant given in Lemma 4.4. Then for λ Λ 0 there exists a solution u λ of (S λ ) satisfying u λ D µ λ Φc λ.

20 Vol. 59 (2008) Nonlinear Schrödinger equations 829 Proof. We argue indirectly and assume that Φ λ (u) has no critical oints in D µ λ Φc λ. Since Φ λ(u) satisfy Palais Smale condition, there exists a constant d λ > 0 such that Φ λ(u) λ d λ for all u D µ λ Φc λ and from Lemma 4.4 we have Φ λ(u) λ σ 0 for all u (D 2µ λ \ Dµ λ ) Φc λ. Let ϕ : E R be a Lischitz continuous function such that 1 for u D 3µ 2 λ ϕ(u) =, 0 for u D 2µ λ and 0 ϕ(u) 1 for any u E. For any u Φ c λ, we define Φ λ V (u) = ϕ(u) (u) Φ : Φ c λ (u) λ E. λ Here we identity E and E by the Riesz reresentation theorem. We consider the following deformation flow η : [0, ) Φ c λ Φc λ defined by dη dt = V (η(t,u)), η(0,u) = u Φc λ. η(t,u) has the following roerties: d dt Φ λ(η(t.u)) = ϕ(u) Φ λ(u) λ 0, (4.15) dη dt λ 1 for all t,u, (4.16) η(t,u) = u for all t 0 and u Φ c λ \ D2µ λ. (4.17) Let γ 0 (s 1,s 2,...,s l ) Γ be a ath defined in (4.7) and we consider η(t,γ 0 (s 1,s 2,...,s l )) for large t. Since for all (s 1,s 2,...,s l ) ([0,1] l ), γ 0 (s 1,s 2,...,s l ) D 2µ λ, thus we have by (4.17) that η(t,γ 0 (s 1,s 2,...,s l )) = γ 0 (s 1,s 2,...,s l ) for all (s 1,s 2,...,s l ) ([0,1] l ) and η(t,γ 0 (s 1,s 2,...,s l )) Γ for all t 0. Since suγ 0 (s 1,s 2,...,s l )(x) Ω for all (s 1,s 2,...,s l ) ([0,1] l ) and hence Φ λ (γ 0 (s 1,s 2,...,s l )(x)) and γ 0 (s 1,s 2,...,s l )(x) λ,ω etc. do not deend on λ 0. On the other hand Φ λ (γ 0 (s 1,s 2,...,s l )(x)) c for all (s 1,s 2,...,s l ) [0,1] l and Φ λ (γ 0 (s 1,s 2,...,s l )(x)) = c if and only if s = 1, that is R γ 0 (s 1,s 2,...,s l )(x) Ω = ω for all. Thus we have m 0 := maxφ λ (u) : u γ 0 ([0,1] l ) \ D µ λ } (4.18)

21 830 Z. Tang ZAMP is indeendent of λ and m 0 < c. We claimed that for large T, max Φ λ (η(t,γ 0 (s 1,s 2,...,s l )(x))) max (s 1,s 2,...,s l ) [0,1] l where σ 0 and m 0 are given in (4.11) and (4.18). In fact, if γ 0 (s 1,s 2,...,s l )(x) D µ λ, then by (4.18) we have m 0,c 1 } 2 σ 0µ (4.19) Φ λ ( η(t,γ0 (s 1,s 2,...,s l )(x)) ) m 0 and thus (4.19) holds. Now we consider the case γ 0 (s 1,s 2,..., s l )(x) D µ λ, we consider the behavior of η(t) := η(t,γ 0 (s 1,s 2,...,s l )). We set d λ := mind λ,σ 0 } and T = σ 0µ 2 d λ We consider two cases: 1) η(t) D 3µ 2 λ 2) η(t 0 ) D 3µ 2 λ for all t [0,T]. for some t 0 [0,T]. When 1) holds, we have ϕ( η(t)) 1 and Φ λ ( η(t)) λ d λ for all t [0,T]. Thus by (4.15), we have Φ λ ( η(t)) = Φ λ (γ 0 (s 1,s 2,...,s l )) + = Φ λ (γ 0 (s 1,s 2,...,s l )) T c 0 d λ ds = c d λ T = c 1 2 σ 0µ. T 0 T When 2) holds, there exists 0 t 1 < t 2 T such that 0 d ds Φ λ( η(t)) ϕ( η(s))) Φ λ( η(s)) λds η(t 1 ) D µ λ, (4.20) η(t 2 ) D 3µ 2 λ, (4.21) We will rove that η(t) D 3µ 2 λ \ Dµ λ for all t [t 1,t 2 ]. (4.22) η(t 1 ) η(t 2 ) λ 1 µ. (4.23) 2 To see (4.23), we set ω 1 = η(t 1 ) and ω 2 = η(t 2 ). It follows from (4.21) that ω 2 λ,r N \Ω = 3µ (1 2 or ω 2 λ,ω ) 1 c 0 = 3µ 2 for some 0.

22 Vol. 59 (2008) Nonlinear Schrödinger equations 831 We only see the later case, the former case can be dealt with the similar way. By (4.20), (1 ω 1 λ,ω ) 1 c 0 µ. Thus we have ω 1 ω 2 λ,ω (1 ω 2 λ,ω ) 1 c 0 (1 ω 1 λ,ω ) 1 c µ. Thus ω 1 ω 2 λ ω 1 ω 2 λ,ω 1 µ and we roved (4.23). 2 By (4.16),(4.23) and mean value theorem, we have t 2 t 1 1 µ. Using (4.11) 2 we have Φ λ ( η(t)) = Φ λ (γ 0 (s 1,s 2,...,s l )(x)) T t2 c σ 0 ds = c σ 0 (t 1 t 2 ) t 1 c 1 2 σ 0µ 0 ϕ( η(s))) Φ λ( η(s)) λds and thus (4.19) is roved. We recall that η(t) = η(t,γ 0 (s 1,s 2,...,s l )) Γ. Thus b λ, Φ λ ( η(t)) max m 0,c 1 } 2 σ 0µ. (4.24) However, by Corollary 4.3, we have b λ, c as λ. This is a contradiction with (4.24) and thus Φ λ (u) has critical oint u λ (x) D µ λ for large λ and by Proosition 3.4, u λ (x) is a solution of the original roblem (S λ ). Now we give the roof of main results. Proof of Theorem 2.2. Let u λ (x) be a solution the roblem (S λ ) obtained in Proosition 4.5, alying Proosition 3.3, for any given sequence λ n, we can extract a subsequence, still denote it by λ n which satisfies the conclusion of Proosition 3.3. With the same argument in the roof of Lemma 4.4, we can extract a subsequence of u λn still denote u λn such that u λn u in E (HA 1 (RN )) and ( uλn +ia(x)u λn 2 +(λ n a(x)+1) u λn 2) ( ) 1 1 = c for all, lim n λ n,ω 2 1 (4.25)

23 832 Z. Tang ZAMP ( lim uλn + ia(x)u λn 2 + (λ n a(x) + 1) u λn 2) = 0. n R N \Ω (4.26) Since the limit in (4.25) and (4.26) do not deend on the choice of sequence λ n, thus we have (2.1) and (2.2). and the limit function u(x) satisfies 1) u(x) 0 for x R N \ Ω, 2) u(x) Ω is least energy solution of ( + ia(x)) 2 u(x) + u(x) = u(x) 2 u(x), x Ω, u(x) H 0,1 A (Ω ) for. This comletes the roof of Theorem 2.2. References [1] A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal. 140 (1997), [2] A. Ambrosetti, A. Malchiodi and S. Secchi, Multilicity results for some nonlinear Schrödinger equations with otentials, Arch. Ration. Mech. Anal. 159 (2001), [3] G. Arioli and A. Szulkin, A semilinear Schrödinger equation in the resence of a maganetic field, Arch. Rational Mech. Anal. 170 (2003), [4] T. Bartsch, E. N. Dancer and S. Peng, On multi-bum semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields. Adv. Differential Equations. 11 (2006), [5] T. Bartsch and Z. Q. Wang, Multile ositive solutions for a nonlinear Schrödinger eqaution, Z. angew. Math. Phys. 51 (2000), [6] D. Cao and Z. Tang, Existence and uniqueness of multi-bum bound states of nonlinear Schrödinger equations with electromagnetic fields,. Diff. Equat. 222 (2006), [7] S. Cingolani, Semiclassical stationary states of nonlinear Schrödinger eqaution with external magnetic field,. Diff. Equat. 188 (2003), [8] S. Cingolani and M. Lazzo, Multile ositive solutions to nonlinear Schrödinger equations with cometing otential functions,. Diff. Equat. 160 (2000), [9] S. Cingolani and S. Secchi, Semiclassical limit for nonlinear Schrödinger eaution with electromagnetic fields,. Math. Anal. Al. 275 (2002), [10] S. Cingolani and M. Nolasco, Multi-eaks eriodic semiclassical states for a class of nonlinear Schrödinger equations, Proc. Royal Soc. Edinburgh. 128 (1998), [11] M. Del Pino, P. Felmer, Semi-classical states for nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré 15 (1998), [12] M. Del Pino, P. Felmer, Multi-eak bound states for nonlinear Schrödinger equations,. Funct. Anal. 149 (1997), [13] Y. Ding and K. Tanaka, Multilicity of ositive solutions of a nonlinear Schrödinger equation, Manuscrita Math. 112 (2003), [14] M. Esteban and P. L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external maganetic field, in: Partial Differential equations and the Calculus of Variations, Essays in Honor of Ennio De Giorgi, , [15] A. Floer and A. Weinstein, Nonsreading wave ackets for the cubic Schrödinger equation with a bounded otential,. Funct. Anal. 69 (1986), [16] D. Gilbarg, N. Trudinger, Ellitic artial differential equations of second order, second edition, Sringer-Verlag, New York, 1983.

24 Vol. 59 (2008) Nonlinear Schrödinger equations 833 [17] K. Kurata, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields, Nonlin. Anal., TMA 41 (2000), [18] P. L. Lions, The concentration-comactness rincile in the calculus of variations. The locally comact case. Part I, Ann. I. H. Poincaré, Anal. non linéaire 1 (1984), [19] Y.-G. Oh, On ositive multi-lum bound states of nonlinear Schrödinger equations under multile well otential, Comm. Math. Phys. 131 (1990), [20] Y.-G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with otentials of class (V ) a, Comm. Part. Diff. Equat. 13 (1988), [21] Z. Tang, On the least energy solutions of nonlinear Schrödinger equations with electromagnetic fields, Com. Math. Al., to aear. Zhongwei Tang School of Mathematical Sciences Beiing Normal University Beiing, P.R. of China tangzw@bnu.edu.cn (Received: March 21, 2007) Published Online First: August 25, 2007 To access this ournal online: