Multi-bump solutions for a strongly indefinite semilinear Schrödinger equation without symmetry or convexity assumptions

Nonlinear Analysis 68 (2008) 3067–3102 www.elsevier.com/locate/na

Multi-bump solutions for a strongly indefinite semilinear Schr¨odinger equation without symmetry or convexity assumptions Shaowei Chen ∗ LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, PR China Department of Mathematics, Fujian Normal University, Fuzhou 350007, PR China Received 4 October 2006; accepted 5 March 2007

Abstract In this paper, we study the following semilinear Schr¨odinger equation with periodic coefficient: −∆u + V (x)u = f (x, u),

u ∈ H 1 (R N ).

The functional corresponding to this equation possesses strongly indefinite structure. The nonlinear term f (x, t) satisfies some superlinear growth conditions and need not be odd or increasing in t. Using a new variational reduction method and a generalized Morse theory, we proved that this equation has infinitely many geometrically different solutions. Furthermore, if the solutions of this equation under some energy level are isolated, then we can show that this equation has infinitely many m-bump solutions for any positive integer m ≥ 2. c 2007 Elsevier Ltd. All rights reserved.

MSC: 35J20; 35J70 Keywords: Semilinear Schr¨odinger equation; Multi-bump solutions; Critical group; Reduction methods

1. Introduction and main results In this paper, we consider the following problem: −∆u + V (x)u = f (x, u), where x = (x1 , x2 , . . . , x N ) ∈

RN

u ∈ H 1 (R N ),

(1.1)

(N ≥ 1) and V , f satisfy the following conditions:

(V1 ) V ∈ L ∞ (R N ) is 1-periodic in each xi , i = 1, . . . , N . (V2 ) The linear operator L : H 2 (R N ) → L 2 (R N ), u 7→ −∆u + V u is invertible and 0 lies in a gap of the spectrum of L. (f1 ) f (x, t) is a Caratheodory function and is 1-periodic in each xi , i = 1, . . . , N . f t0 (x, t) exists for every t ∈ R and for almost all x ∈ R N . And f t0 (x, t) is a Caratheodory function. ∗ Corresponding address: Department of Mathematics, Fujian Normal University, Fuzhou 350007, PR China.

E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.03.001

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(f2 ) For some 2 < q < p < 2∗ :=



2N , N ≥3 N −2 ∞, N = 1, 2

| f t0 (x, t)| ≤ C(|t|q−2 + |t| p−2 ),

and C > 0,

for any (x, t) ∈ R N × R.

(f3 ) There exists γ > 2 such that for every t 6= 0 and x ∈ R N , 0 < γ F(x, t) ≤ t f (x, t), Rt where F(x, t) = 0 f (x, s)ds. Note that the power nonlinearity f (x, t) = h(x)|u| p−2 u, with positive 1-periodic h ∈ L ∞ (R N ), h(x) ≥ 0, h 6≡ 0 and 2 < p < 2∗ , satisfies all the assumptions. Under the assumptions (V1 )–(V2 ) and (f1 )–(f3 ) the functional Z Z 1 (|∇u|2 + V (x)u 2 )dx − F(x, u)dx (1.2) J (u) = 2 RN RN is of class C 2 on the Sobolev space H 1 (R N ) and critical points of (1.2) correspond to weak solutions of Eq. (1.1). The operator L = −∆ + V (on L 2 (R N )) has purely continuous spectrum which is bounded below and consists of closed disjoint intervals [16, Theorem XIII.100]. We denote by |L|1/2 the square root of the absolute value of L. The R domain of |L|1/2 is the space X := √ H 1 (R N ). On X , we choose the inner product (u, v) X = R N |L|1/2 u · |L|1/2 vdx and the corresponding norm kuk = (u, u) X . There exists an orthogonal decomposition X = Y ⊕ Z such that Z and Y are the positive and negative spaces corresponding to the spectral decomposing of L. They are invariant under the action of Z N , i.e., for any u ∈ Y or u ∈ Z and for any k = (n 1 , . . . , n N ) ∈ Z N , u(·−k) is also in Y or Z . Furthermore, Z ∀u ∈ Y, (|∇u|2 + V u 2 )dx = −(u, u) X = −kuk2 , (1.3) RN Z ∀u ∈ Z , (|∇u|2 + V u 2 )dx = (u, u) X = kuk2 . (1.4) RN

Since 0 lies in a gap of spectrum of L, the dimensions of Y and Z are both infinity. In this case, Eq. (1.1) is called strongly indefinite. By (1.3) and (1.4), we have Z 1 1 F(x, u)dx, u ∈ X. (1.5) J (u) = kQuk2 − kPuk2 − 2 2 RN It is easy to verify that if v is a solution of Eq. (1.1), then v(· − k) is also a solution of Eq. (1.1) for any k ∈ Z N . Let u and v be two solutions of Eq. (1.1). They are called geometrically different if u(· − k) 6= v for any k ∈ Z N . Let v1 , . . . , vn be of Eq. (1.1) such that their barycenters are sufficiently separated. Solutions of Eq. (1.1) that Psolutions m are close to i=1 vi are called m-bump solutions. The main result of this paper is the following theorem: Theorem 1.1. Assume (V1 )–(V2 ) and (f1 )–(f3 ). Eq. (1.1) has infinitely many, geometrically different solutions. Furthermore, if the condition (∗ ) (see Section 2 for its definition) holds, then for any positive integer m ≥ 2, Eq. (1.1) has infinitely many, geometrically different, m-bump solutions. Eq. (1.1) arises from studying of steady state and standing wave solutions of time-independent nonlinear Schr¨odinger equations. Readers can consult [13] for more physical background and applications of Eq. (1.1). The semilinear Schr¨odinger equation with periodic potential has been studied by many authors in the past decade. In the celebrated papers [7] and [8], Conti Zelati and Rabinowitz used variational gluing methods to obtain multi-bump type solutions for the Hamilton ODE and elliptic PDEs with periodic potential. The linear parts of the Hamilton ODE and elliptic PDEs they studied are positive definite and the functionals corresponding to these Hamilton ODE and elliptic PDEs have Mountain Pass structures. Conti Zelati and Rabinowitz used the solutions obtained from the Mountain Pass theorem as basic building blocks to construct multi-bump solutions. Readers can consult [2,11,12] and references therein for more recent development in this direction. In [17], S´er´e considered some Hamiltonian systems whose linear parts are strongly indefinite, i.e., the dimensions of the positive and negative spaces corresponding to the spectral decomposing are both infinity. He constructed multi-bump solutions for these Hamiltonian systems. But

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he imposed some convexity conditions on the nonlinear terms of these Hamiltonian systems and then transformed them by dual variational methods into some equivalent systems whose variational functionals are bounded below. For the strong indefinite semilinear Schr¨odinger equation (1.1), Alama and Li constructed multi-bump solutions in [3] by dual variational methods under the assumption that f (x, t) increases strictly in t, i.e., the function F(x, t) is convex. The first work directly dealing with Eq. (1.1) without the convexity assumptions on nonlinear term f was done by Troestler and Willem in [18]. They obtained results on the existence of nontrivial solutions to Eq. (1.1). In [10], Kryszewski and Szulkin obtained the result that there exist infinitely many geometrically different solutions to Eq. (1.1) whenever f is odd in u. In a very recent paper [1], Ackermann provided an interesting abstract framework in which multi-bump solutions can be obtained in many situations. It reduces the problem of constructing multi-bump solutions to the problem of finding an isolated solution with nontrivial topology in a specific sense. Using the abstract results of [1], a very general result on the existence of multi-bump solutions to strong indefinite periodic semilinear Schr¨odinger equations (even with nonlocal nonlinearities) is obtained. However in [1], the result on the existence of multi-bump solutions for Eq. (1.1) was obtained under the assumptions that the nonlinear term f (x, t) is C 2 and convex in t. Therefore, the question of the existence of infinitely many geometrically different solutions for Eq. (1.1) without the assumption that f is odd in t or F(x, t) is convex in t was left open. Theorem 1.1 of the present paper gives an affirmative answer to this open problem. In the present paper we shall show that Eq. (1.1) has a solution u 0 6= 0 which has the properties that after reducing the corresponding functional of Eq. (1.1) in a neighborhood of u 0 , the critical group of the reduction function in the critical point u 0 is nontrivial. Then using u 0 as a basic building block, we constructed multi-bump solutions for Eq. (1.1) by a perturbation technique stemming from Chang and Ghoussoub (see [6]). To obtain such u 0 , we consider the approximation problem first: −∆u + V (x)u = f (x, u),

1 u ∈ Hper (Q k ),

(1.6)

1 (Q ) denotes the space of H 1 (Q )-functions which are kwhere Q k is a cube of R N with edge length k ∈ N and Hper k k periodic in xi , i = 1, 2, . . . , N . The variational functional corresponding to (1.6) satisfies the Palais–Smale condition and has linking structure (see [14]). Secondly, using the linking theorem (one can see [19] for reference), we can get a solution u k of (1.6) which satisfies that there exist finite many nontrivial solutions u i , i = 1, . . . , n, of Eq. (1.1) and j sequences {bki }, i = 1, 2, . . . , n, such that |bki − bk | → ∞, i 6= j, as k → ∞ and

n

X

u i (· − bki ) → 0.

u k −

1 i=1 H (Q k )

Finally, we show that at least one of u i , i = 1, . . . , n, has the properties that its critical group of the reduction function is nontrivial. This paper is organized as follows. From Sections 2–4, we use an approximation method, reduction methods and critical point theory to obtain the existence of a special nontrivial solution of Equation of (1.1) which has the properties we mentioned above. In Section 5, we give the proof of Theorem 1.1. In the Appendix, we provide the detailed proofs of some lemmas stated in Section 3. Notation. R, Z and N denote the sets of real numbers, integers and positive integers respectively. B E (a, ρ) denotes the open ball in E centered at a and having radius ρ. The closure of a set A is denoted by A or cl(A). By → we denote the strong and by * the weak convergence. dist(a, A) denotes the distance from the point a to the set A. diam(A) denotes the diameter of the set A. By ker A we denote the null space of the operator A. If f is a C 2 functional defined on a Hilbert space H , ∇ f (or d f ) and ∇ 2 f denote the gradient of f and the second differential of f respectively. And for a, b ∈ R, we denote as f a := {u ∈ H : f (u) ≤ a} and f b := {u ∈ H : f (u) ≥ b} the sublevel and superlevel n sets i= j a of the functional f ; moreover, f b := {u ∈ H : b ≤ f (u) ≤ a}. δi, j denotes the Kronecker notation: δi, j = 1, 0, i 6= j . If H is a Hilbert space and W is a closed subspace of H , we denote the orthogonal complement space of W in H by W ⊥ . For a subset A ⊂ H , span{A} denotes the subspace of H generated by A. 2. A periodic approximation problem Associated with Eq. (1.1), the approximation problem in cubes Q k of R N with edge length k ∈ N

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S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

−∆u + V (x)u = f (x, u)

1 in Q k , u ∈ E k := Hper (Q k ),

(2.1)

1 (Q ) denotes the space of H 1 (Q )-functions which are k-periodic in x , i = 1, 2, . . . , N , is studied. The where Hper k k i operator −∆ + V on L 2per (Q k ) has discrete spectrum with eigenvalues λk,1 ≤ λk,2 ≤ · · · → +∞ and there exists a finite min{i : λk,i > 0}. Moreover, every eigenvalue λk,i is contained in the spectrum of −∆ + V on the whole space. This follows from the spectral gap around 0 assumed by Reed and Simon (see [16]). Therefore, if (−α, β), α, β > 0 denotes the spectral gap around 0 assumed in (V2 ). We claim that λk,i 6∈ (−α, β) for every k, i ∈ N. We denote by φk,i the corresponding eigenfunctions. Let j (k) = min{i : λk,i > 0} − 1. Now we define an orthogonal decomposition of E k by E k = Yk ⊕ Z k , where

Yk = span{φk,1 , . . . , φk, j (k) },

Z k = Yk⊥ .

The associated energy functional to (2.1) is Z Z 1 Jk (u) = (|∇u|2 + V (x)u 2 )dx − F(x, u)dx. 2 Qk Qk We may define a new inner product (·, ·)k on E k with corresponding norm k · kk such that Z (|∇u|2 + V u 2 )dx = −(u, u)k = −kuk2k , ∀u ∈ Yk ,

(2.2)

(2.3)

Qk

∀u ∈ Z k ,

Z Qk

(|∇u|2 + V u 2 )dx = (u, u)k = kuk2k .

If we denote by Pk : E k → Yk and Tk : E k → Z k the orthogonal projections, our functional becomes Z 1 F(x, u)dx. Jk (u) = (kTk uk2k − kPk uk2k ) − 2 Qk

(2.4)

(2.5)

For convenience, we assume that Q k = (− k2 , k2 ) N , k ∈ N; then 1 Hper (Q k ) (

    k k = u ∈ H 1 (Q k ) : u x1 , . . . , xi−1 , − , xi+1 , . . . , x N = u x1 , . . . , xi−1 , , xi+1 , . . . , x N , 2 2 )  N −1  k k , i = 1, . . . , N . (x1 , . . . , xi−1 , xi+1 , . . . , x N ) ∈ − , 2 2

Let Zk = Z/kZ. For b ∈ ZkN and u ∈ E k , the action of b on u, we still denote it by u(·+b), is defined in the following i

way: For b = (0, . . . , 0, 1, 0 . . . , 0) ∈ ZkN and u ∈ E k ,  k k  − ≤ xi ≤ − 1 u(x1 , . . . , xi−1 , xi + 1, xi+1 , . . . , x N ), 2 2 u(x + b) = k k  u(x , . . . , x , x + 1 − k, x , . . . , x ), − 1 ≤ xi ≤ , N 1 i−1 i i+1 2 2  k k  − + 1 ≤ xi ≤ u(x1 , . . . , xi−1 , xi − 1, xi+1 , . . . , x N ), 2 2 u(x − b) = k k  u(x , . . . , x , x − 1 + k, x , . . . , x ), − ≤ x ≤ − + 1. N i 1 i−1 i i+1 2 2 Since V (x) and f (x, t) is 1-periodic in xk , k = 1, . . . , N , we deduce that Jk is invariant under the action of ZkN . Lemma 2.1 ([14, Lemma 2]). There exist constants C1 > 0 and C2 > 0 which are independent of k such that for any u ∈ E k , C1 kuk H 1 (Q k ) ≤ kukk ≤ C2 kuk H 1 (Q k ) .

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By the conditions (f1 )–(f3 ), we have the following lemma (one can see [15] for reference): Lemma 2.2. For any k ∈ N, Jk satisfies the (PS) conditions. Lemma 2.3 ([13, Lemma 3.3] or [14, Lemma 4]). There exists 1 > 0 not depending on k such that ku k kk ≥ 1 , kuk ≥ 1 holds for any nontrivial critical point u k of Jk and u of J . In addition, there exists 2 > 0 not depending on k such that Jk (u k ) ≥ 2 , J (u) ≥ 2 holds for any nontrivial critical point u k of Jk and u of J . Lemma 2.4 ([14, Lemma 8]). There exist real numbers δ > 0 and r > 0 which are independent of k such that infu∈Nk Jk (u) ≥ δ, where Nk = {z ∈ Z k : kzkk = r }. Now for each k, we fix a function z k ∈ Z k with kz k kk = 1. For ρ > 0, we define the sets Mk = {y + t z k : ky + t z k kk ≤ ρ, t ≥ 0, y ∈ Yk }. Lemma 2.5 ([14, Lemma 9]). There exists a ρ > r which is independent of k such that sup Jk (u) = 0. u∈∂ Mk

Lemma 2.6 ([13, Theorem 3.4] or [14, Lemma 10]). The number ck = inf sup J (h(u)) h∈Γk u∈Mk

is a critical value of Jk and there exists positive number M which is independent of k such that 0 < δ ≤ ck ≤ M < ∞, where Γk = {h ∈ C(E k , E k ) : h|∂ Mk = id}. Let χk be cut-off functions such that 0 ≤ χk ≤ 1, χk ≡ 1 on Q k−1 , χk ≡ 0 outside of Q k and |∇χk | ≤ C, k = 1, 2, . . . . Lemma 2.7 ([13, Theorem 5.1] or [14, Theorem 11]). Under the assumptions (f1 )–(f3 ). Let vk ∈ E k be a uniformly bounded sequence which satisfies Jk0 (vk ) → 0 and e ck = Jk (vk ) → e c > 0. Then there exist critical points v i , j i = 1, 2, . . . , ν, of J and sequences dki ∈ Z N such that as k → ∞, |dki − dk | → ∞, i 6= j,

ν

X

i i v (· + dk ) → 0

vk −

i=1 k Pν and i=1 J (v i ) = e c. Let K and K k be the sets of critical points of J and Jk , k = 1, 2, . . ., respectively. For a, b ∈ R, let K a := K ∩ J a , K a := K ∩ Ja , K ab := K ∩ J a ∩ Jb and K ka = Jka ∩ K k . Let c0 = sup ck ,

(2.6)

k

where ck is the minimax value defined in Lemma 2.6. Now we impose the following condition on Eq. (1.1): (*) There exists α0 > 0 such that K c0 +α0 /Z N is finite. Lemma 2.8. If the condition (∗ ) holds, then the following three statements hold: (1) There exists β0 ∈ (0, α0 ) such that inf{k∇ J (u)k : u ∈ X, J (u) = c0 + β0 } > 0

(2.7)

and there exists constant 3 > 0 not depending on k such that inf{k∇ Jk (u)kk : u ∈ E k , Jk (u) = c0 + β0 } > 3 if k is large enough.

(2.8)

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S. Chen / Nonlinear Analysis 68 (2008) 3067–3102 c +β0

(2) There exist δ0 > 0 and k0 ∈ N such that if k ≥ k0 , then for any u k ∈ K k 0 B E k (u k , δ0 ) ∩

,

c +β Kk 0 0

c +β = B E k (u k , δ0 /2) ∩ K k 0 0 . c +β k0 , for any u 1k , u 2k ∈ K k 0 0 , either

It follows that when k ≥ ku 1k − u 2k kk ≤ δ0 /2 or ku 1k − u 2k kk ≥ δ0 . c +β (3) For any  > 0, there exists k ∈ N such that if k ≥ k , then for any u k ∈ K k 0 0 , c +β0

B E k (u k , δ0 ) ∩ K k 0

c +β0

= B E k (u k , ) ∩ K k 0

, c +β0

where δ0 is the constant that appeared in (2). It follows that when k ≥ k , for any u 1k , u 2k ∈ K k 0 ku 1k − u 2k kk ≤  or ku 1k − u 2k kk ≥ δ0 .

, either

0 Proof. By Lemma 2.3, we know that there exists 2 > 0 such that for any u ∈ K , J (u) ≥ 2 . Let l = [ α0+c ] + 1 and 2 let ( j ) X Fl (K c0 +α0 ) := vi (· − bi ) : 1 ≤ i ≤ j, 1 ≤ j ≤ l, vi ∈ K c0 +α0 , bi ∈ Z N .

i=1

If c ∈ (c0 , c0 + α0 ) satisfies that there exists a sequence {u m } such that as m → ∞, J (vm ) → c,

k∇ J (vm )k → 0,

(2.9) K c0 +α0 , i

then by Proposition 1.24 of [7], we deduce that there exist at most l nontrivial solutions vi ∈ P j and l sequence {dmi } ⊂ Z N such that as m → ∞, |dmi − dm | → ∞, i 6= j, c = li=1 J (v i ) and



l X

i i

u m − v (· − dm ) → 0.

i=1

= 1, . . . , l,

(2.10)

By the condition (*), we know that ) ( j X c0 +α0 A= J (u i ) : 1 ≤ i ≤ j, 1 ≤ j ≤ l, u i ∈ K i=1

is a finite set. It follows that the possible c ∈ (c0 , c0 + α0 ) which satisfies (2.9) is finite. If we choose β0 ∈ (0, α0 ) such that c0 + β0 ∈ (c0 , c0 + α0 ) \ A, then (2.7) holds. We are ready to prove that (2.8) holds. If not, then there exists a sequence {u k } such that u k ∈ E k , Jk (u k ) = c0 + β0 , k = 1, 2, . . . and k∇ Jk (u k )kk → 0 as k → ∞. Then by Lemma 2.7 we deduce that there exist at most l nontrivial solutions vi ∈ K c0 +α0 , i = 1, . . . , l, and l sequence {dmi } ⊂ Z N such that as m → ∞, P j |dmi − dm | → ∞, i 6= j, c0 + β0 = li=1 Jk (v i ) and



l X

i i

u m − v (· − d ) m → 0.

i=1 k

It follows that c0 + β0 ∈ A. This is a contradiction. Thus (2.8) holds. By the condition (*) and Proposition 1.55 of [7], we know that µ = µ(Fl (K c0 +α0 )) := inf{kx − yk : x 6= y ∈ Fl (K c0 +α0 )} > 0. c +β Choose δ0 = µ/2. If there exist two sequences {u 1k } and {u 2k } such that u ik ∈ K k 0 0 , k = 1, 2, . . . , i δ0 1 2 1 2 c0 +α0 ) such that as k 2 < ku k − u k k < δ0 , then by Lemma 2.7, we deduce that there exist vk , vk ∈ Fl (K

ku ik − vki kk → 0,

(2.11) = 1, 2 and →∞

i = 1, 2.

It follows that kvk1 − vk2 k ≤ δ0 < µ when k is large enough. This contradicts the definition of µ. Thus the result of (2) holds. The proof of result (3) is similar. 

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Remark 2.9. In fact, by Lemma 2.7, we get that as k → ∞, c +β0

dist(K k 0

, Fl (K c0 +α0 )) → 0. c +β0

By (2.11), we know that Fl (K c0 +α0 ) is a discrete set. Thus K k 0 which are disjoint each other.

can be decomposed into a union of its subsets

By Lemma 2.8, we have the following lemma: (i)

Lemma 2.10. Suppose that the condition (∗ ) holds. There exist positive integer m k > 0 and m k subsets K k , i = c +β 1, 2 . . . , m k , of K k 0 0 such that if k ≥ k0 , then c +β0

Kk 0

=

mk [

(i)

Kk ,

i=1 (i)

( j)

(i)

dist(K k , K k ) ≥ δ0 , i 6= j and diam(K k ) ≤ δ0 /2, i = 1, 2 . . . , m k , where δ0 , k0 and β0 are the constants that appeared in Lemma 2.8. Furthermore, for any 1 ≤ i ≤ m k , as k → ∞, (i)

diam(K k ) → 0. c +β

(i)

Proof. By Lemma 2.8, we know that if k ≥ k0 , then K k 0 0 can be decomposed into a union of its subsets K k which are disjoint from each other. We show that the number of these subsets is finite. If not, by Lemma 2.8 there S∞ c +β (i) (i) (i) ( j) (i) exist K k , i = 1, 2, . . ., such that K k 0 0 = i=1 K k and dist(K k , kk ) ≥ δ0 , i 6= j. Choose u i,k ∈ K k . Then u i,k satisfies that ku i,k − u j,k k ≥ δ0 ,

(2.12)

i 6= j.

By Lemma 2.2, we get that there exists a subsequence {u im ,k } of {u i,k } and u ∈ E k such that ku im ,k − ukk → 0 as (i) m → ∞. This contradicts (2.12). Therefore, the number of K k is finite. We denote it by m k . Finally, by the result (i) (3) of Lemma 2.8, we get that diam(K k ) → 0 as k → ∞.  Let H∗ (A, B) be the ∗-th singular homology group with coefficient Z2 . By the definition of ck = infh∈Γk maxu∈Mk Jk (h(u)), the Linking Theorem (one can see [15] or [19] for reference) and the proof of Theorem 7.5 of [4], we have the following lemma: Lemma 2.11. Let b δ = min{δ/2, 2 } where δ and 2 are the constants that appeared in Lemma 2.4 and Lemma 2.3 respectively; then c +β0

H j (k)+1 (Jk 0

, Jkδ ) 6= 0.

(2.13)

b

(i)

By Lemma 2.8 we know that K k , i = 1, 2, . . . , m k , are isolated critical sets of Jk . In [6], Chang and Ghoussoub provided a definition of a critical group for an isolated critical set. In [9,5], the authors defined the Gromoll–Meyer pair (for short the GM-pair) for an isolated critical point for a C 1 functional. And in [6], Chang and Ghoussoub extended the definition of the GM-pair into a dynamically isolated critical set (see Definition I.10 of [6]). Let f be a C 1 functional on a Finsler manifold M (the Banach space is a special case of a Finsler manifold) with critical set K f . And let V be a pseudo-gradient vector field V with respect to d f on M. A pseudo-gradient flow associated with V is the unique solution of the following ordinary differential equation in M: η˙ = V1 (η(x, t)),

η(x, 0) = x,

g(x) kVV (x) (x)k

where V1 (x) = and g(x) = min{dist(x, K f ), 1}. A subset W of M is said to have the mean value property (for short (MVP)) if for any x ∈ M and any t0 < t1 we have η(x, [t0 , t1 ]) ⊂ W whenever η(x, ti ) ∈ W, i = 1, 2.

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Definition 2.12 (Definition I.10 of [6]). Let f be a C 1 functional on a Finsler manifold M. A subset S of the critical set K of f is said to be a dynamically isolated critical set if there exist a closed neighborhood O of S and regular values a < b of f such that O ⊂ f −1 [a, b]

(2.14)

and e ∩ K ∩ f −1 [a, b] = S, cl(O) S e= where O t∈R η(O, t). (O, a, b) is called an isolating triplet for S.

(2.15)

After providing the definition of a dynamically isolated critical set, the authors of [6] give the definition of a critical group for a dynamically isolated critical set as follows: Definition 2.13. Let S be a dynamically isolated critical set of a C 1 functional f and let (O, a, b) be any isolating triplet for S. For each integer q, we shall call the qth homology group e+ , f a ∩ O e+ ) Cq ( f, S) = Hq ( f b ∩ O S e+ = the qth critical group for S, where O t≥0 η(O, t). e+ , f a ∩ O e+ ). Remark 2.14. In [6], the critical group is defined as the cohomology group of the topology pair ( f b ∩ O Here we use the homology group instead. All results of [6] still hold for a homology group since the properties of cohomology that the authors used in [6] are the excision property and the homotopy property. Definition 2.15 (Definition III.1 of [6]). Let f be a C 1 functional on a Finsler manifold M and let S be a subset of the critical set K f for f . A pair (W, W− ) of subsets is said to be a GM-pair for S associated with a pseudo-gradient vector field V , if the following conditions hold: (1) W is a closed (MVP) neighborhood of S satisfying W ∩ K = S and W ∩ f α = ∅ for some α. (2) W− is an exit set for W , i.e., for each x0 ∈ W and t1 > 0 such that η(x0 , t1 ) 6∈ W , there exists t0 ∈ [0, t1 ) such that η(x0 , [0, t0 ]) ⊂ W and η(x0 , t0 ) ∈ W− . (3) W− is closed and is a union of a finite number of sub-manifolds that are transverse to the flow η. In [6], the authors proved the following theorem which can be seen as another definition of the critical group for a dynamically isolated critical set. Lemma 2.16 (Theorem III.3 of [6]). Let f be a C 1 functional on a C 1 Finsler manifold M and let S be a dynamically isolated critical set for f . Then for any GM-pair (W, W− ) for S, we have e+ , f a ∩ O e+ ) = C∗ ( f, S), H∗ (W, W− ) ∼ = H∗ ( f b ∩ O where (O, a, b) is an isolating triplet for S. By [6] and Lemma 2.10, we have the following lemma. Lemma 2.17. If k large enough, then there exist index i k satisfying 1 ≤ i k ≤ m k such that (i )

C j (k)+1 (Jk , K k k ) 6= 0.

(2.16)

c +β c +β δ) is an isolating triplet for the isolated critical set K k 0 0 , where Proof. It is easy to verify that ((Jk )bδ0 0 , c0 + β0 , b b δ is the constant appeared in Lemma 2.11 and c +β0

(Jk )bδ0

= {u ∈ E k : b δ ≤ Jk (u) ≤ c0 + β0 }.

By Definition 2.13 and Lemma 2.11, we have c +β0

C j (k)+1 (Jk , K k 0

c +β0

) = H j (k)+1 (Jk 0

, Jkδ ) 6= 0. b

(2.17)

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Let ( C=

j X

) J (vi )|1 ≤ i ≤ j, 1 ≤ j ≤ l, vi ∈ K

c0 +β0

∩ (−∞, c0 + β0 ].

i=1

By the condition (∗) we know that C is a finite set. Without loss of generality, we may assume that C = {c1 , . . . , cn 0 } and c1 < c2 < · · · < cn 0 . Thus by Lemma 2.7, we get that c +β0

distR (Jk (K k 0 Choose 0 =

1 2

), C) → 0,

(2.18)

k → ∞.

min{ci − ci−1 |i = 2, . . . , n 0 }. For 0 <  < 0 , let

(i, j) Kk

(i)

= Kk ∩

(i, j) Kk

c + (Jk )c jj − ,

j = 1, . . . , n 0 , i =

1, . . . , m k . By Lemma 2.8 and Remark 2.9, we know that if k large enough, is independent of . By Proposition 2.2 of [7], we get that for any 0 < δ < µ (for the definition of µ see (2.11) in the proof of Lemma 2.8), inf{k∇ J (u)k|u ∈ J c0 +α0 \ Nδ (Fl (K c0 +α0 ))} > 0.

(2.19)

Then by Lemma 2.7, we deduce that there exists constant ςδ > 0 which is independent of k, such that c +α0

inf{k∇ Jk (u)k|u ∈ Jk 0

c +α0

\ Nδ (K k 0

)} > ςδ > 0.

(2.20)

(i)

(i)

( j)

By Lemma 2.8, we know that diam(K k ) → 0 as k → ∞ and dist(K k , K k ) ≥ δ0 , i 6= j, when k large enough. Thus by (2.18), (2.20), Section 2 of [9] or page 49 and page 50 of [5], we know that we can choose δ ∈ (0, δ0 ) and (i, j)  ∈ (0, 0 ) small enough such that there exist GM-pair (Wi, j , Wi,−j ) of K k such that when k large enough, (i, j)

c +

Wi, j ⊂ {u ∈ E k |dist(u, K k ) ≤ δ/4} ∩ (Jk )c jj − , j = 1, . . . , n 0 , i = 1, 2, . . . , m k , Sm k Sm k c + c +β and ( i=1 Wi, j , i=1 Wi,−j ) is a GM-pair of K k 0 0 ∩ (Jk )c jj − . Therefore, by Lemma 2.16, ! mk mk [ [ c j + c0 +β0 − Wi, j , Wi, j , Cq (Jk , K k ∩ (Jk )c j − ) = Hq i=1

=

mk M

i=1

Hq (Wi, j , Wi,−j )

i=1

=

mk M

(i, j)

Cq (Jk , K k

),

∀q.

(2.21)

i=1

Pn 0 c + b c +β c +β Let Mq = dim j=1 Cq (Jk , K k 0 0 ∩ (Jk )c jj − ) and βq = dim Hq (Jk 0 0 , Jkδ ), ∀q. Then by Morse inequality (see [5]), we get that Mq ≥ βq ,

∀q.

By Lemma 2.11, we have c +β0

M j (k)+1 ≥ β j (k)+1 = H j (k)+1 (Jk 0

, Jkδ ) 6= 0. b

(2.22)

It follows that there exists j0 such that c +β0

C j (k)+1 (Jk , K k 0

c j +

∩ (Jk )c j0 − ) 6= 0.

(2.23)

0

By (2.21) and (2.23), we get that there exists i k such that (i , j0 )

C j (k)+1 (Jk , K k k

) 6= 0.

This completes the proof of this Lemma.

(2.24) 

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3. A reduction method (i )

(i )

Let K k k satisfy (2.16) and u k ∈ K k k , k = 1, 2 . . . . By Lemma 2.7, we know that there exist a positive integer n, n functions u i ∈ K c0 +β0 , i = 1, . . . , n, and n sequences {bki } ⊂ Z N , i = 1, . . . , n, such that as k → ∞,

n

X

j |bki − bk | → +∞, for any i 6= j and u k − u i (· + bki ) → 0. (3.1)

1 i=1 H (Q k )

Without loss of generality, we may assume that |bki | → +∞ as k → ∞, i = 1, . . . , n. The proofs of the following four lemmas will be provided in the Appendix. For convenience, we denote f t0 (x, t) by f 0 (x, t). Lemma 3.1. The following limit holds uniformly for any ψk , ϕk ∈ E k which satisfy kψk kk ≤ 1, kϕk kk ≤ 1, ! Z n X 0 0 i i lim u (· + bk ) · |ψk | · |ϕk |dx = 0. f (x, u k ) − f x, k→∞ Q k i=1 Lemma 3.2. If {e vk } and {vk } are two bounded sequences in H 1 (R N ) which satisfy that ke vk − vk k → 0 as k → ∞, then the following two results hold: (1) The limit Z RN

| f (x,e vk ) − f (x, vk )| · |ϕ| → 0,

k→∞

holds uniformly for ϕ ∈ H 1 (R N ) which satisfies kϕk ≤ 1. (2) The limit Z | f 0 (x,e vk ) − f 0 (x, vk )| · |ϕ · ψ| → 0, k → ∞ RN

holds uniformly for ϕ, ψ ∈ H 1 (R N ) which satisfy kϕk ≤ 1, kψk ≤ 1. Lemma 3.3. As k → ∞, the limit ! Z n n X X 0 u i (· + bki ) − f 0 (x, u i (· + bki )) · |ϕk · ψk |dx → 0 f x, Qk i=1 i=1 holds uniformly for any ψk , ϕk ∈ E k which satisfy that kϕk kk ≤ 1, kψk kk ≤ 1. j

Lemma 3.4. (1) Suppose vki ∈ E k , aki ∈ Z N , i = 1, 2, . . . , n, which satisfy that for any i 6= j, |aki − ak | → ∞ as k → ∞. Then as k → ∞, the limit ! Z n n X X 0 0 i i i i vk (· + ak ) − f (x, vk (· + ak )) · |ϕk · ψk |dx → 0 f x, Qk i=1 i=1 holds uniformly for any ψk , ϕk ∈ E k which satisfy kϕk kk ≤ 1, kψk kk ≤ 1. j (2) Suppose v i ∈ H 1 (R N ), aki ∈ Z N , i = 1, 2, . . . , n, which satisfy that for any i 6= j, |aki − ak | → ∞ as k → ∞. Then as k → ∞, the limit ! Z n n X X 0 0 i i i i v (· + ak ) − f (x, v (· + ak )) · |ϕ · ψ|dx → 0 f x, N R i=1 i=1 holds uniformly for any ψ, ϕ ∈ H 1 (R N ) which satisfy kϕk ≤ 1, kψk ≤ 1.

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Lemma 3.5. Suppose that vk ∈ ker ∇ 2 Jk (u k ) and kvk kk = 1, k = 1, 2, . . . . Then there exist v i ∈ ker ∇ 2 J (u i ), i = 1, . . . , n, such that as k → ∞,

n

X

v i (· + bki ) → 0.

vk −

1 i=1 H (Q k )

Proof. Since vk ∈ ker ∇ 2 Jk (u k ), we have −∆vk + V (x)vk = f 0 (x, u k )vk ,

in E k .

(3.2)

By Lemmas 3.1 and 3.3, we know that the limit Z n X 0 0 i i lim f (x, u (· + bk )) · |ϕk | · |ψk |dx = 0 f (x, u k ) − k→∞ Q k i=1 1 (R N ) as holds uniformly for ϕk , φk ∈ E k which satisfy kϕk kk ≤ 1, kψk kk ≤ 1. Assume that vk (· − bki ) * v i in Hloc k → ∞. By kvk kk = 1, we deduce that v i ∈ H 1 (R N ). By

−∆vk (· − bki ) + V (x)vk (· − bki ) = f 0 (x, u k (· − bki ))vk (· − bki ) u k (· − bki )

*

ui

and

vk (· − bki )

*

vi

−∆v i + V (x)v i = f 0 (x, u i )v i vi

in

1 (R N ), Hloc

in E k ,

we get that

in H 1 (R N ),

(3.3)

ker ∇ 2 J (u i ), i

i.e. ∈ = 1, . . . , n. Recall that χk are cut-off functions satisfying that 0 ≤ χk ≤ 1, χk ≡ 1 on Q k−1 , χk ≡ 0 outside of Q k and |∇χk | ≤ C. Set vki = χk v i , we have vki ∈ E k . If ϕ ∈ E k , then by (3.3), we have Z Z Z Z Z Z i i i i i ∇vk · ∇ϕ + V (x)vk ϕ = ∇v · ∇(χk ϕ) + V (x)v · (χk ϕ) − ϕ∇v ∇χk + v i ∇χk ∇ϕ Qk Qk Qk Qk Qk Qk Z Z Z 0 i i i = f (x, u ) · v · (χk ϕ) − ϕ∇v ∇χk + v i ∇χk ∇ϕ Qk Qk Qk Z Z Z 0 i i i = f (x, u ) · (χk v ) · ϕ − ϕ∇v ∇χk + v i ∇χk ∇ϕ Qk Qk Qk Z Z Z 0 i i i = f (x, u ) · vk · ϕ − ϕ∇v ∇χk + v i ∇χk ∇ϕ. R N \Q k

Qk

It follows that for any ϕ ∈ E k , Z Z ∇vki (· + bki )∇ϕ + V (x)vki (· + bki )ϕ Qk Qk Z Z 0 i i = f (x, u (· + bk )) · vki (· + bki ) · ϕ − Qk

Z + Qk

Qk

R N \Q k

(∇v i (· + bki ) · ∇χk (· + bki )) · ϕ

v i (· + bki )∇χk (· + bki ) · ∇ϕ.

(3.4)

By (3.4), we get that Z Z Z ∇vki (· + bki )∇ϕ + V (x)vki (· + bki )ϕ = f 0 (x, u i (· + bki )) · vk · ϕ Qk Qk Q Z Z k 0 i i i i + f (x, u (· + bk )) · vk (· + bk ) · ϕ − f 0 (x, u i (· + bki )) · vk · ϕ Qk

Z + Qk

Qk

v i (· + bki )∇χk (· + bki ) · ∇ϕ −

Z Qk

(∇v i (· + bki ) · ∇χk (· + bki )) · ϕ.

(3.5)

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By (3.2) and (3.5), we get that for any ϕ ∈ E k , ! ! Z Z n n X X i i i i ∇ vk − vk (· + bk ) · ∇ϕ + V (x) · vk − vk (· + bk ) · ϕ Qk

Qk

i=1

Z

f 0 (x, u k ) −

= Qk

−

n X

! f 0 (x, u i (· + bki )) · vk · ϕ −

i=1

n Z X i=1

Qk

i=1 n Z X i=1

v i (· + bki )∇χk (· + bki ) · ∇ϕ +

n Z X i=1

Qk

Qk

f 0 (x, u i (· + bki )) · (vki (· + bki ) − vk ) · ϕ

(∇v i (· + bki ) · ∇χk (· + bki )) · ϕ.

(3.6)

1 (R N ) as k → ∞, we get that the limit Since vk (· − bki ) * v i in Hloc Z lim f 0 (x, u i (· + bki )) · (vki (· + bki ) − vk ) · ϕ = 0

(3.7)

k→∞ Q k

holds uniformly for ϕ ∈ E k which satisfies kϕkk ≤ 1. By Lemmas 3.1 and 3.3, we know that the following limit holds uniformly for ϕ ∈ E k which satisfies kϕkk ≤ 1: ! Z n X 0 i i 0 f (x, u k ) − f (x, u (· + bk )) · vk · ϕ = 0. (3.8) lim k→∞ Q k

i=1

Furthermore, the following limits hold uniformly for ϕ ∈ E k which satisfies kϕkk ≤ 1: Z Z lim v i (· + bki )∇χk (· + bki ) · ∇ϕ = lim v i (· + bki )∇χk (· + bki ) · ∇ϕ = 0, k→∞ Q k

Z lim

k→∞ Q k

(3.9)

k→∞ R N \Q k

(∇v

i

(· + bki ) · ∇χk (· + bki )) · ϕ

Z = lim

k→∞ R N \Q k

(∇v i (· + bki ) · ∇χk (· + bki )) · ϕ = 0.

(3.10)

By (3.6)–(3.10), we deduce that the following equality holds uniformly for ϕ ∈ E k which satisfies kϕkk ≤ 1: ! ! Z Z n n X X i i i i ∇ vk − vk (· + bk ) · ∇ϕ + V (x) · vk − vk (· + bk ) · ϕ = o(1), as k → ∞. (3.11) Qk

Qk

i=1

i=1

Recall that Pk : E k → Yk and Tk : E k → Z k are the orthogonal projections. Choose ϕ = Tk (vk − Pn and ϕ = Pk (vk − i=1 vki (· + bki )) respectively in (3.11); we get that as k → ∞,

! ! n n



X X



i i i i vk (· + bk ) → 0, vk (· + bk ) → 0.

Pk vk −

Tk vk −



i=1 i=1 Therefore, as k → ∞, kvk −

i i i=1 vk (· + bk )kk

i i i=1 vk (· + bk ))

k

k

Pn

Pn

→ 0.



The following notation will be used in this and later sections. • Let Ni = ker ∇ 2 J (u i ) = span{ei,1 , . . . , ei,li }, where li = dim Ni , i = 1, 2, . . . , n and (ei,s , ei, j ) X = δs, j . Sn • Let Nik = span{(χk ei,1 )(· + bki ), . . . , (χk ei,li )(· + bki )} ⊂ E k , i = 1, 2, . . . , n, Λk = span{ i=1 Nik } ⊂ E k and ⊥ Πk = Λk , the orthogonal complement space of Λk in E k . Sn ek = (e ek = span{ei,1 (· + bi ), . . . , ei,l (· + bi )}, i = 1, . . . , n, e ek } ⊂ X and Π • Let N Λk = span{ i=1 N Λk )⊥ , the i i k k i orthogonal complement space of e Λk in X . bk = span{χk ei,1 , . . . , χk ei,l } ⊂ E k , i = 1, 2, . . . , n. • Let N i i • For convenience, denote (χk u i )(· + bki ), χk u i , (χk ei, j )(· + bki ) and ei, j (· + bki ) by u ik , b u ik , ei,k j and e ei,k j respectively, j = 1, 2, . . . , li , i = 1, 2, . . . , n. Lemma 3.6. There exists k0 ∈ N, δ0 > 0 and η > 0 which are independent of k such that:

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(1) if k ≥ k0 , then for any u ∈ B E k (u k , δ0 ), the operator ek ◦ ∇ 2 Jk (u)|Π : Πk → Πk T k is invertible and ek ◦ ∇ 2 Jk (u)|Π )−1 k ≤ η, k = 1, 2, . . . , k(T k e where Tk : E k → Πk is the orthogonal projection. (2) if k ≥ k0 , then for any u ∈ B E k (b u ik , δ0 ), the operator bk )⊥ → ( N bk )⊥ P( Nbk )⊥ ◦ ∇ 2 Jk (u)|( Nbk )⊥ : ( N i i i

i

is invertible and k(P( Nbk )⊥ ◦ ∇ 2 Jk (u)|( Nbk )⊥ )−1 k ≤ η, i

i

k = 1, 2, . . . ,

bk )⊥ is the orthogonal projection. where P( Nbk )⊥ : E k → ( N i i

Proof. We only give the proof of the result (1), since the proof of the result (2) is similar. If the result (1) is not true, then there exists a sequence {e u k } such that e u k ∈ E k and as k → ∞, ek ◦ ∇ 2 Jk (e kT u k )|Πk k → 0.

ku k − e u k kk → 0,

(3.12)

By (3.12), we know that there exists vk ∈ Πk satisfying kvk kk = 1 and as k → ∞, ek (∇ 2 Jk (e kT u k )vk )kk → 0.

(3.13)

1 (R N ), then v i ∈ ker ∇ 2 J (u i ), i Step 1. We shall prove that if vk (· − bki ) * v i in Hloc Choose ϕ ∈ (ker ∇ 2 J (u i ))⊥ . Let ϕk = χk ϕ. Then ϕk ∈ E k . Assume that

= 1, . . . , n.

ϕk (· + bki ) = e ϕk + b ϕk , ϕk ∈ Λk . Since limk→∞ |bki | = +∞, we deduce for any u ∈ ker ∇ 2 J (u i ), as k → ∞, the limit where e ϕk ∈ Πk and b (ϕk (· + bki ), χk u)k → 0. It follows that kb ϕk kk → 0 as k → ∞. Thus as k → ∞, ke ϕk (· − bki ) − ϕk k H 1 (Q k ) → 0. By (3.13), we get that as k → ∞, Z Z (∇ 2 Jk (e u k )vk , e ϕk ) k = ∇vk · ∇e ϕk + Qk

Qk

V (x)vk · e ϕk −

Z Qk

f 0 (x, e u k ) · vk · e ϕk → 0.

(3.14)

1 (R N ), e 1 (R N ) and e 1 (R N ), by (3.14) and Note that vk (· − bki ) * v i in Hloc u k (· − bki ) * u i in Hloc ϕk (· − bki ) → ϕ in Hloc Z Z lim f 0 (x, e u k (· − bki )) · vk (· − bki ) · e ϕk (· − bki ) = f 0 (x, u i ) · v i · ϕ, k→∞ Q k

we get that Z RN

∇v i · ∇ϕ +

RN

Z RN

V (x)v i · ϕ −

Z RN

f 0 (x, u i )v i · ϕ = 0.

(3.15)

Since ϕ is an arbitrary function in (ker ∇ 2 J (u i ))⊥ , by (3.15), we have v i ∈ ker ∇ 2 J (u i ). Step 2. We shall prove that v i = 0, i = 1, . . . , n. By the result of Step 1 and the definition of Λk , we know that χk v i (· + bki ) ∈ Λk . Since vk ∈ Πk , we get that 1 (R N ) and χ v i → v i (vk , χk v i (· + bki ))k = 0, i.e., (vk (· − bki ), χk v i )k = 0, k = 1, 2 . . . . By vk (· − bki ) * v i in Hloc k 1 N in Hloc (R ), we get that as k → ∞, (vk (· − bki ), χk v i )k → kv i k2H 1 (R N ) .

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Thus v i = 0, i = 1, . . . , n. Step 3. We shall prove that as k → ∞, k∇ 2 Jk (e u k )vk kk → 0. ek (∇ 2 Jk (e ek , to prove this claim we only need to prove that Since kT u )v )k → 0 as k → ∞, by the definition of T k k k Sn k for any u ∈ i=1 Ni , as k → ∞, (∇ 2 Jk (e u k )vk , u)k → 0. Without loss of generality, we may assume that u = χk ϕ(· + bki ), where ϕ ∈ Ni for some i. Then (∇ 2 Jk (e u k )vk , χk ϕ(· + bki ))k Z Z Z = ∇vk ∇(χk ϕ(· + bki )) + V (x)vk · χk ϕ(· + bki ) − f 0 (x, e u k )vk · χk ϕ(· + bki ) Qk Qk Qk Z Z Z = ∇(vk (· − bki )) · ∇(χk ϕ) + V (x)vk (· − bki ) · χk ϕ − f 0 (x, e u k (· − bki )) · vk (· − bki ) · χk ϕ. Qk

Qk

Qk

Since vk (· − bki ) * 0 χk ϕ → ϕ

1 in Hloc (R N ), (by Step 2)

1 in Hloc (R N ),

e u k (· − bki ) * u i

1 in Hloc (R N ), (by (3.12) and (3.1))

by (3.15), we get that as k → ∞, lim (∇ 2 Jk (e u k )vk , χk ϕ(· + bki ))k =

k→∞

Z

∇v i · ∇ϕ + RN

Z RN

V (x)v i · ϕ −

Z RN

f 0 (x, u i )v i · ϕ = 0.

This proves the result of this step. Step 4. We are ready to prove that kvk kk → 0 as k → ∞. Then it induces a contradiction since kvk kk = 1 for any k. We prove kPk vk kk → 0 as k → ∞ firstly. Since Z (∇ 2 Jk (e u k )vk , Pk vk )k = (vk , Pk vk ) − f 0 (x, e u k )vk · (Pk vk ), (3.16) Qk

by (3.12) and (3.1) and Lemma 3.3, we get that as k → ∞, Z n Z X f 0 (x, e u k )vk · (Pk vk ) = f 0 (x, u i (· + bki ))vk · (Pk vk ) + o(1). Qk

i=1

(3.17)

Qk

1 (R N ) (see Step 2), we get that as k → ∞, for i = 1, . . . , n, By (3.17) and the fact that vk (· − bki ) * 0 in Hloc Z Z f 0 (x, u i (· + bki ))vk · (Pk vk ) = f 0 (x, u i )vk (· − bki ) · (Pk vk )(· − bki ) → 0. Qk

Qk

Thus as k → ∞, Z f 0 (x, e u k )vk · (Pk vk ) = o(1).

(3.18)

Qk

By Step 3, we get that as k → ∞, (∇ 2 Jk (e u k )vk , Pk vk )k → 0.

(3.19)

By (3.16), (3.18) and (3.19), we get that as k → ∞, −kPk vk k2k = (vk , Pk vk )k → 0. In the same way, we can prove that kTk vk k2k → 0 as k → ∞. Thus kvk kk = This is a contradiction. This completes the proof of this lemma. 

q

kTk vk k2k + kPk vk k2k → 0 as k → ∞.

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Pn Lemma 3.7. There exists k0 > 0 such that for k ≥ k0 , dim Λk = i=1 li . Pn Proof. By the definition of Λk , we know that dim Λk ≤ i=1 li . Note that if i 6= j, then for any u ik ∈ Nik , vik ∈ N kj j

which satisfy that ku ik k = 1 and kvik k = 1, (u ik , vik )k → 0 as k → ∞, since |bki − bk | → ∞, i 6= j. Thus there exists k0 >P0 such that when k ≥ k0 , χk ei, j (· + bki ), j = . . . , li , i = 1, . . . , n, are linearly independent. Hence P1, n n dim Λk ≥ i=1 li when k ≥ k0 . It follows that dim Λk = i=1 li when k ≥ k0 .  ek , N bk , e By Lemma 3.7, we can define an equivalent norm on Λk (and Nik , N i i Λk as well) by v u n li uX X |||h||| = t xi,2 j , i=1 j=1

Pi where h = i=1 lj=1 xi, j (χk ei, j )(· + bki ) ∈ Λk . In the left part of this section and the next section, we use ||| · ||| as ek , N bk , e the norm of Λk (and Nik , N i i Λk as well). Pn

Lemma 3.8. There exist k0 > 0, δ0 > 0 and τ0 > 0 which are independent of k such that when k ≥ k0 , the following five statements hold: (1) There exists a C 1 mapping wk : BΛk (0, δ0 ) → BΠk (0, τ0 ) ek ∇ Jk (u k + wk (h) + h) = 0, where T ek : E k → Πk is the such that wk (0) = 0 and for any h ∈ BΛk (0, δ0 ), T orthogonal projection. (2) There exists a C 1 mapping w bki : B Nbk (0, δ0 ) → B( Nbk )⊥ (0, τ0 ) i

= 0 and P( Nbk )⊥ ∇ Jk (b u ik + w bki (h) + h) = 0 for any h ∈ B Nbk (0, δ0 ), i = 1, 2, . . . , n, where i i bk )⊥ is the orthogonal projection. : Ek → ( N i

such that P( Nbk )⊥ i

i

w bki (0)

(3) There exists a C 1 mapping ωi : B Ni (0, δ0 ) → B(Ni )⊥ (0, τ0 ) such that ωi (0) = 0 and P(Ni )⊥ ∇ J (u i + ωi (h) + h) = 0 for any h ∈ B Ni (0, δ0 ), i = 1, 2, . . . , n, where P(Ni )⊥ : X → (Ni )⊥ is the orthogonal projection. (4) There exists a C 1 mapping wki : B N k (0, δ0 ) → B(N k )⊥ (0, τ0 ) i

i

such that wki (0) = 0 and P(N k )⊥ ∇ Jk (u ik + wki (h) + h) = 0 for any h ∈ B N k (0, δ0 ), i = 1, 2, . . . , n, where i

P(N k )⊥ : E k → (Nik )⊥ is the orthogonal projection.

i

i

(5) There exists a C 1 mapping w ek : BΛ ek (0, δ0 ) → BΠ ek (0, τ0 ) such that w ek (0) = 0 and for any h ∈ BΛ ek (0, δ0 ), ! n X i i PΠ u (· + bk ) + w ek (h) + h = 0, ek ∇ J i=1

ek is the orthogonal projection. where PΠ ek : X → Π Proof. We only give the proof of result (1), since the proofs of the other results are similar. Set Ik (w + h) = ek ∇ Jk (u k + w + h); then Ik (0 + 0) = 0. By Lemma 3.6, we know that there exist k0 > 0 and δ0 > 0 such T ek ∇ Ik |Π = T ek ∇ 2 Jk (u k + w + h)|Π is invertible if kw + hkk ≤ δ0 and there exists η > 0 such that when k ≥ k0 , T k k

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S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

ek ∇ Ik |Π )−1 k ≤ η for any k ≥ k0 . Then by the implicit functional theorem, we get that there exist τ0 > 0 and that k(T k 1 a C mapping wk : BΛk (0, δ0 ) → BΠk (0, τ0 ) such that wk (0) = 0 and Ik (wk (h) + h) = 0.



Remark 3.9. Since Jk is invariant under the action of ZkN , we get that !! ! li li X X i i i k w bk xi, j χk ei, j (· + bk ) = wk xi, j ei, j . j=1

(3.20)

j=1

If f : E → F is a C 1 map between two Banach spaces E and F, we denote the derivative operator of f at u by f 0 (u) and the action of f 0 (u) on v ∈ E is denoted by f 0 (u)v. The proofs of the following two lemmas will be given in the Appendix. Lemma 3.10. For any 1 ≤ i ≤ n, the following two statements hold: (1) As k → ∞, v   ! ! u li li li  

X X X u

i

xi, j χk ei, j − ωi sup w bk xi,2 j ≤ δ0 → 0. :t xi, j ei, j  

j=1 j=1 j=1 H 1 (Q k )

(2) As k → ∞, for any 1 ≤ s ≤ li and 1 ≤ i ≤ n,  ! ! li li 

X

i 0 X

wk ) xi, j χk ei, j (χk ei,s ) − (ωi )0 sup (b xi, j ei, j ei,s 

j=1 j=1 For h ∈ Λk , h =

Pn

i=1

Pli

k j=1 x i, j ei, j ,

we define h i =

Pli

k j=1 x i, j ei, j , i

H 1 (Q k )

v  u li  uX xi,2 j ≤ δ0 → 0. :t  j=1

= 1, . . . , n.

Lemma 3.11. (1) As k → ∞,

! ) ( n n n

X X X

sup wk hi − wki (h i ) : h = h i ∈ BΛk (0, δ0 ) → 0.

i=1 i=1 i=1 k

(2) As k → ∞, for any 1 ≤ j ≤ li , 1 ≤ i ≤ n,

! ( ) n n n

X X

0 X s 0 k k (wk ) (h s )ei, j : h = h s ∈ BΛk (0, δ0 ) → 0. h s ei, j − sup wk

s=1 s=1 s=1 k

4. Critical groups of reduction functions For x = (x1,1 , . . . , x1,l1 , . . . , xn,1 , . . . , xn,ln ) ∈ B

R

Ik (x) = Jk u k + wk

li n X X

! xi, j ei,k j

+

i=1 j=1

where B

Pn R i=1 li

(0, δ0 ) = {x ∈ R

Pn

i=1 li

Pn i=1 li

li n X X

! xi, j ei,k j

,

i=1 j=1

:

qP

n Pli 2 i=1 j=1 x i, j

Denote xi = (xi,1 , . . . , xi,li ) ∈ BRli (0, δ0 ) and ! ! li li X X i i i I (xi ) = J u + ω xi, j ei, j + xi, j ei, j , j=1

(0, δ0 ), we define

< δ0 }.

i = 1, 2, . . . , n.

j=1

By (3.1) and (3.20), Lemmas 3.11 and 3.10, we have the following lemma.

(4.1)

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S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

Lemma 4.1. As k → ∞, kIk (x) −

Pn

i=1

I i (xi )kC 1 (B

R

Pn i=1 li

(0,δ0 /2))

→ 0.

Proof. Let Iki (xi )

= Jk

li X

+ wki

u ik

! xi, j ei,k j

li X

+

j=1

Then by Lemma 3.11, we have

n

X

Iki (xi )

Ik (x) −

1 i=1

! i = 1, 2, . . . , n.

j=1

→ 0,

C (B

,

xi, j ei,k j

n P

Ri=1

li

(4.2)

k → ∞.

(0,δ0 /2))

Let li X

b Iki (xi ) = Jk b u ik + w bki

! xi, j χk ei, j

+

li X

j=1

! xi, j χk ei, j ,

i = 1, 2, . . . , n.

j=1

Then by Lemma 3.10, we get kb Iki − I i kC 1 (B

Rli

→ 0,

(0,δ0 /2))

k → ∞, i = 1, . . . , n.

(4.3)

By the invariance of the ZkN action on Jk and (3.20), we have Iki (xi ) ≡ b Iki (xi ),

i = 1, 2, . . . , n.

(4.4) 

By (4.2)–(4.4) we get the result of this lemma.

By the properties of wk and ωi , we have the following lemma. (0, δ0 /2) is a critical point of Ik , then ! li li n X n X X X xi,0 j ei,k j + xi,0 j ei,k j

Lemma 4.2. (1) If x 0 ∈ B

R

u k + wk

Pn i=1 li

i=1 j=1

i=1 j=1

is a critical point of Jk . n l (0, δ0 /2) is a critical point of I i , then (2) If x 0 ∈ B Pi=1 i R ! li li X X i i 0 u +ω xi, j ei, j + xi,0 j ei, j j=1

j=1

is a critical point of J . n l (0, δ0 /2) is a Proof. We only give the proof of result (1), since the proof of result (2) is similar. If x 0 ∈ B Pi=1 i R critical point of Ik , then for any xs,t ,

0= =

∂ Ik (x 0 ) ∂ xs,t ∇ Jk u k + wk

li n X X

! xi,0 j ei,k j

+

i=1 j=1

+

∇ Jk u k + wk

li n X X i=1 j=1

li n X X

! xi,0 j ei,k j

, wk0

i=1 j=1

! xi,0 j ei,k j

+

li n X X i=1 j=1

li n X X i=1 j=1

!

!

! xi,0 j ei,k j

k es,t k

!

k xi,0 j ei,k j , es,t

. k

(4.5)

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S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

Since wk0 (

Pn

i=1

Pli

0 k k j=1 x i, j ei, j )es,t

∇ Jk u k + wk

∈ Πk , by the result (1) of Lemma 3.8, we get that ! ! ! ! li li li n X n X n X X X X k xi,0 j ei,k j + xi,0 j ei,k j , wk0 xi,0 j ei,k j es,t = 0. i=1 j=1

i=1 j=1

i=1 j=1

By (4.5) and (4.6), we get that for any 1 ≤ t ≤ ls , 1 ≤ s ≤ n, ! ! ! li li n X n X X X 0 k 0 k k ∇ Jk u k + wk xi, j ei, j + xi, j ei, j , es,t = 0. i=1 j=1

i=1 j=1

(4.6)

k

(4.7)

k

By (4.7) and the result (1) of Lemma 3.8, we know that ! li li n X n X X X u k + wk xi,0 j ei,k j + xi,0 j ei,k j i=1 j=1

is a critical point of Jk .

i=1 j=1



Remark 4.3. By the condition (*) and Lemma 4.2, we know that 0 is the unique critical point of I i (xi ) in BRli (0, δ0 ), i = 1, . . . , n. ek ◦ ∇ 2 Jk (u k ) is a bounded, invertible and self-adjoint operator in Hilbert space Πk . By Lemma 3.6, we know that T + − Let Pk (resp. Pk ) be the orthogonal projection from E k into the positive (resp. negative) subspace Πk+ (resp. Πk− ) ek ◦ ∇ 2 Jk (u k ). By Lemma 1 of [9], we have the following lemma: with respect to the spectral decomposition of T Lemma 4.4. For any u ∈ B E k (u k , δ0 /2), u has the unique decomposition u = u k + w + h where w ∈ Πk and h ∈ Λk . There exists a diffeomorphism Ψk : B E k (u k , δ0 /2) → E k which satisfies that Ψ (u k ) = u k such that for any Pn Pli k u = u k + w + i=1 j=1 x i, j ei, j ∈ B E k (u k , δ0 /2), ! ! li li n X n X X X + − k k 2 2 xi, j ei, j + xi, j ei, j . Jk (Ψk (u)) = kPk wkk − kPk wkk + Jk u k + wk i=1 j=1

i=1 j=1 (i )

Remark 4.5. By Lemma 2.8, we know that if k is large enough, then K k k ⊂ B E k (u k , δ0 /2). Let Kk := {x ∈ Pn Pli (i ) k n l (0, δ0 /2) : there exists u ∈ K k such that PΛ (u − u k ) = B Pi=1 i=1 k i k j=1 x i, j ei, j }, where PΛk : E k → Λk is R

n l (0, δ0 /2). By Lemma 2.8, we the orthogonal projection. Then by Lemma 4.2, Kk is the critical set of Ik in B Pi=1 i R know that diam(Kk ) → 0 as k → ∞. Pn bk which satisfies that 0 ≤ m bk ≤ Lemma 4.6. If k is large enough, then there exists integer m i=1 li such that Cm bk (Ik , Kk ) 6= 0.

Proof. Let (W1 , W1− ) and (W2 , W2− ) be the GM-pairs for the functionals Jk (w + h) = kwk2k − khk2k in the unique critical point 0 and Ik in the critical set Kk respectively. Then by [5, Lemma 5.1], we know that (W1 × W2 , (W1− × (i ) W2 ) ∪ (W1 × W2− )) is a GM-pair for the isolated critical set Ψk−1 (K k k ) of the functional Jk ◦ Ψk . Then by Theorem 5.5 of [5], we get that (i )

C∗ (Jk ◦ Ψk , Ψk−1 (K k k )) = C∗ (Jk , 0) ⊗ C∗ (Ik , Kk ).

(4.8)

Since Cq (Jk , 0) = δq,dim Π e − Z2 , by (4.8), we get that k

(i )

−1 k Cq (Ik , Kk ) = Cq+dim Π e − (Jk ◦ Ψk , Ψk (K k )). k

(i )

(4.9)

(i )

Since C∗ (Jk ◦ Ψk , Ψk−1 (K k k )) ∼ = C∗ (Jk , K k k ), by (4.9), we get that (i k ) Cq (Ik , Kk ) ∼ = Cq+dim Π e − (Jk , K k ). k

(4.10)

S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

3085

(i k ) ∼ e − +1. bk = j (k)−dim Π By Lemma 2.17 and (4.10), we have C j (k)−dim Π e − +1 (Ik , Kk ) = C j (k)+1 (Jk , K k ) 6= 0. Let m k k Pn P n Notice thatPIk is a function defined in some subset of R i=1 li , we get that Cq (Ik , Kk ) = 0 if q > i=1 li . Thus n bk ≤ i=1 0≤m li . 

Pn i Lemma i=1 I (x i ) with respect to Pn 4.7.i Let (W, W− ) be Pa GM-pair of the isolated critical point 0 of n l (0, δ0 /2). If k is large enough, then (W, W− ) is also a GM-pair of the isolated −d( i=1 I (xi )) and W ⊂ B i=1 i R critical set Kk of Ik with respect to certain pseudo-gradient vector field of Ik . Proof. Since diam(Kk ) → 0, as k → ∞, we know that there exists r > 0 such that if k is large enough, then n l (0, r ) ⊂ int(W ), the interior of W , and Kk ⊂ B Pn l (0, r/4). Note that B Pi=1 i R R i=1 i ) ( ! n

X

i β := inf d I (xi ) : x ∈ W \ B Pn (0, r/2) > 0. li

i=1 Ri=1

Pn

Define ρ ∈ C 2 (R  1,    ρ(x) =   0,

i=1 li

, R) satisfying

x∈B

n P

Ri=1

x 6∈ B

n P

Ri=1

li

li

(0, r/2) (0, r ),

with 0 ≤ ρ(x) ≤ 1 and a vector field !! n X 3 i ρ(x)d Ik (x) + (1 − ρ(x))d I (xi ) . V (x) = 2 i=1 Choosing 0 <  < β/4, by Lemma 4.1, we know that if k is large enough, then

k

X

i I (xi ) < .

Ik (x) −

1 i=1 C (B

n P

Ri=1

li

(4.11)

(0,δ0 ))

2 We shall Pn prove that kV (x)k ≤ 2kd Ik (x)k and (V (x), d Ik (x)) ≥ kd(Ik (x))k , where (·, ·) denotes the inner product l in R i=1 i . n l (0, r/2), By (4.11), we know that for any x 6∈ B Pi=1 i

R

! n

X

i kd Ik (x)k ≥ d I (xi ) −  ≥ β −  > 3.

i=1 Thus we have (V (x), d Ik (x)) = =

!! ! n X 3 i ρ(x)d Ik (x) + (1 − ρ(x))d I (xi ) , d Ik (x) 2 i=1 ! !! ! n X 3 i d Ik (x) + (1 − ρ(x)) d I (xi ) − d Ik (x) , d Ik (x) 2 i=1

3 (kd Ik (x)k2 − kd Ik (x)k) (by (4.11)) 2  3 1 ≥ kd Ik (x)k2 − kd Ik (x)k2 2 3

≥

= kd Ik (x)k2

(4.12)

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S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

and

! ! n

3

X

i kV (x)k = (1 − ρ(x))d I (xi ) + ρ(x)d Ik (x)

2

i=1 3 (kd Ik (x)k + ) 2 ≤ 2kd Ik (x)k (by (4.12)).

≤

(0, r/2), V (x) = d Ik (x), the verification is trivial. Pn n l (0, r ) ⊂ int(W ). It is not difficult to verify that Notice that V (x) = −d( i=1 I i (xi )) outside a ball B Pi=1 i R (W, W− ) is a GM-pair of Ik (x) with respect to V .  Pn Lemma 4.8. There exist an index i 0 satisfying 1 ≤ i 0 ≤ n and a nonnegative integer m 0 satisfying 0 ≤ m 0 ≤ i=1 li such that Cm 0 (I i0 , 0) 6= 0. Pn i Proof. Let (W, W − ) be a GM-pair of i=1 I (x i ) for the isolated critical point 0 which satisfies that W ⊂ Pn B i=1 li (0, δ0 /4). Then by Lemma 4.7, we know that (W, W − ) is also a GM-pair of Ik for the isolated critical R set Kk if k is large enough. Thus by Lemmas 2.16 and 4.6, we get that ! n X i Cm I (xi ), 0 = Hm (4.13) bk bk (W, W− ) = C m bk (Ik , Kk ) 6= 0. Since for any x ∈ B

R

Pn i=1 li

i=1

By [5, Theorem 5.5], we get that ! n n X O i I (xi ), 0 = C∗ (I i , 0). C∗ i=1

(4.14)

i=1

The result of this lemma follows from (4.13) and (4.14).



5. Proof of Theorem 1.1 If the condition (*) does not hold, then Eq. (1.1) has infinitely many geometrically different solutions and the proof terminates. In the following, we always assume that the condition (*) holds. We denote u i0 by u 0 , where i 0 is the index that appeared in Lemma 4.8. By the condition (*), we know that u 0 is an isolated critical point of J . We will use u 0 as a basic “one-bump” solution to construct multi-bump solutions for Eq. (1.1). N For positive integer m ≥ 2 and bi ∈ PmZ , i = 1, 2, . . . , m, define u bi = u 0 (x − bi ), i = 1, 2, . . . , 2m. For N m k = (b1 , . . . , bm ) ∈ (Z ) , define u k = i=1 u bi and let lk = min{|bi −b j | : i 6= j}. Let N be the kernel of ∇ J (u 0 ) and let N = span{e1 , . . . , el }, where l = dim N and ei , i = 1, . . . , l, satisfy (ei , e j ) X = δi, j . Let Nbi be the kernel of ∇ 2 J (u bi ), i = 1, 2, . . . , m, then Nbi = span{e1 (· − bi ), . . . , el (· − bi )}. Let Nk = span{Nbi : i = 1, 2, . . . , m} and Z k = (Nk )⊥ ⊂ X . By the same argument as for Lemmas 3.8 and 4.2, we have the following two lemmas: Lemma 5.1. There exist δ > 0 and a C 1 mapping ω : BRl (0, δ) → N ⊥ such that P (1) PN ⊥ ∇ J (u 0 + ω(x1 , . . . , xl ) + li=1 xi ei ) = 0 and ω(0) = 0, where PN ⊥ : X → N ⊥ is the orthogonal projection. P P (2) If (x10 , . . . , xl0 ) is a critical point of J (u 0 + ω(x1 , . . . , xl ) + li=1 xi ei ), then u 0 + ω(x1 , . . . , xl ) + li=1 xi ei is a critical point of J . Lemma 5.2. There exist L > 0 and δ > 0 such that if lk > L, then there is a C 1 mapping wk : BRml (0, δ) → Z k satisfying that: Pm Pl (1) PZ k ∇ J (u k + wk (x1,1 , . . . , x1,l , . . . , xm,1 , . . . , xm,l ) + i=1 j=1 x i, j e j (· − bi )) = 0.

3087

S. Chen / Nonlinear Analysis 68 (2008) 3067–3102 0 , . . . , x 0 ) is a critical point of (2) If (x1,1 m,l

J u k + wk (x1,1 , . . . , x1,l , . . . , xm,1 , . . . , xm,l ) +

m X l X

! xi, j e j (· − bi )

i=1 j=1 0 , . . . , x0 , . . . , x0 , . . . , x0 ) + then u k + wk (x1,1 m,l 1,l m,1

Pm Pl i=1

0 j=1 x i, j e j (· − bi )

is a critical point of J .

Pm Pl Let Hk (x1,1 , . . . , xm,l ) = J (u k + wk (x1,1 , . . . , xm,l ) + i=1 j=1 x i, j e j (· − bi )) and H (x 1 , . . . , xl ) = J (u 0 + Pl ω(x1 , . . . , xl ) + j=1 x j e j ). Let x = (x1,1 , . . . , xm,l ) and xi = (xi,1 , . . . , xi,l ), i = 1, 2, . . . , m. Remark 5.3. By the condition (*), we know that u 0 is an isolated critical point of J . Then by Lemma 5.1, we know that 0 is the unique critical point of H in BRl (0, δ). By the same argument as for Lemma 4.1, we have the following lemma: Lemma 5.4. There exists δ > 0 such that as lk → ∞,

m

X

H (xi ) → 0.

Hk (x) −

1 i=1 C (BRml (0,δ))

Note that H (x) = I i0 (x) for x ∈ BRl (0, δ). By Lemma 4.8, we know that Cm 0 (I i0 , 0) 6= 0. Thus we have the following lemma: Lemma 5.5. Cm 0 (H, 0) 6= 0. Pm Nm By Lemma 5.5 and Cmm 0 ( i=1 H (xi ), 0) = i=1 Cm 0 (H (xi ), 0) (see (4.14) for reference), we get the following lemma: Pm Lemma 5.6. Cmm 0 ( i=1 H (xi ), 0) 6= 0. By Lemmas 5.6 and 5.4, we have the following lemma. Lemma 5.7. If lk is large enough, then Hk has at least a critical point x k ∈ BRml (0, δ) which satisfies that x k → 0 as lk → ∞. Pm Proof. PmLet (W, W− ) be a GM-pair of the isolated critical point 0 of i=1 H (xi ) with respect to the gradient field −d( i=1 H (xi )). By Lemma 5.4 and the proof of Lemma 4.7, we know that when lk is large enough, if η is the flow generalized by the following ordinary differential equation: η˙ = V1 (η(x, t)),

η(x, 0) = x,

Pn Pn (x) where V1 (x) = g(x) kVV (x)k , V (x) = 32 (ρ(x)d Hk (x) + (1 − ρ(x))d( i=1 H (xi ))), ρ ∈ C 2 (R i=1 li , R) satisfying 0 ≤ ρ(x) ≤ 1 for any x and ( 1, x ∈ BRml (0, δ/4) ρ(x) = 0, x 6∈ BRml (0, δ/2), n min{dist(x, K ), 1}, if K Hk 6= ∅ and g(x) = 1, if K H = ∅Hk , then (W, W− ) satisfies the following conditions: k

(1) W has the (MVP) property with respect to the flow η. (2) W− is an exit set for W , i.e., for each x0 ∈ W and t1 > 0 such that η(x0 , t1 ) 6∈ W , there exists t0 ∈ [0, t1 ) such that η(x0 , [0, t0 ]) ⊂ W and η(x0 , t0 ) ∈ W− . (3) W− is closed and is a union of a finite number of sub-manifolds that are transverse to the flow η.

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S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

If Hk has no critical point in W , then for any x ∈ BRml (0, δ) g(x) ≥ ι

and

kV (x)k ≥ ι

(5.1)

for some ι > 0. Thus for any x ∈ W , there exists t ≥ 0 such that η(x, t) 6∈ W . In fact, if there exists x ∈ W such that for any t ≥ 0, η(x, t) ∈ W , then by the fact that W is a bounded closed set in finite dimension space Rml , we deduce that there exists a sequence {tn } such that tn → +∞ as n → ∞ and η(x, tn ) converges to some point x0 ∈ W . Then x0 must satisfy that V1 (x0 ) = 0. This contradicts (5.1). Thus for any x ∈ W , tx = inf{t 0 ≥ 0 : η(x, t 0 ) ∈ W− } < +∞. It is easy to verify that tx = 0 for any x ∈ W− . Define H : W × [0, 1] → W , H(x, s) = η(x, stx ). It follows that W− is aPdeformation retract of W . Thus Hq (W, W− ) = 0, ∀q. But (W, W− ) is a GM-pair of the isolated critical point 0 m of i=1 H (xi ); by Lemmas 2.16 and 5.6, we get that ! m X ∼ Hmm (W, W− ) = Cmm H (xi ), 0 6= 0. 0

0

i=1

This is a contradiction. Thus if lk is large enough, then Hk has at least a critical point x k in BRml (0, δ). Finally, by Lemma 5.4 and Remark 5.3, we get that x k → 0 as lk → ∞.  By Lemmas 5.7 and 5.2, we get the following result: P Pl k , . . . , x k )+ m k Theorem 5.8. u k +wk (x1,1 i=1 j=1 x i, j e j (·−bi ) is a critical point of J . Furthermore, as lk → +∞, m,l P P m l k k , . . . , x k ) → 0 and wk (x1,1 i=1 j=1 x i, j e j (· − bi ) → 0. m,l Acknowledgement The author acknowledges the support of NNSF of China (No. 10526041). Appendix In this section, we shall give the proofs of Lemmas 3.1, 3.3, 3.10 and 3.11. The proof of Lemma 3.2 is similar to the proof of Lemma 3.1 and the proof of Lemma 3.4 is similar to the proof of Lemma 3.3. Proof of Lemma 3.1. For convenience, we set vk =

n X

u i (· + bki ) − u k ,

i=1

and by (3.1), we have kvk k H 1 (Q k ) → 0

as k → ∞.

(A.1)

Let 0 Ω,k = {x ∈ Q k : |u k (x)| ≤ 1/},

1 Ω,k = {x ∈ Q k : |u k (x)| > 1/},

and 0 U,k = {x ∈ Q k : |vk (x)| < },

1 U,k = {x ∈ Q k : |vk (x)| ≥ }.

By kvk k H 1 (Q k ) → 0 as k → ∞, we get that for every  > 0, 1 mes(U,k )→0

as k → ∞

(A.2)

and by the fact that {u k } is bounded in H 1 (Q k ), we get that as  → 0, 1 mes(Ω,k ) → 0,

holds uniformly for k ∈ N, where mes(A) denotes the Lebesgue measure of the set A.

(A.3)

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S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

i and U i , i = 0, 1, we have By the definition of vk and Ω,k ,k ! Z Z n X 0 i i 0 u (· + bk ) − f (x, u k ) · |ψk | · |ϕk |dx = | f 0 (x, u k + vk ) − f 0 (x, u k )| · |ψk | · |ϕk |dx f x, Qk Q k i=1 Z Z Z . (A.4) + ≤ + 0 ∩Ω 0 U,k ,k

1 Ω,k

1 U,k

By the condition (f2 ) and the H¨older inequality, we get Z | f 0 (x, u k + vk ) − f 0 (x, u k )| · |ϕk | · |ψk |dx 1 U,k

Z

|u k + vk |

q−2

≤C 1 U,k

Z +C 1 U,k

Z · |ϕk | · |ψk |dx + C

|u k |q−2 · |ϕk | · |ψk |dx + C

Z 1 U,k

1 U,k

|u k + vk | p−2 · |ϕk | · |ψk |dx

|u k | p−2 · |ϕk | · |ψk |dx

and Z 1 U,k

|u k + vk |q−2 · |ϕk | · |ψk |dx Z

≤

!1 r

1r dx

U1

!

Z

q+(q−2)δ 0

· 1 U,k

where q + (q − 2)δ 0 ≤ 2∗ and

1 r

|u k + vk |

+

1 q δ 0 + q−2

+

1 δ0 +

q q−2

Z · 1 U,k

2 q

!1 q

|ϕk |q

Z · 1 U,k

!1 q

|ψk |q

,

= 1. It follows that there exists C10 > 0 which is independent of k, ψk

and ϕk such that Z 1 1 |u k + vk |q−2 |ψk · ϕk |dx ≤ C10 (mes(U,k )) r . 1 U,k

Thus there exists C1 > 0 which is independent of k, ψk and ϕk such that Z 1 1 | f 0 (x, u k + vk ) − f 0 (x, u k )| · |ψk · ϕk |dx ≤ C1 (mes(U,k )) r .

(A.5)

In the same way, there exists C2 > 0 which is independent of k, ψk and ϕk such that Z 1 1 | f 0 (x, u k + vk ) − f 0 (x, u k )| · |ψk · ϕk |dx ≤ C2 (mes(Ω,k )) r .

(A.6)

1 U,k

1 Ω,k

R Choose η ∈ C0∞ (R) which satisfies that 0 ≤ η ≤ 1, |η0 (t)| ≤ 2, R η(t)dt = 1, η ≡ 1 in (− 21 , 12 ) and η ≡ 0 in R \ (−1, 1). Let ηδ (t) = 1δ η( δt ) and Z gδ (x, t) = f s0 (x, s)ηδ (t − s)ds. R

Since f 0 (x, t) is a Caratheodory function, we deduce that for almost all x ∈ R N and for all t ∈ R, lim gδ (x, t) = f 0 (x, t).

δ→0

We shall prove that for every  ∈ (0, 1), the following limit holds uniformly for k ∈ N: Z q lim | f 0 (x, u k + vk ) − gδ (x, u k + vk )| q−2 dx = 0. δ→0 Ω 0 ∩U 0 ,k ,k

(A.7)

(A.8)

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S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

If not, then there exist 0 > 0, η0 > 0 and sequences {δm }, {km } which satisfy that δm → 0 and Z q | f 0 (x, u km + vkm ) − gδm (x, u km + vkm )| q−2 dx > η0 , m = 1, 2, . . . .

(A.9)

Ω0 ,km ∩U0 ,km 0

0

By the condition (f2 ) and the definition of gδm , we know that there exist C, C 0 > 0 which are independent of m such that q

| f 0 (x, u km (x) + vkm (x)) − gδm (x, u km (x) + vkm (x))| q−2 ≤ C 0 |u km (x) + vkm (x)|q ! 1 q , ≤C q + |vk (x)| 0

∀x ∈ Ω10 ,km ∩ U00 ,km . (A.10)

By (A.7), we get that q

lim | f 0 (x, u km (x) + vkm (x)) − gδm (x, u km (x) + vkm (x))| q−2 = 0

a.e.

m→∞

(A.11)

Moreover, by the fact that there exists constant C > 0 which is independent of m such that Z Z q q 0 0 |u km |q ≤ C 0 mes(Ω0 ,km ∩ U0 ,km ) ≤ |u km | ≤ Ω0 ,km ∩U0 ,km 0

0

(A.12)

Qk

we get that there exists constant C > 0 which is independent of m such that mes(Ω00 ,km ∩ U00 ,km ) ≤ C.

(A.13)

By (A.13), we may assume that the limit limm→∞ mes(Ω00 ,km ∩ U00 ,km ) exists. By (A.10) and the Fatou theorem, we get that ! ! Z q 1 q − | f 0 (x, u km (x) + vkm (x)) − gδm (x, u km (x) + vkm (x))| q−2 dx lim inf C q + |vk | 0 Ω0 ,km ∩U0 ,km m→∞ 0 0 ! ! Z q 1 0 q C − | f (x, u km (x) + vkm (x)) − gδm (x, u km (x) + vkm (x))| q−2 dx. ≤ lim inf q + |vk | m→∞ Ω 0 0 0  ,km ∩U ,km 0

0

(A.14) By the fact that limk→∞ C

R Qk

1 0 q lim mes(Ω0 ,km 0 m→∞

|q

= 0 and (A.14), we get that Z q ∩ U00 ,km ) − lim | f 0 (x, u km + vkm ) − gδm (x, u km + vkm )| q−2 dx

|vk

Ω0 ,km ∩U0 ,km m→∞

0 0 Z q 1 ≤ C q lim mes(Ω00 ,km ∩ U00 ,km ) − lim sup | f 0 (x, u km + vkm ) − gδm (x, u km + vkm )| q−2 dx. m→∞ 0 0 0 m→∞ Ω ,k ∩U ,k 0 m 0 m

(A.15) By (A.15) and (A.11), we get that Z q lim sup | f 0 (x, u km + vkm ) − gδm (x, u km + vkm )| q−2 dx = 0.

(A.16)

Ω0 ,km ∩U0 ,km

m→∞

0

0

This contradicts (A.9). Thus (A.8) holds uniformly for k ∈ N. Then by (A.8) and Z | f 0 (x, u k + vk ) − gδ (x, u k + vk )| · |ψk · ϕk |dx 0 ∩U 0 Ω,k ,k

Z

| f (x, u k + vk ) − gδ (x, u k + vk )| 0

≤ 0 ∩U 0 Ω,k ,k

q q−2

! q−2 Z q dx RN

|ψk |q

 1 Z q

RN

|ϕk |q

1 q

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S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

Z

| f (x, u k + vk ) − gδ (x, u k + vk )| 0

≤C 0 ∩U 0 Ω,k ,k

q q−2

! q−2 q

dx

(A.17)

we get that the following limit holds uniformly for k ∈ N, kϕk kk ≤ 1 and kψk kk ≤ 1: Z | f 0 (x, u k + vk ) − gδ (x, u k + vk )| · |ψk · ϕk |dx = 0. lim δ→0 Ω 0 ∩U 0 ,k ,k

(A.18)

By the same argument as for (A.18), we get that the following limit holds uniformly for k ∈ N, kϕk kk ≤ 1 and kψk kk ≤ 1: Z | f 0 (x, u k ) − gδ (x, u k )| · |ψk · ϕk |dx = 0. (A.19) lim δ→0 Ω 0 ∩U 0 ,k ,k

By the definition of gδ (x, t) and the condition (f2 ), we get that for all t ∈ [−2/, 2/], there exists constant C > 0 such that Z ∂gδ ∂ηδ 0 = ≤ C /δ. f (x, s) (A.20) (x, t) (t − s)ds ∂t ∂s R 0 ∩ U0 , By (A.20), we get that for all x ∈ Ω,k ,k

|gδ (x, u k (x) + vk (x)) − gδ (x, u k (x))| ≤

C |vk (x)|. δ

(A.21)

If q − 2 ≤ 1, then there exists constant C() > 0 such that |gδ (x, u k (x) + vk (x)) − gδ (x, u k (x))| ≤

C() C |vk (x)| ≤ |vk (x)|q−2 , δ δ

0 0 ∀x ∈ Ω,k ∩ U,k .

Then Z 0 ∩U 0 Ω,k ,k

C() ≤ δ ≤

|gδ (x, u k + vk ) − gδ (x, u k )| · |ψk · ϕk |dx ≤ ! q−2 q

Z

q

0 ∩U 0 Ω,k ,k

|vk |

!1 q

Z

q

0 ∩U 0 Ω,k ,k

|ϕk |

Z C() |vk |q−2 |ϕk · ψk |dx 0 0 δ Ω,k ∩U,k !1 Z q

0 ∩U 0 Ω,k ,k

|ψk |q

C2 () q−2 kvk k H 1 (Q ) , k δ

(A.22)

where C2 () > 0 is a constant which is independent of ψk , ϕk and k. If q − 2 > 1, then Z Z C |vk | · |ϕk · ψk |dx |gδ (x, u k + vk ) − gδ (x, u k )| · |ψk · ϕk |dx ≤ 0 ∩U 0 0 ∩U 0 δ Ω,k Ω,k ,k ,k !1 Z !1 Z !1 Z 3 3 3 C 3 3 3 |ψk | |ϕk | ≤ |vk | 0 ∩U 0 0 ∩U 0 0 ∩U 0 δ Ω,k Ω,k Ω,k ,k ,k ,k 1 ! Z 3 C 0 () ≤ 2 |vk |3 0 ∩U 0 δ Ω,k ,k ≤

C200 () kvk k H 1 (Q k ) , δ

where C200 () > 0 is a constant which is independent of ψk , ϕk and k.

(A.23)

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By (A.22) and (A.23), we get that the following limit holds uniformly for kϕk kk ≤ 1 and kψk kk ≤ 1: Z lim |gδ (x, u k + vk ) − gδ (x, u k )| · |ψk · ϕk |dx = 0. k→∞ Ω 0 ∩U 0 ,k ,k

(A.24)

By (A.18), (A.19) and (A.24), we get that the following limit holds uniformly for kϕk kk ≤ 1 and kψk kk ≤ 1: Z lim | f 0 (x, u k + vk ) − f 0 (x, u k )| · |ψk · ϕk |dx = 0. (A.25) k→∞ Ω 0 ∩U 0 ,k ,k

Finally, by (A.2), (A.3), (A.5), (A.6) and (A.25), we get that the limit Z lim | f 0 (x, u k + vk ) − f 0 (x, u k )| · |ψk · ϕk |dx = 0 k→∞ Q k



holds uniformly for kϕk kk ≤ 1 and kψk kk ≤ 1.

Proof of Lemma 3.3. Firstly, we shall prove that as k → ∞, the limit ! ! Z n n X X 0 1 i i 1 0 i i 1 0 1 u (· + bk − bk ) − f x, u (· + bk − bk ) − f (x, u ) · |ϕk · ψk | → 0 f x, u − Qk i=2 i=2

(A.26)

holds uniformly for ϕk , ϕk ∈ E k which kϕk kk ≤ 1, kψk kk ≤ 1. Pn satisfy For convenience, we set vk = i=2 u i (· + bki − bk1 ). 1 = {x ∈ Q : |v (x)| ≤ 1/}, Ω 2 = {x ∈ Q : |v (x)| > 1/}, Let Ω,k k k k k ,k U1 = {x ∈ Q k : |u 1 (x)| ≤ 1/},

U2 = {x ∈ Q k : |u 1 (x)| > 1/}.

Then the limit 2 lim mes(Ω,k )=0

(A.27)

→0

holds uniformly for k ∈ N and mes(U2 ) → 0, Note that Z

as  → 0.

(A.28)

| f (x, u + vk ) − f (x, vk ) − f (x, u )| · |ϕk · ψk |dx ≤ 0

1

0

0

1

Qk

Z

Z

Z

+ Ω1 ∩U1

+ Ω2

U2

.

(A.29)

By the same argument as for (A.5) and (A.6), we know that there exist constants r > 0, C1 > 0 and C2 > 0 which are independent of k, ϕk and ψk such that Z 1 2 (A.30) | f 0 (x, u 1 + vk ) − f 0 (x, vk ) − f 0 (x, u 1 )| · |ϕk · ψk |dx ≤ C1 (mes(Ω,k )) r , 2 Ω,k

Z U2

1

| f 0 (x, u 1 + vk ) − f 0 (x, vk ) − f 0 (x, u 1 )| · |ϕk · ψk |dx ≤ C2 (mes(U2 )) r .

(A.31)

By the same argument as for (A.18) and (A.19), we know that for every  ∈ (0, 1), the following limits hold uniformly for k ∈ N, kϕk kk ≤ 1 and kψk kk ≤ 1: Z lim | f 0 (x, u 1 + vk ) − gδ (x, u 1 + vk )| · |ϕk · ψk |dx = 0, (A.32) δ→0 Ω 1 ∩U 1  ,k

Z lim

δ→0 Ω 1 ∩U 1  ,k

Z lim

δ→0 Ω 1 ∩U 1  ,k

| f 0 (x, vk ) − gδ (x, vk )| · |ϕk · ψk |dx = 0,

(A.33)

| f 0 (x, u 1 ) − gδ (x, u 1 )| · |ϕk · ψk |dx = 0.

(A.34)

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S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

1 ∩ U 1 , by the same argument as for (A.21), we deduce that there exists constant M For x ∈ Ω,k ,δ > 0 which  depends only on  and δ such that

|gδ (x, vk (x) + u 1 (x)) − gδ (x, vk (x)) − gδ (x, u 1 (x))| ≤ M,δ |u 1 (x)| + C(|u 1 (x)|q−2 + |u 1 (x)| p−2 ). We shall prove that for any  > 0, as k → ∞, the limit Z |gδ (x, vk + u 1 ) − gδ (x, vk ) − gδ (x, u 1 )| · |ϕk | · |ψk | → 0

(A.35)

(A.36)

1 ∩U 1 Ω,k 

holds uniformly for ϕk , ψk ∈ E k which satisfy kϕk kk ≤ 1 and kψk kk ≤ 1. We distinguish two cases: e,δ > 0 such that Case 1. 0 < q − 2 ≤ 1. In this case, by (A.35), there exists M e,δ |u 1 |q−2 , |gδ (x, vk + u 1 ) − gδ (x, vk ) − gδ (x, u 1 )| ≤ M

1 ∀x ∈ Ω,k ∩ U1 .

Then q

q

e q−2 |u 1 |q , |gδ (x, vk + u 1 ) − gδ (x, vk ) − gδ (x, u 1 )| q−2 ≤ M ,δ

1 ∀x ∈ Ω,k ∩ U1 .

Since as k → ∞, q

|gδ (x, vk + u 1 ) − gδ (x, vk ) − gδ (x, u 1 )| q−2 → 0

a.e.,

by the Lebesgue convergence theorem, we get that as k → ∞, Z q |gδ (x, vk + u 1 ) − gδ (x, vk ) − gδ (x, u 1 )| q−2 → 0. 1 ∩U 1 Ω,k 

It follows that as k → ∞, the limit Z |gδ (x, vk + u 1 ) − gδ (x, vk ) − gδ (x, u 1 )| · |ϕk · ψk |dx 1 ∩U 1 Ω,k 

Z

|gδ (x, vk + u ) − gδ (x, vk ) − gδ (x, u )| 1

≤ 1 ∩U 1 Ω,k 

1

q q−2

! q−2 Z q

q

 1 Z q

q

|ϕk |

1 q

|ψk |

Qk

→0

Qk

holds uniformly for ϕk , ψk ∈ E k which satisfy kϕk kk ≤ 1, kψk kk ≤ 1. b,δ > 0 such that Case 2. q − 2 > 1. In this case, by (A.35), we get that there exists M b,δ |u 1 |. |gδ (x, vk + u 1 ) − gδ (x, vk ) − gδ (x, u 1 )| ≤ M By the Lebesgue convergence theorem, we deduce that, as k → ∞, the limit Z |gδ (x, vk + u 1 ) − gδ (x, vk ) − gδ (x, u 1 )|3 → 0 1 ∩U 1 Ω,k 

holds uniformly for ϕk , ψk ∈ E k which satisfy kϕk kk ≤ 1, kψk kk ≤ 1. Therefore, as k → ∞, the limit Z |gδ (x, vk + u 1 ) − gδ (x, vk ) − gδ (x, u 1 )| · |ϕk · ψk |dx 1 ∩U 1 Ω,k 

Z

|gδ (x, vk + u ) − gδ (x, vk ) − gδ (x, u )| 1

≤ 1 ∩U 1 Ω,k 

1 3

! 1 Z 3

3

 1 Z 3

3

|ϕk | Qk

|ψk |

1 3

→0

Qk

holds uniformly for ϕk , ψk ∈ E k which satisfy kϕk kk ≤ 1, kψk kk ≤ 1. Thus (A.36) holds. By (A.27)–(A.34) and (A.36), we get (A.26). Finally, by an inductive argument, we can get the desired result of this lemma. 

3094

S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

Proof of Lemma 3.10. (1) By Lemma 3.8, we get that for any ϕ ∈ (Nik )⊥ , ! ! Z li li X X 0= ∇ b u ik + w bki xi, j χk ei, j + xi, j χk ei, j · ∇ϕ Qk

j=1

Z + Qk

j=1 li X

V (x) · b u ik + w bki

xi, j χk ei, j

+

j=1

Z

x, b u ik

f

−

!

Qk

li X

+w bki

li X

! xi, j χk ei, j

·ϕ

j=1

! xi, j χk ei, j

+

j=1

!

li X

xi, j χk ei, j ϕ,

(A.37)

j=1

and for any ψ ∈ Ni⊥ , ! ! Z li li X X i i xi, j ei, j + xi, j ei, j · ∇ψ 0= ∇ u +ω RN

j=1

Z

V (x) · u + ω i

+ RN

j=1 i

li X

! xi, j ei, j

+

li X

j=1

Z

x, u + ω i

f

− RN

li X

i

! xi, j ei, j

+

2 j=1 x i, j

·ψ

li X

! xi, j ei, j ψ.

(A.38)

→ 0,

(A.39)

j=1

Note that the limit

! ! li li

X

i X

i xi, j ei, j − χk ω xi, j ei, j

ω

j=1 j=1 Pli

xi, j ei, j

j=1

j=1

holds uniformly for

!

k→∞

H 1 (R N )

≤ δ02 and

kei, j − χk ei, j k H 1 (R N ) → 0,

ku i − b u ik kk → 0,

(A.40)

k → ∞.

By (A.38)–(A.40) and Lemma 3.2, we get that as k → ∞, ! ! Z li li X X i i o(1) = ∇ b u + χk ω xi, j ei, j · ∇ψ xi, j ei, j + χk Qk

j=1

Z

V (x) · b u + χk ω i

+ Qk

i

! + χk

xi, j ei, j

j=1

Z

x, b u + χk ω i

f

−

j=1 li X

Qk

i

li X

! ·ψ

xi, j ei, j

j=1

! xi, j ei, j

li X

+ χk

j=1

li X

! xi, j ei, j ψ

(A.41)

j=1

Pi holds uniformly for ψ ∈ Ni⊥ satisfying kψk ≤ 1 and lj=1 xi,2 j ≤ δ02 . It is easy to verify that the limit ! ! li X i χk ω xi, j ei, j , χk ei, j → 0, k → ∞ j=1

holds uniformly for

(A.42)

k

Pli

2 j=1 x i, j bk )⊥ ; (N i

≤ δ02 , since (ωi (

Pli

j=1 x i, j ei, j ), ei, j )

= 0. Recall that P( Nbk )⊥ is the orthogonal

projection from E k into by (A.42) and (A.39), we get that the limit

! !! li li

X

i X

i xi, j ei, j − P( Nbk )⊥ χk ω xi, j ei, j → 0, k → ∞

ω

i

j=1 j=1 H 1 (R N )

i

(A.43)

3095

S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

Pli

≤ δ02 . By (A.43) and (A.41), we get that as k → ∞, !! ! li li X X i i ∇ b u + P( Nbk )⊥ χk ω xi, j ei, j + χk xi, j ei, j · ∇ψ

holds uniformly for Z o(1) =

2 j=1 x i, j

i

Qk

Z

j=1

j=1

V (x) · b u + P( Nbk )⊥ χk ω i

+ Qk

x, b u + P( Nbk )⊥ χk ω

li X

i

i

Qk

+ χk

xi, j ei, j

li X

j=1

i

f

−

i

i

Z

!!

li X

! xi, j ei, j

·ψ

j=1

!! + χk

xi, j ei, j

li X

j=1

! xi, j ei, j ψ.

(A.44)

j=1

Pi Pi xi, j ei, j )) and let ψ ∈ H 1 (R N ) be an extension of ϕ. We Choose ϕ = w bki ( lj=1 xi, j χk ei, j ) − P( Nbk )⊥ (χk ωi ( lj=1 i bk )⊥ . Then subtracting (A.37) from (A.44), by the mean value theorem, we get that have ϕ ∈ ( N i



P( Nbk )⊥ (∇ 2 Jk (u i,k,t ))ϕ, ϕ



i

holds uniformly for u i,k,t = b u ik

Pli

2 j=1 x i, j

→ 0,

k

(A.45)

k→∞

≤ δ02 , where li X

+ (1 − t)b wki

! xi, j χk ei, j

+ t P( Nbk )⊥ χk ω

li X

i

i

j=1

!! xi, j ei, j

+

j=1

li X

xi, j χk ei, j

j=1

and t ∈ [0, 1]. By Lemma 3.6, we know that the operator P( Nbk )⊥ (∇ 2 Jk (u i,k,t )|( Nbk )⊥ ) is invertible and there exists i i constant η > 0 such that k(P( Nbk )⊥ (∇ 2 Jk (u i,k,t )|( Nbk )⊥ ))−1 k ≤ η. i

(A.46)

i

By (A.45) and (A.46), we get that kϕkk → 0,

k→∞ (A.47) Pli holds uniformly for j=1 xi,2 j ≤ δ02 . Thus the result (1) of this lemma follows from (A.47) and (A.43) directly. (2) Differentiating the equalities (A.37) and (A.38) for the variable xi,s , we get that for any ϕ ∈ (Nik )⊥ , ! ! Z li X i 0 0= ∇ (b wk ) xi, j χk ei, j χk ei,s + χk ei,s · ∇ϕ Qk

j=1

Z

V (x) ·

+ Qk

f

0

Qk

!

xi, j χk ei, j χk ei,s + χk ei,s

x, b u ik

+w bki

!

li X

xi, j χk ei, j

+

j=1

Z f

−

!

·ϕ

j=1

Z −

li X

(b wki )0

0

Qk

x, b u ik

+w bki

li X

! xi, j χk ei, j

·

(b wki )0

j=1

!

li X

xi, j χk ei, j

+

j=1

li X

li X

!

!

xi, j χk ei, j χk ei,s

·ϕ

j=1

! xi, j χk ei, j

· (χk ei,s ) · ϕ

(A.48)

j=1

and for any ψ ∈ Ni⊥ , Z

∇ (ω )

i 0

0= RN

li X

!

!

xi, j ei, j ei,s + ei,s

Z · ∇ψ + RN

j=1

Z f

− RN

0

x, u + ω i

i

li X j=1

! xi, j ei, j

+

li X j=1

V (x) · (ω )

i 0

! xi, j ei, j ei,s + ei,s

j=1

! xi, j ei, j

li X

· (ω )

i 0

li X j=1

! xi, j ei, j ei,s

! ·ψ

! ·ψ

3096

S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

Z f

−

0

RN

x, u + ω i

!

li X

i

xi, j ei, j

+

!

li X

j=1

Note that the limit

! ! li li

X

i 0 X

i 0 xi, j ei, j ei,s − χk (ω ) xi, j ei, j ei,s

(ω )

j=1 j=1 holds uniformly for as k → ∞, Z f

Pli

2 j=1 x i, j

x, u + ω i

0

RN

! xi, j ei, j

+

j=1

Z f

=

x, b u ik

0

Qk

(A.49)

→ 0,

(A.50)

k→∞

H 1 (R N )

≤ δ02 . By (A.39), (A.40) and (A.50), the result (1) of Lemmas 3.10 and 3.2, we get that

li X

i

· ei,s · ψ.

xi, j ei, j

j=1

li X

!

li X

· (ω )

i 0

xi, j ei, j

j=1 li X

+w bki

!

! ·ψ

xi, j ei, j ei,s

j=1

! xi, j χk ei, j

+

li X

j=1

!

li X

· χk (ω )

xi, j χk ei, j

i 0

!

! xi, j ei, j ei,s

· ψ + o(1) (A.51)

j=1

j=1

and Z f

x, u + ω i

0

RN

li X

i

! xi, j ei, j

j=1

Z f

=

x, b u ik

0

Qk

+

li X

! xi, j ei, j

· ei,s · ψ

j=1 li X

+w bki

! xi, j χk ei, j

+

li X

j=1

! · (χk ei,s ) · ψ + o(1).

xi, j χk ei, j

(A.52)

j=1

By (A.39), (A.40) and (A.48)–(A.52), we can get that ! ! Z li X i 0 ∇ (b wk ) xi, j χk ei, j χk ei,s + χk ei,s · ∇ϕ Qk

j=1

Z

V (x) ·

+ Qk

f

0

Qk

Z f

=

!

!

·ϕ

xi, j χk ei, j χk ei,s + χk ei,s

j=1

Z −

li X

(b wki )0 +w bki

x, b u ik

li X

! xi, j χk ei, j

+

j=1

x, b u ik

0

Qk

li X

+w bki

li X

! xi, j χk ei, j

·

li X

(b wki )0

j=1

! xi, j χk ei, j

+

li X

j=1

!

!

xi, j χk ei, j χk ei,s

·ϕ

j=1

! xi, j χk ei, j

· (χk ei,s ) · ϕ

(A.53)

j=1

and Z

li X

∇ χk (ω )

i 0

Qk

!

!

xi, j ei, j ei,s + χk ei,s

Z · ∇ψ + Qk

j=1

Z f

− Qk

Z f

= Qk

0

0

i

x, b u

+w bki

li X

! xi, j χk ei, j

+

j=1 i

x, b u

V (x) · χk (ω )

i 0

+w bki

li X j=1

+ χk

li X j=1

!

!

xi, j ei, j ei,s + χk ei,s

·ψ

j=1

! xi, j χk ei, j

j=1

! xi, j χk ei, j

li X

li X

· χk (ω )

i 0

li X

! xi, j ei, j ei,s

! ·ψ

j=1

! xi, j ei, j

· χk ei,s · ψ + o(1).

(A.54)

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S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

Since ((ωi )0 (

Pli

j=1 x i, j ei, j )ei,s , ei,t ) li X

(χk ω )

i 0

= 0, we get that

!

!

xi, j ei, j ei,s , χk ei,t

= o(1),

(A.55)

k→∞

j=1

holds uniformly for

Pli

≤ δ02 . By (A.55) and (A.50), we get that the limit

2 j=1 x i, j

! ! ! li li

X

i 0 X

i 0 xi, j ei, j ei,s − P( Nbk )⊥ χk (ω ) xi, j ei, j ei,s

(ω ) i

j=1 j=1 holds uniformly for

Pli

P( Nbk )⊥

∇

i

Z

≤ δ02 . By (A.54) and (A.56), we get that ! ! ! li X i 0 xi, j ei, j ei,s + χk ei,s · ∇ψ χk (ω ) j=1

V (x) ·

+

P( Nbk )⊥ χk (ω ) i

Z f Qk

Z f

=

0

i

x, b u

xi, j ei, j ei,s

i

li X

+w bki

x, b u

Qk

+w bki

+ χk ei,s

·ψ

j=1

! xi, j χk ei, j

+

j=1

0

!

!

!

li X

i 0

Qk

−

(A.56)

k→∞

2 j=1 x i, j

Z Qk

→ 0, H 1 (R N )

li X

li X

! xi, j χk ei, j

j=1

! xi, j χk ei, j

+ χk

j=1

li X

·

P( Nbk )⊥ χk (ω )

i 0

i

xi, j ei, j ei,s

!! ·ψ

! xi, j ei, j

· χk ei,s · ψ + o(1).

(A.57)

j=1

Pli

of ϕ. We have ϕ ∈   P( Nbk )⊥ (∇ 2 Jk (u i,k ))ϕ, ϕ → 0, k

i

!

j=1

Pli i 0 bk )⊥ (χk (ω ) ( j=1 x i, j χk ei, j )χk ei,s − P( N j=1 x i, j ei, j )ei,s ) and let ψ i k ⊥ b ( Ni ) . Then subtracting (A.53) from (A.57), we get that

Choose ϕ = (b wki )0 (

li X

∈ H 1 (R N ) be an extension

(A.58)

k→∞

Pi Pi Pi xi, j ei, j . By Lemma 3.6, we holds uniformly for lj=1 xi,2 j ≤ δ02 , where u i,k = b ui + w bki ( lj=1 xi, j χk ei, j ) + χk lj=1 2 know that the operator P( Nbk )⊥ (∇ Jk (u i,k )|( Nbk )⊥ ) is invertible and there exists constant M > 0 such that i

i

k(P( Nbk )⊥ (∇ Jk (u i,k )|( Nbk )⊥ )) 2

i

−1

i

(A.59)

k ≤ M.

By (A.58) and (A.59), we get that kϕkk → 0,

k→∞ (A.60) Pli holds uniformly for j=1 xi,2 j ≤ δ02 . Thus the result (2) of this lemma follows from (A.60) and (A.56) directly.  ek (wi (h i )), i = 1, 2, . . . , n. By (3.20) and the fact that Proof of Lemma 3.11. (1) Let θki (h i ) = wki (h i ) − T k j i |bk − bk | → ∞ as k → ∞ for i 6= j, we get that ! ! !! ! li li X X i k k i i k wk xi,s ei,s , e j,t = w bk xi,s χk ei, j (· + bk ), e j,t s=1

s=1

k

w bki

=

li X

k

!! xi,s χk ei, j

! j , (χk e j,t )(· + bk

s=1

− bki ) k



0, i = j = o(1), as k → ∞, i 6= j.

(A.61)

3098

S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

By (A.61), we get that the limit kθki (h i )kk → 0

as k → ∞

(A.62)

holds uniformly for h i satisfying |||h i ||| ≤ δ0 . Thus by the result (1) of Lemma 3.8, we know that as k → ∞, the limit ek (∇ Jk (u i + T ek (wi (h i )) + h i )) → 0 T k k Pn u ik k H 1 (Q k ) = 0, holds uniformly for h i satisfying |||h i ||| ≤ δ0 . By Lemma 3.4 and the fact that limk→∞ ku k − i=1 we get that as k → ∞, the limit ! Z n n n X X X i i i ek (w (h i )) + e T h − f (x, u + T (w (h )) + h ) f x, u k + i k i · |ϕk | → 0 k k k i Qk i=1 i=1 i=1 Pn holds uniformly for h = i=1 h i satisfying |||h||| ≤ δ0 and ϕk ∈ E k satisfying kϕk kk ≤ 1. Thus we deduce that as k → ∞, the limit !! n n X X ek (wi (h i )) + ek ∇ Jk u k + →0 (A.63) T hi T k

i=1

holds uniformly for h = we get that as k → ∞,

i=1

Pn

i=1 h i

ek (∇ Jk (u k + wk ( satisfying |||h||| ≤ δ0 . Since T

n X

!

ek ∇ Jk u k + wk T

hi

+

n X

i=1

(Z

1

=

!! ek ∇ Jk u k + −T

hi

n X

ek ∇ Jk u k + (1 − t)wk T

0

×

! +t

hi

i=1 n X

ek (wi (h i )) − wk T k

i=1

i=1 h i ) +

ek (wi (h i )) + T k

i=1

i=1

2

n X

n X

Pn

n X

ek (wi (h i )) + T k

i=1

n X

Pn

i=1 h i ))

= 0, by (A.63),

!! hi

i=1 n X

!!

) dt

hi

i=1

!! hi

i=1

(A.64)

= o(1) holds uniformly for h ∈ BΛk (0, δ0 ). By Lemma 3.6, we know that when k ≥ k0 ,

! !! !−1

Z 1 n n n X X X

i

≤η

ek (ω (h i )) + ek ∇ Jk u k + (1 − t)wk dt hi + t T hi T k



0 i=1 i=1 i=1 Pn holds uniformly for h = i=1 BΛk (0, δ0 ). Thus we get that as k → ∞,

( ! ) n n n

X X X

i e hi − Tk (wk (h i )) : h = h i ∈ BΛk (0, δ0 ) → 0. sup wk

i=1 i=1 i=1

(A.65)

k

The result (1) of this lemma follows from (A.65) and (A.62) directly. (2) By Lemma 3.8, we know that for any ϕ ∈ Πk , ! ! ! n n X X e 0 = Tk ∇ Jk u k + wk hs + hs , ϕ s=1

Z

∇ u k + wk

= Qk

n X

s=1

! hs

+

s=1

Z f

− Qk

x, u k + wk

n X

k

! hs

Z

n X s=1

! hs

+

V (x) · u k + wk

· ∇ϕ + Qk

s=1 n X s=1

! hs

· ϕ.

n X s=1

! hs

+

n X s=1

! hs

·ϕ

3099

S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

Differentiating the above equality for the variable xi, j , we get that ! ! ! ! Z Z n n X X 0 0 k k k k ∇ wk h s ei, j + ei, j · ∇ϕ + V (x) · wk h s ei, j + ei, j · ϕ 0= Qk

Qk

s=1

Z f

−

x, u k + wk

0

!

n X

Qk

hs

+

n X

s=1

Z f

− Qk

hs

·

n X

wk0

s=1

!

n X

x, u k + wk

0

s=1

!

hs

+

! ·ϕ

ei,k j

hs

s=1

n X

s=1

!

! · ei,k j · ϕ.

hs

(A.66)

s=1

Since ϕ ∈ Πk , we have (ei,k j , ϕ)k = 0. Thus by (A.66), we get that ! ! ! ! Z Z n n X X 0 k 0 k ∇ wk h s ei, j · ∇ϕ + V (x) · wk h s ei, j · ϕ Qk

Qk

s=1

Z

n X

f 0 x, u k + wk

− Qk

Z

hs

n X

+

s=1

f 0 x, u k + wk

=

s=1

!

Qk

n X

!

n X

· wk0

hs

hs

+

s=1

n X

! ·ϕ

h s ei,k j

s=1

s=1

!

!

! · ei,k j · ϕ.

hs

(A.67)

s=1

Ps k )ek = 0 if s 6= i, we know that for any ϕ ∈ Π , By the same argument and noting that (wks )0 ( lt=1 xs,t es,t k i, j ! ! ! ! Z Z ls ls X X k k ∇ (wks )0 xs,t es,t ei,k j · ∇ϕ + V (x) · (wks )0 xs,t es,t ei,k j · ϕ Qk

Qk

t=1

Z f

−

x, u sk

0

Qk

= δs,i

ls X

+ wks

! k xs,t es,t

+

f

0

Qk

x, u sk

+ wks

! k xs,t es,t

ls X

(wks )0

·

t=1

t=1

Z

t=1

ls X

ls X

! k xs,t es,t

! ·ϕ

ei,k j

t=1

ls X

+

! k xs,t es,t

! · ei,k j · ϕ.

k xs,t es,t

(A.68)

t=1

t=1

By Lemma 3.4, we get that as k → ∞, the following two equalities: ! ! Z ls ls n n X X X X 0 s s k k f x, uk + wk xs,t es,t + xs,t es,t · ei,k j · ϕ Qk

=

s=1 n Z X Qk

s=1

s=1

t=1

t=1

f 0 x, u sk + wks

ls X

! k xs,t es,t

+

ls X

! · ei,k j · ϕ + o(1)

k xs,t es,t

(A.69)

t=1

t=1

and Z f

0

x,

Qk

=

n X s=1

n Z X s=1

Qk

u sk

+

n X

ls X

wks

s=1

! k xs,t es,t

+

f 0 x, u sk + wks

! k xs,t es,t

·

(wks )0

t=1

t=1 ls X

ls X

! k xs,t es,t

+

ls X

ls X

! ei,k j

·ϕ

t=1

! k xs,t es,t

· (wks )0

ls X

t=1

t=1

! k xs,t es,t

! k xs,t es,t ei,k j

! · ϕ + o(1)

(A.70)

t=1

hold uniformly for h ∈ BΛk (0, δ0 ) and ϕ ∈ Πk satisfying kϕk ≤ 1. Furthermore, if s 6= i, then by (3.20) and the fact that |bki − bks | → ∞ as k → ∞, we get that the following two limits: ! ! ! ! Z li ls ls X X X 0 s s k k i 0 k k lim f x, u k + wk xs,t es,t + xs,t es,t · (wk ) xi,t ei,t ei, j · ϕ = 0 (A.71) k→∞ Q k

t=1

t=1

t=1

3100

S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

and Z lim

k→∞ Q k

f

! k xs,t es,t

+

ls X

! · ei,k j · ϕ = 0

k xs,t es,t

(A.72)

t=1

t=1

Pls

≤ δ02 . By (A.69)–(A.72), we get that as k → ∞, the following two equalities: ! n n n X X X s s x, uk + wk (h s ) + h s · ((wki )0 (h i )ei,k j ) · ϕ

hold uniformly for Z

ls X

+ wks

x, u sk

0

f0 Qk

2 t=1 x s,t

s=1

Z

s=1

s=1

f 0 (x, u ik + wki (h i ) + h i ) · ((wki )0 (h i )ei,k j ) · ϕ + o(1)

= Qk

(A.73)

and Z f

0

x,

Qk

n X

u sk

+

n X

s=1

!

n X

wks (h s ) +

s=1

hs

Z

·ϕ =

· ei,k j

Qk

s=1

f 0 (x, u ik + wki (h i ) + h i ) · ei,k j · ϕ + o(1)

(A.74)

Pn hold uniformly for s=1 h s ∈ BΛk (0, δ0 ) as k → ∞. By (A.68), (A.73) and (A.74) and the fact that P s k )ek = 0 if s 6= i, we get that (wks )0 ( lt=1 xs,t es,t i, j n X

Z ∇ Qk

(wks )0

ls X

s=1

!

Z

ei,k j

li X

(wki )0

Qk

∇ Qk

! k xi,t ei,t

!

f

=

x, u ik

0

Qk

li X

+ wki

Qk

! +

t=1

Z f

+

0

Qk

Z f

=

x, u ik

x,

Qk

n X

u sk

+

s=1

Z f

+

0

x,

Qk

n X

! k xi,t ei,t

u sk

+

s=1

! ·ϕ

ei,k j

t=1 li X

(wki )0

! k xi,t ei,t

! ei,k j

·ϕ

t=1

· ei,k j · ϕ

k xi,t ei,t

+

wks

!

li X

k xi,t ei,t

ls X

! k xs,t es,t

s=1

! ei,k j

· ϕ.

+

ls X

! k xs,t ei,t

· ei,k j · ϕ

t=1

ls X

wks

! k xs,t es,t

t=1

t=1

n X

·

ls X

(wks )0

t=1

s=1

n X

! k xs,t es,t

!

li X

t=1

0

ls X

t=1

li X

+ wki

V (x) ·

· ∇ϕ +

k xi,t ei,t

(wks )0

s=1

Z

ei,k j

t=1

Z

n X

V (x) ·

· ∇ϕ +

t=1

Z =

! k xs,t es,t

! k xs,t es,t

+

ls X

! k xs,t es,t

·

t=1

t=1

n X

(wks )0

s=1

ls X

! k xs,t es,t

(A.75) Pn

s s=1 u k kk

Hence by (A.75), limk→∞ ku k − n X

∇ Qk

(wks )0

s=1

Z

ls X

!

Qk

holds uniformly for

n X

Qk

!

n X s=1

Pn

s=1 h s

hs

n X

hs

+

!

n X

s=1

f 0 x, u k + wk

=

= 0 and the result (1) of this lemma, we have that ! ! Z ls n X X k · ∇ϕ + V (x) · (wks )0 xs,t es,t ei,k j · ϕ

t=1

Qk

Z

!

k xs,t es,t ei,k j

f 0 x, u k + wk

−

·ϕ

t=1

+ o(1).

Z

! ei,k j

! hs

s=1

+

·

t=1

s=1 n X

! (wks )0 (h s ) · ϕ

s=1

! hs

· ei,k j · ϕ + o(1)

s=1

∈ BΛk (0, δ0 ) as k → ∞.

(A.76)

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S. Chen / Nonlinear Analysis 68 (2008) 3067–3102

Subtracting (A.67) from (A.76), we get that for any ϕ ∈ Πk , ! !! ! !! ! n n n n X X X X s 0 k 2 0 k ek ∇ Jk u k + wk T hs + hs wk h l ei, j − (wk ) (h s )ei, j ,ϕ s=1

= o(1),

s=1

l=1

s=1

k

as k → ∞.

(A.77)

By (3.20), we deduce that as k → ∞, for any 1 ≤ ν ≤ lt and 1 ≤ t ≤ n, we have that ! n X s 0 k k (wk ) (h s )ei, j , et,ν → 0 s=1

(A.78)

k

holds uniformly for h ∈ BΛk (0, δ0 ). Thus as k → ∞, the limit

! n n

X

X

s 0 k s 0 k e (wk ) (h s )ei, j → 0

(wk ) (h s )ei, j − Tk

s=1 s=1

(A.79)

k

holds uniformly for h ∈ BΛk (0, δ0 ). By (A.79) and (A.77), we get that ! !! ! ! !! n n n n X X X X s 0 k k 0 2 ek (w ) (h s )e ek ∇ Jk u k + wk h s ei, j − T ,ϕ wk hs + hs T k i, j s=1

s=1

s=1

s=1

k

as k → ∞. (A.80) Pn Pn k s k 0 e Choose ϕ = s=1 h s )ei, j − ( s=1 Tk (wk ) (h s )ei, j )) in (A.80); by Lemma 3.6, we deduce that as k → ∞,

! ! ! n n

X

0 X s 0 k k ek (wk ) (h s )ei, j → 0 (A.81) h l ei, j − T

wk

s=1 l=1 = o(1),

(wk0 (

k

holds uniformly for h ∈ BΛk (0, δ0 ). Thus the result (2) of this lemma follows from (A.81) and (A.79) directly.



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