Bound states of schrödinger equations with electromagnetic fields and vanishing potentials

Acta Mathematica Scientia 2017,37B(2):405–424 http://actams.wipm.ac.cn

¨ BOUND STATES OF SCHRODINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS AND VANISHING POTENTIALS∗

nA)

Na BA (

School of Science, Hubei University of Technology, Wuhan 430068, China E-mail : [email protected]

“A
)

Jinjun DAI (

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China E-mail : [email protected] Abstract We study the bound states to nonlinear Schr¨ odinger equations with electro2 p−1 h magnetic fields ih ∂ψ = ( ∇ − A(x)) ψ + V (x)ψ − K(x)|ψ| ψ = 0, on R+ × RN . Let ∂t i p+1

−N

− 2

G(x) = [V (x)] p−1 2 [K(x)] p−1 and suppose that G(x) has k local minimum points. For h > 0 small, we find multi-bump bound states ψh (x, t) = e−iEt/h uh (x) with uh concentrating at the local minimum points of G(x) simultaneously as h → 0. The potentials V (x) and K(x) are allowed to be either compactly supported or unbounded at infinity. Key words

Multi-bump solutions; nonlinear Schr¨ odinger equation; electromagnetic fields; potentials compactly supported or unbounded at infinity

2010 MR Subject Classification

1

35J20; 35J60

Introduction

The linear Schr¨ odinger equation is a basic tool of quantum mechanics and provides a description of the dynamics of a particle in a non-relativistic setting. The nonlinear Schr¨odinger equation arise in different physical theories, for example, the description of Bose-Einstein condensates, plasma physics and nonlinear optics(see [10, 34]), and the presence of many particles leads one to consider nonlinear terms that simulate the interaction effect among them. Both the linear and the nonlinear Schr¨ odinger equations have been widely considered in literature. In this article, we investigate the following nonlinear Schr¨odinger equation 2 ∂Ψ  h ih = ∇ − A(x) Ψ + P (x)Ψ − K(x)|Ψ|p−1 Ψ, (t, x) ∈ R × RN , (1.1) ∂t i where 1 < p < (N + 2)/(N − 2) for N ≥ 3 and 1 < p < ∞ for N = 1, 2. The function Ψ(x, t) takes on complex values, h is the Planck constant, i is the imaginary unit. Here, A : RN → RN ∗ Received

August 2, 2016. Na Ba was supported by National Natural Science Foundation of China (11201132), Scientific Research Foundation for Ph.D of Hubei University of Technology (BSQD12065), and the Scientific Research Project of Education Department of Hubei Province (Q20151401).

406

ACTA MATHEMATICA SCIENTIA

Vol.37 Ser.B

denotes a magnetic potential and the Schr¨ odinger operator is defined by h 2 2h h ∇ − A(x) Ψ := −h2 ∆Ψ − A · ∇Ψ + |A|2 Ψ − ΨdivA. i i i Actually, in general dimension N ≥ 2, the magnetic field B is a 2-form, where Bk,j = ∂j Ak − ∂k Aj ; in the case N = 3, B = curlA. The function P : RN → R represents an electric potential acting on the particle and K(x) a particle-interaction term, which avoids spreading of the wave packets in the time-dependent version of the above equation. We are interested in standing wave solutions for (1.1), that is, solutions of the form Ψ(x, t) = e u(x) for some function u : RN 7→ C. Substituting this ansatz into (1.1), one is led to solve the complex equation in RN , 2 h ∇ − A(x) u + (P (x) − E)u = K(x)|u|p−1 u. (1.2) i − iEt h

For simplicity, let V (x) = (P (x)−E) and assume that V is non-negative on the whole space R . The transition from quantum mechanics to classical mechanics can be formally described by letting h → 0, thus the existence of solutions for h small has physical interest. Standing waves for h small are usually referred as semi-classical bound states (see [25]). N

In recent years, a lot of work has been devoted to studying standing-wave solutions in the case A(x) ≡ 0. In this case, one can be led to look for real-valued solutions u : RN → R of the semilinear equation −h2 ∆u(x) + V (x)u(x) = K(x)|u|p−1 u.

(1.3)

Different approaches were taken to deal with this problem under various hypotheses on the potentials and the nonlinearity (see [12, 19, 22, 30, 31] and so on). When the magnetic potential A(x) 6≡ 0, (1.2) is a complex-valued problem and can not be reduced to a real-valued one. The first work would be appear to be [20], where standing waves were obtained for h > 0 fixed and for special classes of magnetic fields. Concerning semiclassical bound states, Kurata [27] proved that when h > 0 is small, (1.2) admits a least energy solution which concentrates near the global minimum of V . Results concerning bounded vector potentials, existence and concentration phenomena at any nondegenerate critical point, not necessarily a minimum, of V , as h → 0 (that is, V has a manifold of stationary points), were proved by Cingolani and Secchi [17] using a perturbation approach given by Ambrosetti, Malchiodi, and Secchi [3]. Multi-peak semiclassical solutions for (1.2) for bounded vector potentials was constructed in [14] by Cao and Tang. In [18], using a penalization procedure, Cingolani and Secchi extended the result in [17] to the case of a vector potential A, possibly unbounded. The penalization approach was also used in [6, 15] to obtain multi-bump semiclassical bound states for problem (1.1) with more general nonlinearity f (x, Ψ). Concerning other articles on the topic, we mention that Helffer [24, 25] studied asymptotic behavior of the eigenfunctions of the Schr¨odinger operators with magnetic fields in the semiclassical limit. See also [7] for generalization of the results in [26] for potentials which degenerate at infinity. For more recent results, we can refer to [4, 11, 16, 28, 35, 36, 38] and the references therein. However, we should point out that, for the case A(x) 6≡ 0, in all the above mentioned work, the potential V (x) should have a positive lower bound and the potential K(x) > 0 should be bounded from above at infinity. For the case that V (x) vanishes at infinity or K(x) tends to

No.2

¨ N Ba & J Dai: SCHRODINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS

407

infinity at infinity, it seems that there are very few results. In this article, we will consider this type of problem, that is,  2 h   ∇ − A(x) u + V (x)u = K(x)|u|p−1 u, x ∈ RN , i (1.4)   u ∈ H 1 (RN , C),

where V (x) ≥ 0 has a compact support, and K(x) ≥ 0 may tend to zero or infinity as |x| → ∞. For the real-valued case of (1.4) (that is problem (1.3)), it seems that Ambrosetti, Felli, and Malchiodi [1] are the first to consider (1.3) with vanishing potential V (x). More precisely, γ0 c under the condition 1+|x| , where γ0 , γ1 , β0 , c > 0, it is α0 ≤ V (x) ≤ γ1 and 0 < K(x) ≤ 1+|x|β0 proved in [1] that (1.3) has a ground state solution for 0 ≤ α0 < 2 provided that h is sufficiently small and σ < p < (N + 2)/(N − 2), where  4β0  N +2 − , if 0 < β0 < α0 , N − 2 α (N − 2) 0 σ=   1, otherwise.

In [2], this result was generalized to the case 1 < p < (N + 2)/(N − 2) and 0 ≤ α0 ≤ 2. For the case of faster decay of V (x), when V (x) ∼ |x|α at infinity with α ≥ −4, using a constructive argument, Cao and Peng [13] studied the existence of multi-peak bound states with prescribed number of maximum points approaching to a local minimum point of the function G(x) as h → 0, where p+1 N 2 G(x) = [V (x)] p−1 − 2 [K(x)]− p−1 , is referred to as a ground energy function introduced in [37]. Recently, the case that V (x) has compact support, which is a more difficult case, was studied by Fei and Yin [21], and the least energy solution (which is a single-bump solution) was obtained. On the basis of this result, Ba, Deng, and Peng [5] attained multi-bump solutions with higher energy by variational methods and penalization techniques. For more references, we refer to [8–10, 33]. In this article, we try to find multi-bump solutions for the complex-valued problem (1.4) when V (x) has compact support. As we will see later, the presence of the magnetic vector potential A(x) 6≡ 0 leads us to find complex-valued solutions of (1.4) and also makes the problem more complicated. We assume that A(x), V (x), and K(x) satisfy the following conditions: (A) A : RN → RN is locally H¨ older continuous; N (H1 ) V (x) ∈ C(R ) is compactly supported, V (x) ≥ 0, and V (x) 6≡ 0; K(x) ∈ C(RN ) and K(x) ≥ 0; (H2 ) There exist smooth bounded domains Λj of RN , mutually disjoint such that V (x) > 0 ¯ j , j = 1, · · · , k, and and K(x) > 0 on Λ 0 < cj ≡ inf G(x) < inf G(x); x∈Λj

x∈∂Λj

(H3 ) There exist constants k0 > 0, α < 2 such that 0≤K(x)≤k0 (1 + |x|)α in RN . Our main result is

408

ACTA MATHEMATICA SCIENTIA

Vol.37 Ser.B

Theorem 1.1 Assume N ≥ 5 and max(1, (N + α)/(N − 2)) < p < (N + 2)/(N − 2). Under the assumptions (A), (H1 )–(H3 ), there exists an h0 > 0 such that for every 0 < h < h0 , problem (1.4) admits a solution uh ∈ H 1 (RN , C). Moreover, |uh | possesses exactly k local maxima xjh ∈ Λj (j = 1, · · · , k), which satisfy that G(xjh ) → inf G(x) as h → 0. x∈Λj

To obtain peaked solutions, we will use variational methods and penalization techniques. As R R the boundedness of RN V (x)|u|2 can not guarantee that u is a bound state and RN K(x)|u|p+1 may not make sense for u ∈ H 1 (RN , C) due to the fact that V (x) may vanish or K(x) may be unbounded at infinity, we will employ a penalization argument to modify the nonlinear term K(x)|u|p−1 u, which was firstly introduced in [21]. To obtain multi-peak bound states with the desired number of peaks, we must give an exact estimate to the functional energy corresponding to the solution uh of the modified problem; to this end, we add a functional Ph (u) to the functional related to the modified equation, which can help us to restrict the main part of uh in the desired domains. This article is organized as follows: In Section 2, we construct the penalization functionals and prove that the modified functional admits a nontrivial critical point. In Section 3, we will give the important estimate that implies that the critical point has concentration property. In Section 4, we prove that this nontrivial critical point is actually a multi-bump solution of the original problem (1.4). Throughout the whole article, Rew and Imw stand for the real and the imaginary part and w ¯ for the complex conjugate of the complex number w.

2

A Penalized Functional

h h Set Dh u = (D1h u, · · · , DN u), where Djh = ∂j − Aj (x). Let Eh (RN ) be the Hilbert space i defined as follows: Z n o 1,2 u ∈ DA (RN )| (|Dh u|2 + V (x)|u|2 ) < ∞ , RN

n

o 1,2 where DA (RN ) = u ∈ L (RN )|Dh u ∈ L2 (RN ) . For the simplicity, we denote Eh (RN ) by Eh . The scalar product of Eh is denoted by Z h i  h  h (u, v)h = Re ∇u − A(x)u ∇v − A(x)v + V (x)uv i i RN 2N N −2

and the associated norm Z  12 kukh = (|Dh u|2 + V (x)|u|2 ) RN Z Z = [h2 |∇u|2 + (|A(x)|2 + V (x))|u|2 ] − 2Re RN

ihA(x)u∇¯ u

RN

 12

.

It follows the diamagnetic inequality for Dh u(see [29] for example) that |Dh u(x)| ≥ h|∇|u|(x)|

for any u ∈ Eh .

Let Λ = ∪kj=1 Λj . For ξ ∈ Λ, consider the following equation,   −∆u(x) + V (ξ)u(x) = K(ξ)|u|p−1 u(x),  u ∈ H 1 (RN , C).

x ∈ RN ,

(2.1)

(2.2)

No.2

¨ N Ba & J Dai: SCHRODINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS

409

The functional corresponding to (2.2) is defined as Z Z Z 1 V (ξ) K(ξ) Iξ (u) = |∇u|2 + |u|2 − |u|p+1 . 2 RN 2 p + 1 N N R R The following function G(ξ) = inf Iξ (u) u∈Mξ

is referred to as ground energy function of (2.2)(see [37]), where Mξ is the Nehari manifold defined by Mξ = {u ∈ H 1 (RN , C) \ {0} : hIξ′ (u), ui = 0}. For further properties of G(ξ), one can see [37], although u(x) is complex-valued function in our case. By assumptions (H1 ) and (H2 ), we obtain Lemma 2.1 from the Sobolev inequality. Lemma 2.1 Assume that (H1 ), (H2 ) hold true, then for each h ∈ (0, 1], there exists a constant C1 > 0 independent of h such that Z N (p−1) K(x)|u|p+1 ≤ C1 h− 2 kukp+1 ∀ u∈Eh . (2.3) h , Λ

Now, we define n fh (x, |t|2 )t = min K(x)|t|p−1 t,

o h h3 t, , θ N 1 + |x| 0 1 + |x|

where θ0 > 2 will be suitably chosen later on. Define gh (x, |t|2 )t = χΛ (x)K(x)|t|p−1 t + (1 − χΛ (x))fh (x, |t|2 )t, Rt where χΛ (x) represents the characteristic function of Λ. Set Gh (x, t) = 0 gh (x, s)ds. We consider the modified functional Z  Z Z  1 1 1 Lh (u) = |Dh u|2 + V (x)|u|2 − K(x)|u|p+1 − Fh (x, |u|2 ), 2 RN p+1 Λ 2 RN \Λ Rt where Fh (x, t) = (1 − χΛ (x)) 0 fh (x, s)ds. For u ∈ Eh , due to θ0 > 2, we have Z   Fh (x, |u|2 ) + fh (x, |u|2 )|u|2 RN \Λ

≤C

Z

RN \Λ

≤ Ch3

Z

RN

Z  NN−2 2N h3 2 3 N −2 |u| ≤ Ch |u| 1 + |x|θ0 RN \Λ |∇|u||2 ≤ Chkuk2h.

Hence, Lh (u) is well defined in Eh . Assume that G(ξj ) = cj and ξj ∈ Λj . Let σj > 0 satisfy sup G(x) ≤ cj + σj , Λj

and assume that

k X j=1

σj <

1 min{cj |j = 1, · · · , k}. 2

(2.4)

410

ACTA MATHEMATICA SCIENTIA

Vol.37 Ser.B

˜j This can be verified by making Λj smaller if necessary. For mutually disjoint open sets Λ ˜ compactly containing Λj and satisfying V (x) > 0 on the closure of Λj , we define on Eh the functional Z Z Z 1 1 1 Ljh (u) = (|Dh u|2 + V (x)|u|2 ) − K(x)|u|p+1 − Fh (x, |u|2 ). 2 Λ˜ j p + 1 Λj 2 Λ˜ j \Λj Let M be a positive constant that will be determined later, and define Ph (u) = M

k X

1

1

N

{(Ljh (u)+ ) 2 − h 2 (cj + σj ) 2 }2+ .

(2.5)

j=1

Finally, we define the penalized functional Jh : Eh → R, Jh (u) = Lh (u) + Ph (u). Then, Jh (u) is of class C 1 . Moreover, the following compactness result holds. Lemma 2.2 Let {un } be a sequence in Eh such that Jh (un ) is bounded and Jh′ (un ) → 0. Then, {un } has a convergent subsequence for any fixed small h. Proof Firstly, we prove that {un } is bounded in Eh . By (2.4), for a fixed q ∈ (2, p + 1) and fixed small h, 1 ′ Lh (un ) − hLh (un ), un i q Z 1  1 1 Z 1  − (|Dh un |2 + V (x)|un |2 ) + − K(x)|un |p+1 = 2 q RN q p+1 Λ Z Z 1 1 + fh (x, |un |2 )|un |2 − Fh (x, |un |2 ) q RN \Λ 2 RN \Λ Z ≥C (|Dh un |2 + V (x)|un |2 ). RN

Similarly, we have ′ 1 Ljh (un ) − hLjh (un ), un i ≥ C q

Z

˜j Λ

(|Dh un |2 + V (x)|un |2 ),

and

k

X 1 1 N 1 1 Ph (un ) − hPh′ (un ), un i ≥ −M 2 h 2 (cj + σj ) 2 Ph2 (un ). q j=1 Z So, by the fact that Ph (un ) ≤ M (|Dh un |2 + V (x)|un |2 ), we get RN

1 Ch ≥ Jh (un ) − hJh′ (un ), un i q Z Z N ≥C (|Dh un |2 + V (x)|un |2 ) − Ch 2 RN

RN

 12 (|Dh un |2 + V (x)|un |2 ) ,

(2.6)

which implies that {un } is bounded in Eh . As Eh ֒→ H 1 (RN , C), the boundedness of {un } in Eh implies that after passing to a subsequence if necessary, there exists u ∈ Eh such that un → u weakly in Eh , un → u strongly in Lqloc (RN ), for 2 ≤ q < 2N/(N − 2).

¨ N Ba & J Dai: SCHRODINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS

No.2

411

Next, we will prove that kun kh → kukh as n → ∞. From hJh′ (un ), ui → 0, we have Z Z Z Re (|Dh u|2 + V (x)un u¯) − Re K(x)|un |p−1 un u¯ − Re fh (x, |un |2 )un u¯ RN

+M

RN \Λ

Λ

k X

1

−1

1

N

{Ljh (un )+2 − h 2 (cj + σj ) 2 }+ Ljh (un )+ 2

j=1



× Re

Z

˜j Λ

(|Dh u|2 + V (x)un u ¯) − Re

Z

K(x)|un |p−1 un u ¯ − Re

Λj

Z

˜ j \Λj Λ

 fh (x, |un |2 )un u ¯

= on (1).

(2.7)

Alternatively, by hJh′ (un ), un i = o(kun kh ) and the boundedness of un , we see Z Z Z h 2 2 p+1 (|D un | + V (x)|un | ) − K(x)|un | − fh (x, |un |2 )|un |2 RN

RN \Λ

Λ

k X

+M

1

−1

1

N

{Ljh (un )+2 − h 2 (cj + σj ) 2 }+ Ljh (un )+ 2

j=1

×

Z

˜j Λ

(|Dh un |2 + V (x)|un |2 ) −

Z

K(x)|un |p+1 −

Λj

Z

˜ j \Λj Λ

fh (x, |un |2 )|un |2

= on (1).

(2.8)

In addition, we find Z Z lim V (x)(un u ¯ − |un |2 ) = 0 = lim K(x)(|un |p−1 un u¯ − |un |p+1 ), n→∞ RN n→∞ Λ Z Z lim V (x)(un u ¯ − |un |2 ) = 0 = lim K(x)(|un |p−1 un u ¯ − |un |p+1 ), n→∞



n→∞

˜j Λ

lim

n→∞

Z

˜ j \Λj Λ

fh (x, |un |2 )(un u¯ − |un |2 ) = 0,

n→∞

BR (0)\Λ

(2.10)

Λj

and for any fixed large R > 0 satisfying Λ ⊂ BR (0), Z Z lim fh (x, |un |2 )un u¯ = lim n→∞

(2.9)

(2.11)

fh (x, |un |2 )|un |2 .

BR (0)\Λ

We will prove that for any δ > 0, there exists R > 0 such that for all n, Z Z fh (x, |un |2 )un u ¯ < δ, fh (x, |un |2 )|un |2 < δ. RN \BR (0)

RN \BR (0)

We only need prove the first estimate because the second one can be checked similarly. Arguing as (2.4), we see Z Z |un ||u| 2 3 fh (x, |un | )un u ¯ ≤ h 1 + |x|θ0 N N R \BR (0) R \BR (0) h ≤ θ0 −2 kun kh kukh → 0 as R → ∞. R Hence, Z Z fh (x, |un |2 )un u¯ = lim

lim

n→∞

n→∞

RN \Λ

fh (x, |un |2 )|un |2 .

(2.12)

RN \Λ

By the boundedness of {un }, we have 1

N

1

−1

M {Ljh(un )+2 − h 2 (cj + σj ) 2 }+ Ljh (un )+ 2 = βj + on (1),

(2.13)

412

ACTA MATHEMATICA SCIENTIA

Vol.37 Ser.B

where βj ≥ 0, j = 1, · · · , k, are constants. From (2.7) to (2.13), it follows that on (1) =

Z

h

2

h

2

(|D un | − |D u| ) + RN

R

N n→∞ R

n→∞

(βj + on (1))

j=1

which implies that lim lim

k X

Z

|Dh un |2 =

R

RN

Z

˜j Λ

(|Dh un |2 − |Dh u|2 ),

|Dh u|2 and hence

(|Dh un |2 + V (x)|un |2 ) =

RN

Z

(|Dh u|2 + V (x)|u|2 ).

RN

 Now, we can use the critical point theory to find critical points of Jh . We consider the class Γ of continuous functions γ : [0, 1]k → Eh such that there are continuous functions gj : [0, 1] → Eh for j = 1, · · · , k satisfying (i) gj (0) = 0, Lh (gj (1)) < 0; k P gj (τj ) for all τ = (τ1 , · · · , τk ) ∈ ∂[0, 1]k ; (ii) γ(τ1 , · · · , τk ) = j=1

(iii) supp{gj (t)} ⊂ Λj for all t ∈ [0, 1]; k P (iv) Jh (γ(τ )) ≤ hN ( cj −σ0 ) for all τ ∈ ∂[0, 1]k , where 0 < σ0 < j=1

1 2

min{cj |j = 1, · · · , k}.

The min-max value associated with the class Γ is given by Ch = inf

sup Jh (γ(τ )).

γ∈Γ τ ∈[0,1]k

From the proof of Lemma 3.2 in [6], we see that Γ is non-empty. Let Γj be the class of all ˜ j ) such that γj (0) = 0, Lj (γj (1)) < 0 and define continuous path γj : [0, 1] → Eh (Λ h djh = inf

sup Ljh (γj (t)).

γj ∈Γj t∈[0,1]

We have the following estimate for djh and Ch . Lemma 2.3 djh

N

= h (cj + o(1)),

Ch = h

N

k X j=1

Proof

 cj + o(1)

as h → 0.

The proof is similar to Lemmas 3.2 and 3.3 in [6] and thus we omit it.



The functional Jh satisfies the Palais-Smale condition; by Lemma 2.3, there exists a critical ′ point uh ∈ Eh of Jh such that Jh (uh ) = Ch and Jh (uh ) = 0. N

Lemma 2.4 There exists C > 0 such that for h > 0 sufficiently small, kuh kh ≤ Ch 2 . Proof

The proof is similar to that of (2.6).



We define the local weights 1

1

N

1

ρjh = M {(Ljh (uh )+ ) 2 − h 2 (cj + σj ) 2 }+ (Ljh (uh )+ )− 2 , then the functions ρh =

k X j=1

ρjh χΛ˜ j .

¨ N Ba & J Dai: SCHRODINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS

No.2

413

The critical point uh satisfies in the weak sense i 1  h  1 −div (1 + ρh ) h2 ∇uh + Ahuh −(1 + ρh ) hA∇uh − |A|2 uh +(1 + ρh )V uh i i −(1 + ρh )gh (x, |uh |2 )uh = 0 and Re

Z

h

(1 + ρh )(D uh

Dh ϕ

+ V uh ϕ) = Re

RN

Z

(1 + ρh )gh (x, |uh |2 )uh ϕ,

∀ϕ ∈ Eh .

RN

˜ h = {y ∈ RN |hy ∈ Λ ˜ j }, we rescale the function uh as Setting Λhj = {y ∈ RN |hy ∈ Λj }, Λ j vh (y) = uh (hy) for y ∈ RN , then it follows that h  i 1  1 −div (1 + ρh (hy)) ∇vh + A(hy)vh −(1 + ρh (hy)) A(hy)∇vh − |A(hy)|2 vh i i +(1 + ρh (hy))V (hy)vh − (1 + ρh (hy))gh (hy, |vh |2 )vh = 0. (2.14) Also from Lemma 2.4, we have Z  2  1 ∇vh − A(hy)vh +V (hy)|vh |2 ≤ C. i RN

3

(2.15)

An Essential Estimate

In this section, we will prove that ρh ≡ 0 and uh concentrates at some points. To this end, we first give an essential estimate. Lemma 3.1 Let M be as in (2.5). There exists C > 0 such that if M > C, lim Ljhn (uhn )h−N ≤ cj + σj n

n→+∞

for j = 1, · · · , k.

To prove Lemma 3.1, we need some preliminary lemmas. Given R > 0, define NR (Ω) to be the set {y| dist(y, Ω) < R}, for any Ω ⊂ RN . Lemma 3.2 There exists C > 0 such that for any given R > 0, Z 2  C  1 ∇vh − A(hy)vh +V (hy)|vh |2 ≤ i R RN \NR (Λh )

(3.1)

for all h sufficiently small. Proof

h Given R > 0, h > 0, define smooth cut-off functions 0 ≤ ψj,R ≤ 1 satisfying    1, if dist(y, Λhj ) < R/2, h ψj,R (y) = (3.2)   0, if dist(y, Λh ) > R, j

h and |∇ψj,R | ≤ C/R. Set ηR = 1 −

Z

RN \Λh

≤h

3

Z

≤ Ch

j=1

h ψj,R . Similar to (2.4), we verify

2 ηR fh (hy, |vh |2 )|vh |2

RN \Λh

Z

k P

RN \Λh

2 ηR |vh |2 ≤ Ch 1 + |hy|θ0 2 ηR |∇|vh ||2

Z

+ Ch

|∇(ηR |vh |)|2

RN \Λh

Z

NR (Λh )\Λh

|∇ηR |2 |vh |2

414

ACTA MATHEMATICA SCIENTIA

≤ Ch

Vol.37 Ser.B

Z 2 2 1 |∇ηR |2 |vh |2 . ηR ∇vh − A(hy)vh +Ch i NR (Λh )\Λh RN \Λh

Z

2 v in (2.14), one gets Hence, using the test function ηR h Z 2   1 C ∇vh − A(hy)vh +V (hy)|vh |2 i RN \NR (Λh ) Z 2  1  ≤ ∇vh − A(hy)vh +V (hy)|vh |2 i RN \NR (Λh ) Z Z 2 2 1 −Ch ηR |∇ηR |2 |vh |2 ∇vh − A(hy)vh −Ch i RN \Λh NR (Λh )\Λh Z 2   2 1 ≤ (1 + ρh (hy))ηR ∇vh − A(hy)vh +V (hy)|vh |2 − fh (hy, |vh |2 )|vh |2 i RN \Λh Z = −2 (1 + ρh (hy))(∇vh − iA(hy)vh )v h ηR ∇ηR NR (Λh )\Λh Z 1 C C ≤ ∇vh − A(hy)vh |v h | ≤ , R NR (Λh )\Λh i R

because ρh is uniformly bounded by a constant possibly depending on M .



Lemma 3.3 Assume that A0 = (a1 , · · · , aN ) ∈ RN is a constant vector, and v ∈ H 1 (RN , C) ∩ C(RN , C) satisfies the equation 1 2 ∇ − A0 v + v = χ{x1 <0} |v|p−1 v, x ∈ RN . (3.3) i Then, v ≡ 0. We first show that |v| = 0 on {x1 = 0}. Obviously, v also satisfies 1 2 ∇ − A0 v + v¯ = χ{x1 <0} |v|p−1 v¯. (3.4) i Multiplying (3.3) by ∂¯ v /∂x1 and (3.4) by ∂v/∂x1 and integrating over RN , we have Z Z +∞ Z 2  ∂  1 2 ′ 2 dx |v(0, x′ )|p+1 dx′ = 0. ∇v − A0 v +|v| dx1 − i p + 1 RN −1 RN −1 −∞ ∂x1 Proof

The first summand above is zero, so we obtain |v(0, x′ )| ≡ 0. Next, we prove |v(x1 , x′ )| ≡ 0 for x1 > 0. From (3.3) and Kato’s inequality(Theorem X.33 in [32])  v¯  ∇ 2  ∆|v| ≥ −Re − A0 v , |v| i it follows that −∆|v| + |v| ≤ χ{x1 <0} |v|p , where we use |v|χ{x1 >0} ∈ H 1 (RN ) as a test function and obtain |v|χ{x1 >0} ≡ 0. This contradicts the strong maximum principle if |v| 6≡ 0.  In Proof of Lemma 3.1, for the fixed A0 ∈ RN , we will use the fact that the norm Z  2  1 kvk2 = ∇v − A0 v +|v|2 i RN

is equivalent to the usual norm on H 1 (RN , C). One can also check [4] for the equivalence of the norm Z  2  1 kvk2 = ∇v − A(hy)v +V (hy)|v|2 i Ω

No.2

¨ N Ba & J Dai: SCHRODINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS

415

and the usual norm on H 1 (Ω, C) for any compact set Ω in RN . Proof of Lemma 3.1 The main idea is from [6, 19]. Suppose that there exists certain j ∈ {1, · · · , k} such that for a sequence hn → 0, the critical point uhn of Jhn satisfies lim sup Ljhn (uhn )h−N > cj + σj . n n→+∞

We first claim that there exist n0 ∈ Z+ , S > 0, and ρ > 0, such that Z |vhn |2 ≥ ρ, ∀ n ≥ n0 . sup n y∈Λh j

(3.5)

BS (y)

In fact, because lim Ljhn (uhn )h−N > cj + σj , there exist n0 ∈ Z+ and λ > 0 such that n n→+∞

2   1 ∇vhn − A(hn y)vhn +V (hn y)|vhn |2 ≥ 2λ, i ˜ hn Λ j

Z

∀ n ≥ n0 .

Lemma 3.2 implies that for any R > 0 large enough, Z 2   1 ∇vhn − A(hn y)vhn +V (hn y)|vhn |2 ≥ λ. n i NR (Λh j )

(3.6)

Now, assume that (3.5) is false. Arguing as in [6], we can prove that Z |vhn |p+1 → 0 as n → ∞. n NR (Λh j )

hn h Set vnR = ψj,2R vhn , where ψj,R is defined by (3.2). Using vnR as a test function in (2.14), we find Z 2  1  hn ∇vhn − A(hn y)vhn +V (hn y)|vhn |2 ψj,2R hn i ˜ Λ Z j Z   1 hn 2 2 hn = ghn (hn y, |vhn | )|vhn | ψj,2R − Re ∇vhn + A(hn y)vhn v hn ∇ψj,2R i ˜ hn ˜ hn Λ Λ j j Z Z |vhn |2 ≤ K(hn y)|vhn |p+1 + h3n hn 1 + |h y|θ0 n n n Λh N2R (Λh j j )\Λj Z C 1 + ∇vhn − A(hn y)vhn |vhn | R N2R (Λhj n )\NR (Λhj n ) i Z 2 1  Z  1 2 ≤C +V (h y)|v | + h3n ∇v − A(h y)v +C |vhn |p+1 , n hn hn n hn hn R i N R Λj

which contradicts (3.6) if we choose R and n sufficiently large. Thus, (3.5) holds true. Suppose that the sequence yn ∈ Λhj n satisfies Z |vhn |2 ≥ ρ > 0, for n ≥ n0 .

(3.7)

BS (yn )

n Now, we set xn = hn yn , vn (y) = vhn (yn + y) = uhn (xn + hn y), and Λhj,y = {y ∈ RN |y + yn ∈ n hn hn Λj }. We have xn ∈ Λj and for y ∈ Λj,yn , vn satisfies 2 1 ∇ − A(xn + hn y) vn + V (xn + hn y)vn − ghn (xn + hn y, |vn |2 )vn = 0. (3.8) i Now, using Lemma 3.3 and proceeding as done in [6], we can prove that xn → ξ ∈ Λj and {vn } converges along a subsequence to v ∈ C 2 (Ω) for each compact subset Ω ⊂ RN . Moreover, set

416

ACTA MATHEMATICA SCIENTIA

Vol.37 Ser.B

vˇ(x) = e−iA(ξ)x v(x), then vˇ(x) 6≡ 0 and vˇ(x) ∈ H 1 (RN , C) solves −∆ˇ v + V (ξ)ˇ v − K(ξ)|ˇ v |p−1 vˇ = 0.

(3.9)

Now, we distinguish two cases: (i) vˇ is a least energy solution of (3.9); (ii) vˇ is not a least energy solution of (3.9). In Case (i), we suppose cj ≤ Iξ (ˇ v ) ≤ cj + σj . As vn converges strongly in the H 1 -sense over any compact set. Passing to a further subsequence if necessary, we may find a sequence of positive numbers Rn → ∞ such that Z 2  i 1 h 1 lim ∇vn − A(hn y)vn +V (hn y)|vn |2 −Ghn (hn y, |vn |2 ) n→∞ B 2 i Rn (yn ) Z h i 1 1 v ) ≤ cj + σj . (3.10) (|∇ˇ v |2 + V (ξ)|ˇ v |2 ) − K(ξ)|ˇ v |p+1 = Iξ (ˇ = p+1 RN 2 However, limn→+∞ Ljhn (uhn )h−N > cj + σj implies the existence of η > 0 such that for all large n n, Z 2  1  ∇vhn − A(hn y)vhn +V (hn y)|vhn |2 ≥ η > 0. i ˜ hn \BR (yn ) Λ n j So, similar to (3.5), there exist S > 0 and a sequence y˜n ∈ Λhj n \ BRn (yn ) such that Z |vhn |2 ≥ ρ > 0. BS (˜ yn )

2 Thus, up to a subsequence vhn (· + y˜n ) converges in Cloc (RN ) to a nonzero vˆ. Moreover, vˆ solves 1 2 ˜ vˆ + V (ξ)ˆ ˜ v − K(ξ)|ˆ ˜ v |p−1 vˆ = 0 ∇ − A(ξ) i ˜ ¯ j , and v˜(x) = e−iA(ξ)x with ξ˜ ∈ Λ vˆ(x) ∈ H 1 (RN , C) satisfies

˜ v − K(ξ)|˜ ˜ v |p−1 v˜ = 0. −∆˜ v + V (ξ)˜ Hence, Iξ˜(˜ v ) ≥ cj . Next, we claim that lim Ljhn (uhn )h−N ≥ Iξ (ˇ v ) + Iξ˜(˜ v ) ≥ 2cj . n

(3.11)

n→∞

To prove (3.11), we first check that max |vhn (x)| → 0

˜ hn x∈∂ Λ j

as n → ∞.

(3.12)

˜ j, Suppose on the contrary that there exist subsequences, still denoted by {hn } and {¯ yn } ⊂ ∂ Λ ˜ j as n → ∞ and |uhn (¯ such that hn → 0, y¯n → y0 ∈ ∂ Λ yn )| ≥ δ > 0. Choose R0 > 0 such that N k BR0 (y0 ) ⊂ R \ (∪j=1 Λj ). We may assume {¯ yn } ⊂ BR0 (y0 ). Using the above scaling technique on BR0 (y0 ), they are obtained that wn (x) := uhn (¯ yn + hn x) converges in C 2 on any compact 1 N set to some function w ∈ H (R , C), and |w| ≥ δ satisfies 2 −∆w − A(y0 )∇w + |A(y0 )|2 w + V (y0 )w = 0 i

in RN ,

¨ N Ba & J Dai: SCHRODINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS

No.2

which means

417

1 2 ∇w − A(y0 )w +V (y0 )|w|2 = 0. RN i

Z

Hence, w ≡ 0. This is impossible, so (3.12) holds true. ˜ hn the equation Recall that vhn satisfies on Λ j 1 2 ∇ − A(hn y) vhn + V (hn y)vhn − ghn (hn y, |vhn |2 )vhn = 0. i Define φ = vhn [ψ(|y − yn |/R) + ψ(|y − y˜n |/R)],

(3.13)

where ψ is a C ∞ function with ψ(t) = 0 for t ≤ 1 and ψ(t) = 1 for t ≥ 2. Now, we use φ¯ in (3.13) as a test function. Denote NR (yn , y˜n ) = B(yn , R) ∪ B(˜ yn , R), and by (3.12), we have Z 2   1 ∇vhn − A(hn y)vhn +V (hn y)|vhn |2 i ˜ hn \NR (yn ,˜ Λ yn ) Z j Z 2 2 ≥ ghn (hn y, |vhn | )|vhn | − ghn (hn y, |vhn |2 )|vhn |2 ˜ hn \NR (yn ,˜ Λ yn ) j

≥

C + + ohn (1) ZR

˜ hn \NR (yn ,˜ Λ yn ) j

N2R (yn ,˜ yn )\NR (yn ,˜ yn )

Ghn (hn y, |vhn |2 ) + O

1 +C(R) + ohn (1), R

where C(R) → 0 as R → +∞, and we have used Z Z Z 1 2 2 |v | ≤ C V (h y)|v | ≤ C V (hn y)|vhn |2 . hn n hn hn hn 1 + |h y|θ0 hn hn N ˜ ˜ n Λj \Λj Λj \Λj R

(3.14)

Therefore, we obtain 2Ljhn (uhn )h−N n Z 2 h 1  i ≥ ∇vhn − A(hn y)vhn +V (hn y)|vhn |2 −Ghn (hn y, |vhn |2 ) i NR (yn ,˜ yn ) 1 +O +C(R) + ohn (1). R Consequently, Z 2 h 1  1  i 1 j −N lim Lhn (uhn )hn ≥ K(ξ)|v|p+1 ∇v − A(ξ)v +V (ξ)|v|2 − n→∞ i p+1 BR (0) 2 Z 2  i h 1  1 ˜ v +V (ξ)|ˆ ˜ v |2 − 1 K(ξ)|ˆ ˜ v |p+1 + v − A(ξ)ˆ ∇ˆ i p+1 BR (0) 2 1 +O +C(R). R Choosing R large, we get (3.11). Now, we claim that lim inf Lhn (uhn )h−N ≥ 0. (3.15) n n→∞

Indeed, define 0 ≤ ψR ≤ 1 as a smooth cut-off function such that    1, if y ∈ NR (Λhn ), ψR (y) =   0, if y ∈ RN \ N2R (Λhn ),

418

ACTA MATHEMATICA SCIENTIA

Vol.37 Ser.B

and |∇ψR | ≤ C/R. Using in (3.13) the test function vhn ψR , by (3.1) and (3.14), we get Z 2  1  C ∇vhn − A(hn y)vhn +V (hn y)|vhn |2 + i R N (Λhn ) Z R = ghn (hn y, |vhn |2 )|vhn |2 ψR N2R (Λhn ) Z Z ≥ K(hn y)|vhn |p+1 − h3n C V (hn y)|vhn |2 . Λhn

RN

Similarly, using (2.4) and (3.1), we see Z 2  1  1 Lhn (vhn ) = ∇vhn − A(hn y)vhn +V (hn y)|vhn |2 2 NR (Λhn ) i Z Z 1 1 C p+1 K(hn y)|vhn | Fhn (hn y, |vhn |2 ) + − − p + 1 Λhn 2 RN \Λhn R Z 2  1  1 ≥ ∇vhn − A(hn y)vhn +V (hn y)|vhn |2 2 NR (Λhn ) i Z C 1 K(hn y)|vhn |p+1 − hn C + . − p + 1 Λhn R Combing the above two inequalities, for R large and hn small, it follows that Z 2  1 1  C 1  Lhn (vhn ) ≥ − ∇vhn − A(hn y)vhn +V (hn y)|vhn |2 + − Chn > 0, 2 p + 1 NR (Λhn ) i R which concludes the claim. Lemma 2.3, (3.11), and (3.15) give that 2

k X

cj ≥ lim inf Jhn (uhn )h−N ≥ M {(2cj )1/2 − (cj + σj )1/2 }2 . n

j=1

n→∞

(3.16)

Therefore, if we choose 2 M>

k P

cj

j=1

((2cj )1/2 − (cj + σj )1/2 )2

,

(3.16) is impossible. In Case (ii), as the least energy of problem (3.9) is isolated (see Proposition 5.1 in [6]), there exists αj (without loss of generality, we suppose σj < αj ) such that Iξ (ˇ v ) > cj + αj . As we prove (3.16), we can verify 2

k X

cj ≥ lim inf Jhn (uhn )h−N ≥ M {(cj + αj )1/2 − (cj + σj )1/2 }2 . n

j=1

n→∞

(3.17)

Hence, if we choose 2

M>

((cj +

(3.17) is impossible. As a result, the proof is completed.

k P

cj j=1 αj )1/2 − (cj

+ σj )1/2 )2

,



Lemma 3.1 implies that ρh ≡ 0 if M is chosen large enough. In the following, we fix M so large that Lemma 3.1 holds true.

No.2

¨ N Ba & J Dai: SCHRODINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS

419

Lemma 3.4 lim Ljh (uh )h−N = cj

h→0

Proof

for all j = 1, · · · , k.

The proof is similar to Lemma 2.4 in [19].



Lemma 3.4 implies that the concentration of uh must occur around some x ¯j ∈ Λj with G(¯ xj ) = cj . The concentration implies the presence of at least one local maximum xjh in each Λj . Next, we will show that |uh | has exactly k local maxima in RN . Lemma 3.5 |uh | has a unique local maximum xjh in Λj for every j ∈ {1, · · · , k}. Proof We argue by contradiction. Assume that there exists a sequence hn → 0, such that uhn possesses two local maxima x1,n , x2,n ∈ Λj . Then, |uhn (xi,n )| ≥ δ > 0 (i = 1, 2). But, the fact that lim G(xi,n ) = cj < inf G(x) implies that these sequences stay away from the n→∞

∂Λj

boundary of Λj , so, if we let vn = uhn (x1,n + hn x), then after passing to a subsequence vn 1 converges in the C 2 sense over compact sets to a solution v ∈ H 1 (RN , C), and vˆ = e−iA(ξ )x v is a least energy solution of −∆ˆ v + V (ξ 1 )ˆ v − K(ξ 1 )|ˆ v |p−1 vˆ = 0, where lim x1,n = ξ 1 . It is known from [27] that the function |v| has a local maximum at the n→∞

origin and is radially symmetric and radially decreasing, which implies that x ˜n = h−1 n (x2,n − x1,n ) satisfies |˜ xn | → +∞. Now, repeating the process in the proof of Lemma 3.1, we have lim Ljhn (uhn )h−N ≥ 2cj , n

n→∞

which is obviously a contradiction to Lemma 3.4. So, in Λj , the maxima xjh of |uh | are unique and hence uh has exactly k peaks.  The above lemmas indeed give the following proposition. Proposition 3.6 The sequences {xjh } ⊂ Λj , j = 1, · · · , k, satisfy that, for any ν > 0, there exist h1 (ν), ρ1 (ν) > 0 such that, for h < h1 (ν), Z 2   h (3.18) h−N ∇uh − A(x)uh +V (x)|uh |2 < ν, i RN \∪k (xjh ) j=1 B hρ1 (ν)

and

dist(xjh , M j ) < ν,

(3.19)

j

where M = {ξj ∈ Λj |G(ξj ) = cj }, j = 1, · · · , k.

4

Proof of Theorem 1.1 In this section, we will show that uh is indeed a solution of the original problem (1.4). Set d1 = min{dist(∂Λj , M j ), j = 1, · · · , k} > 0 and V1 = min V (x)/2 > 0. Fix two x∈Λ

positive numbers K1 > max{128, 2d1} and c > 0 such that c2 ≥ (128K12 )/(d21 V1 ). Let ν1 = 2 min{d1 /K1 , (8C1 )− p−1 } and h2 = min{h1 (ν1 ), d1 /(K1 ρ1 (ν1 )), (ln 2)/c}, where C1 is defined in (2.3), and h1 (ν1 ) and ρ1 (ν1 ) are given in (3.18) and (3.19). Assume h < h2 and ν < ν1 . Then, d1 d1 dist(xjh , ∂Λj ) > , j = 1, · · · , k and hρ1 (ν1 ) < . (4.1) 2 K1

420

ACTA MATHEMATICA SCIENTIA

Vol.37 Ser.B

Define Ωn,h = RN \ ∪kj=1 BRn,h (xjh ) with Rn,h = echn and let n ˜>n ˆ be integers satisfying Rnˆ −1,h <

d1 ≤ Rnˆ ,h , K1

Rn˜ +2,h ≤

d1 < Rn˜ +3,h . 2

It follows from (4.1) that Rn,h ≥ Rnˆ ,h ≥ d1 /K1 > hρ1 (ν1 ) for n ≥ n ˆ , so Ωn,h ∩ (∪kj=1 Bhρ1 (ν1 ) (xjh )) = ∅.

(4.2)

Choose 0 ≤ χn,h ≤ 1 as a smooth cut-off function such that χn,h (x) = 0 in ∪kj=1 BRn,h (xjh ), χn,h (x) = 1 in Ωn+1,h , and |∇χn,h | ≤ 2/(Rn+1,h − Rn,h ). Lemma 4.1 Assume that (H1 ) and (H2 ) hold true. For small h < h2 , one has Z ln2 |∇|χn˜ ,h uh ||2 ≤ ChN −2 2− ch . RN

Firstly, we claim that for n ˆ≤n≤n ˜, Z Z 2   1 h An,h ≤ ∇uh − A(x)uh +V (x)|uh |2 , 2 Ωn,h i RN 2 where An,h (x) = hi ∇(χn,h uh ) − A(x)(χn,h uh ) +V (x)|χn,h uh |2 . Proof

In fact, from hL′h (uh ), χ2n,h uh i = 0, we get Z Z Z An,h = h2 |∇χn,h |2 |uh |2 + RN

Ωn,h

+

Λ∩Ωn,h

Z

(4.3)

χ2n,h K(x)|uh |p+1

2

(RN \Λ)∩Ωn,h

fh (x, |uh | )χ2n,h |uh |2

:= I1 + I2 + I3 . By the fact that h2 |∇χn,h |2 ≤

16 4h2 ≤ 2 2 |Rn+1,h − Rn,h |2 c Rn+1,h

and that, for n ˆ≤n≤n ˜ and x ∈ (∪kj=1 BRn+1,h (xjh )) \ (∪kj=1 BRn,h (xjh )), 128 ≤ 2 c2 Rn+1,h

128 128K12 d21 V1

d21 K12

·

it follows that for x ∈ RN , h2 |∇χn,h |2 ≤ So, I1 ≤

1 8

= V1 ≤ V (x),

1 V (x). 8

2  h  ∇uh − A(x)uh +V (x)|uh |2 . i Ωn,h

Z

Now, we estimate I2 . Similar to (2.3), we get Z Z N (p−1) K(x)|uh |p+1 ≤ C1 h− 2

Λ∩Ωn,h

Λ∩Ωn,h

(4.4)

 p+1 2 (|Dh uh |2 + V (x)|uh |2 ) .

Moreover, from (4.2), for n ≥ n ˆ , we see Λ ∩ Ωn,h ⊂ RN \ (∪kj=1 Bhρ1 (ν1 ) (xjh )). Thus, from (3.18), we obtain Z  p−1 N (p−1) 2 I2 ≤ C1 h− 2 (|Dh uh |2 + V (x)|uh |2 ) RN \(∪k j=1 Bhρ

1 (ν1 )

(xjh ))

No.2

¨ N Ba & J Dai: SCHRODINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS

×

Z

421

(|Dh uh |2 + V (x)|uh |2 )

Λ∩Ωn,h

≤ C1 ν1 2

Z

1 ≤ 8

(|Dh uh |2 + V (x)|uh |2 ).

p−1

Z

(|Dh uh |2 + V (x)|uh |2 )

Ωn,h

(4.5)

Ωn,h

Finally, similar to (2.4), it follows that Z Z 1 h3 2 (|Dh uh |2 + V (x)|uh |2 ). I3 ≤ |uh | ≤ θ0 8 Ωn,h Ωn,h 1 + |x|

(4.6)

Therefore, (4.3) holds from (4.4)–(4.6) and it implies that Z Z 1 An,h ≤ (|Dh uh |2 + V (x)|uh |2 ) 2 N R Ω Z n,h Z 1 1 h 2 2 ≤ (|D (χn−1,h uh )| + V (x)|χn−1,h uh | ) = An−1,h . 2 RN 2 RN Iterating the above process, we have Z  1 n˜ −ˆn Z  1 n˜ −ˆn+1 Z An˜ ,h ≤ Anˆ ,h ≤ (|Dh uh |2 + V (x)|uh |2 ) 2 2 RN RN Ωn ˆ ,h  1 n˜ −ˆn+1 Z ≤ (|Dh uh |2 + V (x)|uh |2 ) j 2 RN \(∪k B (x )) hρ (ν ) j=1 h 1 1  n˜ −ˆn+1 ln2 N −(˜ n−ˆ n+1)ln2 N 1 = Ch e ≤ ChN 2− ch , ≤ Ch 2 where we use the fact that K1 ech(˜n−ˆn+1) = ech˜n e−ch(ˆn−1) = ech˜n Rn−1 ≥ 2. ˆ −1,h ≥ d1 Alternatively, by the diamagnetic inequality (2.1), we see Z Z Z h2 |∇|χn˜ ,h uh ||2 ≤ |Dh (χn˜ ,h uh )|2 ≤ RN

RN

An˜ ,h .

RN

Thus, we obtain Z

ln2

|∇|χn˜ ,h uh ||2 ≤ ChN −2 2− ch .

RN

The proof is completed.



Lemma 4.2 Under the assumption of Lemma 4.1, for any x ∈ RN \ (∪kj=1 B d1 (xjh )), 2

|uh (x)| ≤ C2 Proof

ln2 − 2ch

.

Set ch (x) = χh (x)K(hx)|vh |p−1 + (1 − χh (x))

h3 , 1 + |hx|θ0

here χh is a characteristic function of Λh = {h−1 x|x ∈ Λ}. Exploiting Kato’s inequality (see [32], Theorem X.33)  v¯  ∇ 2  h ∆|vh | ≥ −Re − A(hx) vh , |vh | i 1 we obtain |vh | = |uh (hx)| ∈ Hloc (RN ), which is a nonnegative weak subsolution of ∆|v| + ch (x)|v| = 0.

422

ACTA MATHEMATICA SCIENTIA

Vol.37 Ser.B

2N Fixing s ∈ ( N2 , (p−1)(N −2) ), as θ0 > 2, we see

h3 kLs 1 + |hx|θ0 2  p−1 Z 1 2 ≤C ∇vh − A(hx)vh +V (hx)|vh |2 i h Λ Z  1s N 1 +Ch3− s ≤ C, θ0 s RN \Λ (1 + |y| )

kch (x)kLs ≤ kχh (x)K(hx)vhp−1 kLs + k(1 − χh (x))

which implies that kch (x)kLs is uniformly bounded with respect to h. By Theorem 8.17 in [23], there is a constant C depending only on d1 , the dimension N , and the Ls bound of ch (x) such that −2 Z  N2N 2N |vh (z)| ≤ C |vh | N −2 , ∀z ∈ RN . (4.7) B cd1 (z) 4

It is noted that for h small, ∀ x ∈ RN \ (∪kj=1 B d1 (xjh )).

B hcd1 (x) ⊂ Ωn˜ +1,h , 4

Thus, for x ∈ RN \ (∪kj=1 B d1 2

|uh (x)| = |vh (h

−1

2

(xjh )),

x)| ≤ C

Z

2N

|vh | N −2

B cd1 (h−1 x)



=C h

−N

≤ Ch−

Z

4

|uh |

2N N −2

B hcd1 (x) 4

N −2 2

Z

−2  N2N

|∇|χn˜ ,h uh ||2

RN

 12

−2  N2N

≤ Ch

− N 2−2

Z

2N

|χn˜ ,h uh | N −2

RN

−2  N2N

ln2

≤ C2− 2ch . 

Now, we complete the proof of Theorem 1.1. Proof of Theorem 1.1 By the fact that (N + α)/(N − 2) < p < (N + 2)/(N − 2), we can choose σ less than but close to N − 2 satisfying

Set U (x) =

2 < θ0 < (p − 1)σ − α,

σp − α > N.

k X

RN \ (∪kj=1 B d1 (xjh )),

j=1

1 |x −

in

xjh |σ

(4.8)

2

and Z(x) = U (x) − h2 |uh (x)|. Then, we can check that for small h, Z(x) ≥ 0 on ∂(∪kj=1 B d1 (xjh )) and Z(x) vanishes at infinity 2

due to (4.7). As σ < N − 2, by Lemma 4.2, for any x ∈ RN \ (∪kj=1 B d1 (xjh )) and h sufficiently 2 small, −∆Z = −∆U + h2 ∆|uh (x)| ≥ σ(N − 2 − σ)

k X j=1

1 |x − xjh |σ+2

+ V (x)|uh | − gh (x, |uh |2 )|uh |

No.2

¨ N Ba & J Dai: SCHRODINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS

≥ σ(N − 2 − σ)

k X j=1

1 |x − xjh |σ+2

− χΛ (x)h − (1 − χΛ (x))

423

h 1 + |x|N

≥ 0. So, |uh (x)| ≤ U (x)/h2 in RN \ (∪kj=1 B d1 (xjh )) by the maximum principle. Hence, for all 2

x ∈ RN \ Λ, it follows that |uh (x)| ≤

k X j=1

1 h2 |x

−

xjh |σ

≤

C . h2 (1 + |x|σ )

(4.9)

Now, we show that uh actually solves problem (1.4). From Lemma 4.2, (4.8), and (4.9), we can choose γ larger than but close to 1 such that for all x ∈ RN \ Λ and small h,  p−γ (γ−1) ln 2 C h3 − 2ch K(x)|uh |p ≤ k0 (1 + |x|α ) 2 2 |u | ≤ |uh |, h h (1 + |x|σ ) 1 + |x|θ0 and  K(x)|uh |p ≤ k0 (1 + |x|α )

h2 (1

p+1−γ (γ−1) ln 2 h C 2− 2ch ≤ . σ + |x| ) 1 + |x|N

Therefore, gh (x, |uh |2 ) ≡ K(x)|uh |p−1 holds in RN \ Λ and uh solves the original problem (1.4). Moreover, as σ is close to N − 2, (4.9) implies that uh ∈ L2 (RN ) for N ≥ 5, which means that uh is a bound state. At last, Lemma 3.5, Proposition 3.6, and (4.9) show that |uh | has exactly k local maxima xjh ∈ Λj (j = 1, · · · , k) satisfying G(xjh ) → inf G(x) as h → 0.  Λj

References [1] Ambrosetti A, Felli V, Malchiodi A. Ground states of nonlinear Schr¨ odinger equations with potentials vanishing at infinity. J Eur Math Soc, 2005, 7: 117–144 [2] Ambrosetti A, Malchiodi A, Ruiz D. Bound states of nonlinear Schr¨ odinger equations with potentials vanishing at infinity. J Anal Math, 2006, 98: 317–348 [3] Ambrosetti A, Malchiodi A, Secchi S. Multiplicity results for some nonlinear Schr¨ odinger equations with potentials. Arch Ration Mech Anal, 2001, 159: 253–271 [4] Arioli G, Szulkin A. A semilinear Schr¨ odinger equation in the presence of a magnetic field. Arch Ration Mech Anal, 2003, 170: 277–295 [5] Ba N, Deng Y, Peng S. Multi-peak bound states for Schr¨ odinger equations with compactly supported or unbounded potentials. Ann Inst H Poincar´ e Anal Non Lin´ eaire, 2010, 27: 1205–1226 [6] Bartsch T, Dancer E N, Peng S. On multi-bump semi-classical bound states of nonlinear Schr¨ odinger equations with electromagnetic fields. Adv Differential Equations, 2006, 7: 781–812 [7] Brummenlhuis R. Expotential decay in the semiclassical limit for eigenfunctions of Schr¨ odinger operators with fields and potentials which degenerate at infinity. Comm Partial Differential Equations, 1991, 16: 1489–1502 [8] Byeon J, Wang Z Q. Spherical semiclassical states of a critical frequency for Schr¨ odinger equations with decaying potentials. J Eur Math Soc, 2006, 8: 217–228 [9] Byeon J, Wang Z Q. Standing wave for nonlinear Schr¨ odinger equations with singular potentials. Ann Inst H Poincar´ e Anal Non Lin´ eaire, 2009, 26: 943–958 [10] Byeon J, Wang Z Q. Standing waves with a critical frequency for nonlinear Schr¨ odinger equations. Arch Ration Mech Anal, 2002, 165: 295–316 [11] Cao D, Noussair E S. Multi-bump standing waves with a critical frequency for nonlinear Schr¨ odinger equations. J Differential Equations, 2004, 203: 292–312 [12] Cao D, Peng S. Multi-bump bound states of Schr¨ odinger equations with a critical frequency. Math Ann, 2006, 336: 925–948

424

ACTA MATHEMATICA SCIENTIA

Vol.37 Ser.B

[13] Cao D, Peng S. Semi-classical bound states for Schr¨ odinger equations with potentials vanishing or unbounded at infinity. Comm Partial Differential Equations, 2009, 34: 1566–1591 [14] Cao D, Tang Z. Existence and uniqueness of multi-bump bound states of nonlinear Schr¨ odinger equations with electromagnetic fields. J Differential Equations, 2006, 222: 381–424 [15] Cingolani S, Clapp M. Intertwining semiclassical bound states to a nonlinear magnetic Schr¨ odinger equation. Nonlinearity, 2009, 22: 2309–2331 [16] Cingolani S, Jeanjean L, Secchi S. Multi-peak solutions for magnetic nonlinear Schr¨ odinger equations without non-degeneracy conditions. ESAIM Control Optim Calc Var, 2009, 15: 653–675 [17] Cingolani S, Secchi S. Semiclassical limit for onlinear Schr¨ odinger equations with electromagnetic fields. J Math Anal Appl, 2002, 275: 108–130 [18] Cingolani S, Secchi S. Semiclassical states for nonlinear Schr¨ odinger equations with magenetic potentials having polynomial growths. J Math Phys, 2005, 46: 1–19 [19] del Pino M, Felmer M. Multi-peak bound states for nonlinear Schr¨ odinger equations. Ann Inst H Poincar´ e Anal Non Lin´ eaire, 1998, 15: 127–149 [20] Esteban M, Lions P L. Stationary solutions of nonlinear Schr¨ odinger equations with an external magnetic field. Partial Differential Equations and the Calculus of Variations, Essays in Honor of Ennio De Giorgi, 1989: 369–408 [21] Fei M, Yin H. Existence and concentration of bound states of nonlinear Schr¨ odinger equations with compactly supported and competing potentials. Pacific J Math, 2010, 244: 261–296 [22] Floer A, Weinstein A. Nonspreading wave pachets for the cubic Schr¨ odinger equation with a bounded potential. J Funct Anal, 1986, 69: 397–408 [23] Gilbarg D, Trudinger N S. Elliptic partial differential equations of second order. New York: Springer Press, 1998 [24] Helffer B. Semiclassical annaysis for Schr¨ odinger operator with magnetic wells//Rauch J, Simon B. Quasiclassical Methods. The IMA Volumes in Mathematics and Its Appications, Vol 95. New York: Springer, 1997 [25] Helffer B. On spectral theory for Schr¨ odinger operator with magnetic potentials. Adv Stud Pure Math, 1994, 23: 113–141 [26] Helffer B, Sj¨ ostrand J. Effet tunnel pour l´ equation de Schr¨ odinger avec champ magn´ etique. Ann Scuola Norm Sup Pisa, 1987, 14: 625–657 [27] Kurata K. Existence and semi-classical limit of the least energy solution to a nonlinear Schr¨ odinger equation with electromagenetic fields. Nonlinear Anal, 2000, 41: 763–778 [28] Li G, Peng S, Wang C. Infinitely many solutions for nonlinear Schr¨ odinger equations with electromagnetic fields. J Differential Equations, 2011, 251: 3500–3521 [29] Lieb E H, Loss M. Analysis, Graduate Studies in Mathematics 14. AMS, 1997 [30] Oh Y G. On positive multi-bump bound states of nonlinear Schr¨ odinger equations under multiple well potential. Commun Math Phys, 1990, 131: 223–253 [31] Rabinowitz P H. On a class of nonlinear Schr¨ odinger equations. Z Angew Math Phys, 1992, 43: 270–291 [32] Reed M, Simon B. Methods of Modern Mathematical Physics. New York: Academic Press, 1972 [33] Sato Y. Multi-peak positive solutions for nonlinear Schr¨ odinger equations with critical frequency. Calc Var Partial Differential Equations, 2007, 29: 365–395 [34] Sulem C, Sulem P L. The Nonlinear Schr¨ odinger equation, Self-Focusing and Wave Collapse. New York: Springer, 1999 [35] Tang Z. Multi-bump bound states of nonlinear Schr¨ odinger equations with electromagnetic fields and critical frequency. J Differential Equations, 2008, 245: 2723–2748 [36] Tang Z. Multiplicity of standing wave solutions of nonlinear Schr¨ odinger equations with electromagnetic fields. Z Angew Math Phys, 2008, 59: 810–833 [37] Wang X, Zeng B. On concentration of positive bound states of nonlinear Schr¨ odinger equations with competing potential functions. SIAM J Math Anal, 1997, 28: 633–655 [38] Yan S, Yang J, Yu X. Equations involving fractional Laplacian operator: Compactness and application. J Funct Anal, 2015, 269: 47–79