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Morse theory for strongly indefinite functional
Nonlinear Analysis 48 (2002) 831 – 851
www.elsevier.com/locate/na
Morse theory for strongly inde"nite functional Y.X. Guoa; ∗ , J.Q. Liub a Institute
of Systems Science, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China b Institute of Mathematics, Peking University, Beijing 100871, People’s Republic of China Received 12 October 1999
Keywords: Morse theory; Inde"nite functional; Critical groups; Morse inequalities; Dynamically isolated set
1. Introduction The classical Morse theory has proved to be very useful in nonlinear analysis, in proving the existence of solutions and estimating the number of solutions to the differential equations. However, because only nondegenerate functional, or nondegenerate critical points, were well studied, and the nondegeneracy of a solution to a partial differential equation occurring in practice is di9cult to verify in advance, sometimes it is even not true. So, the applications of the classical Morse theory to partial di;erential equations were very limited. To extend the applications of classical Morse theory, there are two theories in front of us: one is Gromoll–Meyer theory on isolated critical point which may be degenerate (see [4,6]); the other is Conley’s Morse index theory on isolated invariant sets (see [7]). However, since the Conley index is homotopy class, it is di9cult to "gure out. In this sense, Gromoll–Meyer theory is easier than Conley index theory, and among other things, Gromoll–Meyer theory is su9cient for many applications (see [3]). Recently, in their paper [5], K.C. Chang and N. Ghoussoub, in a variational setting, investigated the connection between the two theories, more precisely, they studied the relationship between the Gromoll–Meyer pair of a dynamically isolated critical set and the Conley index pair of its isolated invariant neighborhoods, and showed that ∗
Corresponding author. E-mail address: [email protected] (Y.X. Guo).
0362-546X/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 0 ) 0 0 2 1 8 - 2
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the Gromoll–Meyer pairs are Conley index pairs; and Conley index coincides with the critical groups. That is the information given by the critical groups of such a dynamically isolated critical set is equivalent to which given by the Conley index of its isolated invariant neighborhoods. Thus, in many cases, we can use Gromoll–Meyer theory only to resolve problem, and the proof is also relatively easier. However, we consider the following functional G(x) = 12 x+ 2 − 12 x− 2 (functional of such type is called strongly inde"nite) de"ned on a Hilbert space E = E + + E − , where E + ; E − are orthogonal decomposition of E, and both of them are in"nite-dimensional subspace. Clearly, is the only critical point of G. According to Morse theory Cq (G; ) = ; q = 0; 1; 2; : : : : This example explains why the previous de"nition for critical group is of little use for strongly inde"nite functional. The purpose of our paper is to construct a Morse theory for inde"nite functional. Here a strongly inde"nite functional is a C 1 map which is neither bounded from above nor from below, not even model a "nite-dimensional subspace. In the study of existence theory for systems of elliptic equations and of periodic solutions of Hamiltonian systems of ordinary di;erential equations, the common feature is that these equations can be derived from variational problems which are inde"nite in a very strong sense, in particular the corresponding functional are strongly inde"nite. To "nd the critical points of strongly inde"nite functional, there are at least two sets of methods that have been developed: (i) Morse theory and its generalizations (Gromoll–Meyer theory, Conley index theory). (ii) Minimax theory. And there have been many papers based on minimax theory (see [1,2,9,12,13,15]), however, all these methods have some disadvantages, because one needs strong hypotheses (such as uniformly bounded, convexity, etc.); in 1992, A. Szukin (see [16]) constructed a di;erent theory for an isolated critical point which turned out to be useful for strongly inde"nite functional; on the other hand, in 1997, Geba [10] presented a new generalization of classical Conley index theory, and gave the de"nition of homotopy Conley index about an isolated neighborhood of an isolated invariant set. Motivated by the ideas of papers [5,10], based on the Gromoll–Meyer theory, we construct a di;erent Morse theory for strongly inde"nite functional on its dynamically isolated set. Our paper is organized as follows: In Section 2, we investigate the properties of Low and vector "eld which we need in developing our theory. In Sections 3–5 we construct a Morse theory for strongly inde"nite functional, including the de"nition of critical groups, Morse-type equalities.
2. Flow and vector eld Let H be a real Hilbert space with an inner product ·; · and norm ·. Let f ∈ C 1 (H; R), ∀x ∈ H , we denote the di;erential of f at x by df(x), x is said to be a critical point of f if df(x) = ; ∀a ∈ R1 , fa = {x ∈ H | f(x) ≤ a} is called the level set; K = {x ∈ H | df(x) = } is called the critical set of f; ∀c ∈ R1 , Kc = {x ∈ K | f(x) = c}, if Kc = ∅, then we say c is a critical value of f; otherwisze c is said to be a regular value. Function f is said to satisfy the Palais–Smale
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(PS for abbreviation) condition, if for any sequence {x n } such that f(x n ) is bounded and df(x n ) → (n → ∞) possesses {x n } has a convergent subsequence. Denition 2.1 (Chang and Ghoussoub [5], Flow). Let M be a metric space, a continuous map : R × M → M is said to be a Low, if it satis"es the following properties: (i) (0; x) = x; ∀x ∈ M (ii) (t1 ; (t2 ; x)) = (t1 + t2 ; x); ∀t1 ; t2 ∈ R; x ∈ M . We say that Low is bounded if it is a bounded continuous map. Recall that the integral curve starting at u of a Locally Lipschitz continuous vector "eld g : X → X on some Banach space X is the maximal C 1 -curve (t; u) de"ned on (t − (u); t + (u)) and satisfying the cauchy problem (t; ˙ u) = g((t; u)); (2.1) (0; u) = u: The existence of such integral curves is well known. If we set D() = {(t; u) ∈ R × X; t − (u) ¡ t ¡ t + (u)}, then D() is open and : D() → X is continuous. The map is called the local Low generated by g. It is natural to ask, when the local Low generated by g is global, that is the Low is well de"ned on R × X . Remark 2.1. According to [14, Chapter 8, Proposition 8:1], if t + (u) ¡ ∞; ([0; t + (u)]; u) is unbounded. In addition, if g satis"es |g(u); u| ≤ 12 cu2 + d;
∀u ∈ X;
(2.2)
for some positive constants c; d ¿ 0, then Low generated by g is well de"ned and bounded on R × X . In this paper, what we are interested is the following vector "eld and Low with special form. Denition 2.2 (Geba et al. [10]). We say that F:H →H is a LS vector "eld, if there exists a completely continuous and locally lipschitz continuous map K : H → H such that F(x) = Ax + K(x);
(2.3)
where A : H → H is a linear bounded self-adjoint operator. Denition 2.3 (Geba et al. [10]). We say that a Low : R × H → H is a LS Low, if (t; x) = etA x + U (t; x);
(2.4)
where U : R × H → H is completely continuous. Theorem 2.1. Suppose that F(x) = Ax + K(x) is a LS vector >eld; and K satis>es condition (2:2). Then the @ow generated by F is a LS @ow.
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Proof. Let be the Low generated by F; then d − A = K(); dt this means that satis"es an inhomogeneous linear equation and therefore it can be t represented as (t; u) = exp(tA)u + W (t; u), where W (t; u) = etA 0 e− A K(( ; u)) d . To see that W is compact, suppose [ ; ] × B ⊂ R × H is bounded. Since is bounded, without loss of generality, we may assume that ([ ; ] × B) ⊂ BR for some R ¿ 0. Where BR denotes a ball with radius R. Therefore, K(([ ; ] × B)) ⊂ K(BR ) ⊂ cl(K(BR )); note that K is compact, it follows that cl(K(BR )) is a compact set. Now let Y = {e−sA z | s ∈ [ ; ]; z ∈ cl(K(BR ))} since the map (s; z) → e−sA z is continuous on compact set [ ; ] × cl(K(BR ), its range Y is compact. Therefore, the closed convex hull Yˆ of Y is also compact. So for any T ¿ 0, T (Yˆ ∪ ) is also compact. Note that z(s) = e−sA K((s; u)) ∈ Y; ∀s ∈ [ ; ]; u ∈ B: t Hence for t ∈ [0; T ]; (1=T ) 0 z(s) ds ∈ Yˆ ; and t S Y= z(s) ds | t ∈ [ ; ]; z(s) ∈ Y ⊂ T (Yˆ ∪ ): 0
It follows that W : R × H → H is compact. We say that continuous map ' : D(' ) → H is a family of local Lows on H indexed by ', if D(' ) is an open subset of R × H × T and ' (where ' (t; x) = (t; x; ')) is a local Low for every ' ∈ T. If D(' ) = R × H × T, then we say that is a family of Lows on H indexed by T. Denition 2.4 (Geba et al. [10]). Let T be a compact metric space. We say that F : H × T → H is a family of LS vector "elds, if there exists a completely continuous and local lipschitz continuous map K : H × T → H such that F(x; ') = Ax + K(x; ');
∀(x; ') ∈ H × T;
(2.5)
where A is a bounded linear self-adjoint operator. Denition 2.5 (Geba et al. [10]). We say that a family of Lows : R × H × T → H is a family of LS Lows, if (t; x; ') = etA x + U (t; x; '); U : R × H × T → H is completely continuous. Similar to De"nition 2.4 and De"nition 2.5, we de"ne
(2.6)
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Denition 2.4 . For any given '0 ∈ T, we say that F˜ : H × ' → H is a family of approximation ('0 ) LS vector "elds, if ˜ ') = Ax + K(x; ') + !(x; ') F(x; satisfying (i) A is a bounded linear self-adjoint operator. (ii) K : H × T → H is a compact and Lipschitz continuous map. (iii) !(x; ') → 0 as ' → '0 , uniformly in x belonging to a bounded set. Denition 2.5 . T; '0 are same as in De"nition 2:4. We say that ˜ : R × H × T → H is a family of approximation ('0 ) LS Lows, if (t; ˜ x; ') = etA x + U (t; x; ') + !(t; x; ') satisfying (i) A is a bounded linear self-adjoint operator. (ii) U : R × H × T → H is compact. (iii) !(t; x; ') → 0 as ' → '0 , uniformly in x belonging to a bounded set and t ¡ T for some T ¿ 0. Use the same arguments as in Theorem 2.1, we can prove Theorem 2.2. Assume that F(x; ') = Ax + K(x; ') is a family of LS vector >elds; and there exist a; b ¿ 0 such that |K(x; '); x| ≤ a|x|2 + b;
∀' ∈ T; x ∈ H
then the @ows generated by F(x; ') is a family of LS @ows. Proof. Similar to Theorem 2.1, Lows ' generated by F' (x) := F(x; ') can be repret sented as (t; u; ') = exp(tA)u + W (t; u; '); where W (t; u; ') = etA 0 e− A K(( ; u); ') d . To see that W is compact, suppose [ ; ] × B × T1 ⊂ R × H × T is bounded. Since ' is bounded for any ' ∈ T, hence ([ ; ] × B × T1 ) ⊂ BR × T2 (for some R ¿ 0), so K(([ ; ] × B × T1 )) ⊂ K(BR × T2 ) ⊂ cl(K(BR × T2 )): Since K is compact, so is cl(K(BR × T2 )). Set Y = {e−sA z | s ∈ [ ; ]; z ∈ cl(K(BR × T2 ))} note that (s; z) → e−sA z is continuous on the compact set [ ; ] × cl(K(BR × T2 )), Y and its closed convex hull Yˆ are also compact. Using the same arguments as in Theorem 2.1, we complete the proof of Theorem 2.2. Theorem 2.3. Let F' be a family of approximation ('0 ) LS vector >elds; then a family of @ows ' generated by F(x; ') is a family of approximation ('0 ) LS @ows. Proof. Suppose that ' is the Low generated by F' , by Theorem 2.2 (t; u; ') = exp(tA)u + w(t; u; ') + !(t; u; ');
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where w(t; u; ') = etA !(t; u; ') = etA
t
0
0
t
e− A K(( ; u); ') d ; e− A !(( ; u); ') d
(2.7)
in the Low, we need only to show !(t; u; ') → 0 as ' → '0 uniformly in u bounded and t ¡ T for some T ¿ 0. In fact, from De"nition 2:5 (iii), this is obviously true. 3. Dynamically isolated critical set and isolating triplet Let H be a Hilbert space, B ⊂ H be a subset, denote by N, (B) the , closed neighborhood of B, and denote by cl(B) the closed closer of B; let be a Low, and f be + a functional, B˜ = t∈R (t; B); B˜ = t∈R+ (t; B) and f−1 [ ; ] = {x | ≤ f(x) ≤ }. We now recall the following key concept due to [5]: Denition 3.1 (Dynamically isolated critical set). A subset S of the critical set K is said to be a dynamically isolated critical set, if there exists a closed neighborhood O of S and regular values ; of functional f, such that ˜ ∩ K ∩ f−1 [ ; ] = S; O ⊂ f−1 [ ; ] cl(O) where O˜ = t∈R (t; O) and is the Low associated with functional f, we shall then say that (O; ; ) is an isolating triplet for S. Obviously, the isolated property of a dynamically isolated critical set is dependent on the Low, we also say it is an isolated critical set associated with some Low. In the following, we will investigate the properties of a dynamically isolated critical set about its nearby Lows. Let S be a dynamically isolated critical set associated with Low '0 , here, the so-called nearby Lows mean that the Lows ' , as ' is su9ciently near to '0 . Lemma 3.1. Suppose f ∈ C 2 (H; R) satisfying the (PS) condition; T is a compact metric space; ' (' ∈ T) is a family of LS @ows generated by F' = A + K' ; S is a dynamically isolated critical set of f; (O; ; ) is one of its isolating triplet. If −df ≡ F'0 ; K'0 satis>es global Lipschitz continuative on any bounded set; and ˜ F' (x) − F'0 (x) → 0 uniformly as to bounded x. Then ∃, ¿ 0; such that cl(O' ) ∩ f−1 [ ; ] (uniformly in ') is bounded; whenever |'−'0 | ¡ ,; where O˜ ' = t∈R ' (t; O). Proof. (1) We will show for ∀0 ¿ 0; ∃, ¿ 0 such that ' (x; t) − '0 (x; t) ¡ 0
whenever |' − '0 | ¡ ,
uniformly holds for any bounded x and t.
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In fact, d' = A' + K' (' ); dt
so
d'0 = A'0 + K'0 ('0 ) dt
d(' − '0 ) = A(' − '0 ) + K' (' ) − K'0 ('0 ) dt = A(' − '0 ) + K' (' ) − K'0 (' ) + K'0 (' ) − K'0 ('0 ) since ' is bounded. Therefore, ∀0 ¿ 0; ∃,1 ¿ 0 such that K' (' ) − K'0 (' ) ¡ 0
for ∀' ∈ T; |' − '0 | ¡ ,1
hence, d(' − '0 ) ≤ c|' − ' | + 0 0 dt and |' − '0 | ≤ ect
0
t
0e−cs ds
noting that, c is bounded and 0 is small enough, (1) is proved. (2) Let S1 = K ∩ f−1 [ ; ], we take S2 as the bounded closed neighborhood of S1 such that O ⊂ S2 . Since f satis"es (PS), there exists T1 ¿ 0, such that cl(O˜ '0 ) ∩ f−1 [ ; ] ⊂ '0 (S2 ; [ − T1 ; T1 ]): Now, we choose f−1 [ ; ] ⊃ S3 ⊃ S˜2 = '0 (S2 ; [ − T1 ; T1 ]), such that d(S˜2 ; @S3 ) ¿ 0. By (PS) condition, there exists 2 ¿ 0 such that F'0 (x) ≥ 22 for ∀x ∈ S3 − S˜2 : Since F' (x) − F'0 (x) → 0
uniformly for bounded x;
there exists ,2 ¿ 0 such that for ∀'; |' − '0 | ¡ ,2 , F' (x) − F'0 (x) ¡ 2; thus F' (x) ≥ 2 for ∀'; |' − '0 | ¡ ,2
and
∀x ∈ S3 − S˜2 :
Now, if ' (x; [0; t]) ⊂ S3 − S˜2 , then t ¡ T2 for some T2 ¿ 0. Hence, there exist , = min(,1 ; ,2 ) ¿ 0, and T = max(T1 ; T2 ), such that ∀' ∈ T, |' − '0 | ¡ ,, cl(O˜ ' ) ∩ f−1 [ ; ] ⊂ ' (S˜2 ; [ − T; T ]). The later is a bounded set. The lemma is proved. Here is the main result of this section. Theorem 3.1. Suppose that f ∈ C 2 (H; R) satis>es (PS); ' (t; x) = (t; x; ') = etA x + U (t; x; ') is a family of LS @ows generated by F' ; the zero eigenvalue of A is isolated in 3(A): Assume '0 is the negative gradient @ow generated by df = A + K'0 and S is a dynamically isolated critical set of f associated with @ow '0 . If K'0 is global Lipschitz continuous on any bounded set and F' (x) − df(x) → 0; uniformly in
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Y.X. Guo, J.Q. Liu / Nonlinear Analysis 48 (2002) 831 – 851
bounded x. Then there exists an isolating triplet (O; ; ) of S and a positive number , ¿ 0 such that ∀' ∈ T; |' − '0 | ¡ , we have cl(O˜ ' ) ∩ K ∩ f−1 [ ; ] = S;
(3.1)
cl(O˜ ' ) ∩ K' ∩ f−1 [ ; ] ⊂ int O;
(3.2)
where K' = {x | F' (x) = 0}. Proof. (1) Let (G; ; ) be one of the isolating triplet of S; then S = cl(G˜ '0 ) ∩ K ∩ f−1 [ ; ]; (3.3) where G˜ '0 = t∈R '0 (t; G). We will show: ∃, ¿ 0 (small enough) and a closed neighborhood G, of S including G such that cl((G˜ , )' ) ∩ K ∩ f−1 [ ; ] = S
for ∀' ∈ T; |' − '0 | ¡ ,:
{Gn }∞ 1
If it is not true, then there exist 1 ⊂ G; Gn = x | d(x; S) ≤ n
and
{'n }∞ 1
(3.4)
satisfying
'n → '0 (n → ∞);
cl((G˜ , )'n ) ∩ K ∩ f−1 [ ; ] = S; that is ∀n ∈ N;
∃yn ∈ cl((G˜ n )'n ) ∩ K ∩ f−1 [ ; ];
and
yn ∈ S;
so yn ∈ Y = cl(G˜ '0 ) ∩ f−1 [ ; ]; now we take a sequence {zn } ∈ (G˜ n )'n such that d(zn ; yn ) → 0 (n → ∞); zn ∈ Y; ∀n. By (PS) condition, {yn } ⊂ K ∩ f−1 [ ; ] has a convergence subsequence denoted by {yn } still. Suppose yn → y (n → ∞), then y ∈ Y . In fact, if y ∈ Y; then y ∈ S. Since G is the closed neighborhood of S, we have {yn } ⊂ G (for n large enough), this is a contradiction. (2) Since 'n is LS Low, which can be written as 'n (x; t) = etA x + U (t; x; 'n ); without loss of generalization, assume zn = 'n (x n ; tn ); tn ¿ 0; x n ∈ Gn and x n → x ∈ S: Note that x n ∈ Y , and zn ∈ Y , there must exist a point on the integral curve starting at x n and ending at zn which belongs to @Y . We suppose wn is the last point in @Y; zn = 'n (wn ; 3n ); x n = 'n (wn ; −sn ), then we claim that 3n ; sn → +∞: If not, 3n is bounded, without loss of generality, we assume that 3n → 30 (n → ∞); thus wn = 'n (zn ; −3n ) → '0 (y; −30 ) = y (note that y ∈ S is a critical point). So, there exists a neighborhood V (y) of y such that {wn } ⊂ V (y) (for n large enough), this is in contradiction with wn ∈ @Y; ∀n. (3) We denote the trajectories starting from x n to zn by 'n (x n zn ) and denote the set of all these trajectories by D. Then Lemma 3.1 implies that D is bounded. (4) We prove that {wn } is compact. By (2), zn = 'n (wn ; 3n ) and x n = 'n (wn ; −sn ) convergence to a critical point in f−1 [ ; ]; moreover 3n ; sn → +∞. Thus for ∀T ¿ 0,
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−1 TA −T ≤t≤T 'n (wn ; 3n ) ⊂ f [ ; ], for n large enough. Therefore un =e wn +U (wn ; T; 'n ) is bounded, let M be the boundary, H =H+ ⊕H0 ⊕H− be the orthogonal decomposition of H according to the positive, negative, zero eigenvalue subspace of A. Since the zero eigenvalue of A is isolated in the spectra set 3(A). Thus for ∀8 ¿ 0, there exists T ¿ 0 such that
etA ≥ 8x;
∀t ≥ T; x ∈ H+ ;
(3.5)
etA ≥ 8x;
∀t ≤ −T; x ∈ H− :
(3.6)
Assume wn = wn+ + wn0 + wn− ∈ H+ ⊕ H0 ⊕ H− ; ∀n. Since H0 is a "nite-dimensional space, so {wn0 } has a convergence subsequence. In the following, without loss of generality, we assume that {wn+ } has no convergence subsequence, then ∃0 ¿ 0
such that wi+ − wj+ ≥ 0;
for i = j;
(3.7)
thus for 8 = 3M=0; ∃T0 ¿ 0, such that (3.5) is true for ∀t ≥ T0 . Therefore, eT0 A wi − eT0 A wj = eT0 A (wi − wj ) ≥ eT0 A (wi+ − wj+ ) ≥
3M wi+ − wj+ ≥ 3M; 0
on the other hand, eT0 A wi − et0 A wj = eT0 A wi + U (T0 ; wi ; 'i ) + U (T0 ; wi ; 'i ) − U (T0 ; wj ; 'j ) + eT0 A wj + U (To ; wj ; 'j ) ≤ 2M + U (To ; wi ; 'i ) − U (T0 ; wj ; 'j ); hence U (T0 ; wi ; 'i ) − U (T0 ; wj ; 'j ) ¿ M;
for i = j
this is a contradiction with the fact that U is compact. Let w be the limit of the convergence subsequence, then w ∈ @Y: At last, we will deal with a contradiction. Since w ∈ @Y; then we have w = '0 (y; t); for ∀y ∈ G; ∀t ∈ R. Otherwise, by the continuous Low, there exists a neighborhood Br (y) ⊂ G of y such that t∈R '0 (Br (y); t) ⊂ Y is the neighborhood of w, this contradicts with w ∈ @Y . So, for 0 ¿ 0 small enough, ∃h ¿ 0 such that f('0 (w; h)) = + 0. Note ∀T ¿ 0, and n large enough, we have f('n (wn ; T )) ≤ ; hence, for T = h; f('n (wn ; h)) ≤ ; let n → ∞, we obtain f('0 (w; h)) ≤ this is a contradiction. Eq. (3.1) is proved. Now we prove (3.2). In fact, by (PS) condition, ∃2 ¿ 0, such that df(x) ≥ 2
∀x ∈ f−1 [ ; ] \ O
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by Lemma 3.1, ∃,1 ¿ 0, such that for all ', |' − '0 | ¡ ,1 , cl(O˜ ' ) ∩ f−1 [ ; ] are included in a bounded set G. So F' (x) − df(x) → 0 uniformly as to x ∈ G: Therefore, ∃,2 ¿ 0, such that 2 F' (x) − df(x) ¡ ; 2
∀x ∈ G
∀'; |' − '0 | ¡ ,2
thus, if |' − '0 | ¡ , = min{,1 ; ,2 }; y' ∈ cl(O˜ ' ) ∩ K' ∩ f−1 [ ; ], and y' ∈ int O, then 2 ≤ df(y' ) ≤ F' (y' ) − df(y' ) ¡ 2=2 this implies that, ∃, ¿ 0, such that cl(O˜ ' ) ∩ K' ∩ f−1 [ ; ] ⊂ int O
for ∀' ∈ T |' − '0 | ¡ ,:
The theorem is proved. Theorem 3.1 shows if S is a dynamically isolated critical set associated with Low '0 (generated by df), then it is also dynamically isolated about to its nearby Lows ' (generated by F' ). Moreover, the zero point set of the vector "eld F' , which includes cl(O˜ ' ) ∩ f−1 [ ; ] will all be in the interior of O, while ' is su9ciently approximating '0 . As to the approximation ('0 )LS, we have Theorem 3.2. Suppose that f satis>es the assumes in Theorem 3:1; '0 is negative LS gradient @ow of f; S is a dynamically isolated critical set of f associated with @ow '0 . ' (' ∈ T) is a family of approximation ('0 ) LS @ows; then there exists an isolate triplet (O; ; ) of S and a real number , ¿ 0 such that cl(O˜ ' ) ∩ K ∩ f−1 [ ; ] = S;
(3.8)
cl(O˜ ' ) ∩ K' ∩ f−1 [ ; ] ⊂ int O for ∀' ∈ T; |' − '0 | ¡ ,:
(3.9)
Proof. We use the same notation as in Theorem 3.1. The proof of Theorem 3.2 is similar to Theorem 3.1 but a little di;erence in step (4), we give the sketch. Indeed, to prove the compactness of {wn }, we denote the boundary of un =eTA wn +U (wn ; T; 'n )+ !(!n ; T; 'n ) by M , since !(!n ; T; 'n ) → 0 ('n → 0) uniformly in bounded !n , we denote its boundary by m. Similar to Theorem 3.1, for ∀8 ¿ 0, there exists T ¿ 0 such that etA ≥ 8x;
∀t ≥ T; x ∈ H+ ;
(3.10)
etA ≥ 8x;
∀t ≤ −T; x ∈ H− ;
(3.11)
moreover, without loss of generality, again we assume that {wn+ } has no convergence subsequence, then ∃0 ¿ 0;
such that wi+ − wj+ ≥ 0;
for i = j;
(3.12)
thus for 8 = 3(M + m)=0; ∃T0 ¿ 0, such that (3.10) is true for ∀t ≥ T0 . Therefore, eT0 A wi − eT0 A wj = eT0 A (wi − wj )
Y.X. Guo, J.Q. Liu / Nonlinear Analysis 48 (2002) 831 – 851
≥
841
3(M + m) wi+ − wj+ 0
≥ 3(M + m): On the other hand, eT0 A wi − et0 A wj = eT0 A wi + U (T0 ; wi ; 'i ) + !(T0 ; !i ; 'i ) + U (T0 ; wi ; 'i ) − U (T0 ; wj ; 'j )) + !(T0 ; !i ; 'i ) − !(T0 ; !j ; 'j ) + eT0 A wj + U (To ; wj ; 'j ) + !(T0 ; !j ; 'j ) ≤ 2(M + m) + U (To ; wi ; 'i ) − U (T0 ; wj ; 'j ): Thus, U (T0 ; wi ; 'i ) − U (T0 ; wj ; 'j ) ¿ M + m;
for any i = j;
this contradicts with the fact that U is compact. The remainder is same as Theorem 3.1. We omit it. The proof of (3.9) is same as (3.2). 4. Critical group Let be a real separable Hilbert space with inner product ·; · and norm · ; and H ∞ H = i=1 Hi with all subspace Hi mutually orthogonal and "nite dimension. Henceforth, we assume that (1) f ∈ C 2 (H; R) with the form f(x) = 12 Ax; x + G(x). (2) A : H → H is a linear, self-adjoint and bounded operator with a "nite-dimensional kernel N , and the zero eigenvalue of A is isolated in 3(A). (3) ∇G:=K(x) is compact, and global Lipschitz continuous on any bounded set. For a simple reason, we assume that (4) There exist a; b ¿ 0 such that |K(x); x| ≤ ax2 + b ∀x ∈ H . Then it is easy to know, df = A + K is a LS vector "eld. Denition 4.1. Let W = {Pn | n = 1; 2; : : :} be a sequence of orthogonal projections. We said W is an approximation scheme with respect to A, if the following properties hold: (1) Pn H = H n is "nite dimensional for ∀n, (2) Pn → I; (n → ∞) (strongly), (3) [Pn ; A] = Pn A − APn → 0 (n → ∞) (in the operator norm). Theorem 4.1. Suppose that S is a dynamically isolated critical set associated with negative gradient @ow generated by df(x)=Ax+K(x), then there exists an isolating triplet (O; ; ) for S and n0 ∈ N such that for all n ≥ n0 (4.1) cl(O˜ n ) ∩ K ∩ f−1 [ ; ] = S;
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cl(O˜ n ) ∩ Kn ∩ f−1 [ ; ] ⊂ int O; (4.2) where O˜ n = t∈R n (O; t); n is the negative gradient @ow generated by Fn = (I − Pn )A(I − Pn ) + Pn (A + K)Pn ; and Kn denotes the zero points set of vector >eld Fn . Proof. We de"ne a family of approximation (0) LS vector "eld F : H × [0; 1] → H as follows: F(x; s) := (I − Pn )A(I − Pn )x + Pn APn (x) + Pn KPn (x);
1 ∀ n+1 ¡ s ≤ 1n ;
F(x; 0) = Ax + K(x); since F(x; s) = Ax + Pn KPn (x) + (I − Pn )A(I − Pn )(x) + Pn APn (x) − Ax and (Pn A − APn )x → 0 (n → ∞) uniformly w.r.t. bounded x, so [(I − Pn )A(I − Pn ) + Pn APn − A](x) → 0
(n → ∞) uniformly in x
hence, for large n; F(x; s) is a family of approximation (0) LS vector "elds satisfying (2.2). Thus, according to Theorem 2.3, the Lows generated by them is a family of approximation (0) LS Lows, denoted by ˜s : R × H × T → H and ˜0 ≡ . Therefore, by Theorem 3.2, there exists an isolating triplet (O; ; ) for S and s0 ∈ [0; 1] such that for all s ∈ [0; s0 ] cl(O˜ s ) ∩ K ∩ f−1 [ ; ] = S; cl(O˜ s ) ∩ Ks ∩ f−1 [ ; ] ⊂ int O; take n0 ¡ 1=s0 , then for all n ≥ n0 , we have cl(O˜ 1=n ) ∩ K ∩ f−1 [ ; ] = S; cl(O˜ 1=n ) ∩ K1=n ∩ f−1 [ ; ] ⊂ int O; note that ˜1=n = n , the theorem is proved. By the de"nition of the vector "eld Fn and F˜ n , the following lemma is obvious. Lemma 4.1. Let ˜n be the @ow generated by F˜ n (x) = Pn (A + K)Pn (x) : H n → H n , n is the @ow generated by Fn (x) = (I − Pn )A(I − Pn )(x) + Pn (A + K)Pn (x) : H → H; then n |H n = ˜n . Here n |H n is the restriction of n on H n . Theorem 4.2. Suppose that S is a dynamically isolated critical set associated with negative gradient @ow generated by the gradient >eld df(x) = Ax+K(x). If (O; ; ) is an isolating triplet for S associated with @ow n (generated by Fn (x)) satisfying Theorem 4:1 for n large enough. Then (O ∩ H n ; ; ) is an isolating triplet for the critical set Sn = O ∩H n ∩Kfn associated with the @ow generated by dfn (x) = Fn (x)|H n , where fn = f|H n ; Kfn is the critical set of fn .
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Proof. By Lemma 4.1, the Low generated by dfn (x) in H n is exactly the restriction of the Low n on H n , which is generated by Fn (x), that is n |H n . Thus, we need only to show cl((O ] ∩ H n )n |H n ) ∩ Kfn ∩ fn−1 [ ; ] ⊂ int (O ∩ H n ) (4.3) since cl(O˜ n )∩K1=n ∩f−1 [ ; ] ⊂ int O and K1=n = {x | (I −Pn )A(I −Pn )(x)+dfn (x) = 0}, if fn has critical point in cl(O˜ n ) ∩ f−1 [ ; ] then it belongs to int(O ∩ H n ). Therefore, for ∀y ∈ cl((O ] ∩ H n )n | n ) ∩ Kfn ∩ fn−1 [ ; ]; H
since cl((O ] ∩ H n )n |H n ) ∩ fn−1 [ ; ] ⊂ cl(O˜ n ) ∩ f−1 [ ; ]; so y ∈ cl(O˜ n ) ∩ f−1 [ ; ] thus y ∈ int(O ∩ H n ). We complete the proof of the theorem. In the following, we need the following de"nitions which will be very useful to what follows: Denition 4.2 (Chang and Ghoussoub [5], Mean value property). Let : R × M → M be a Low, M be a metric space. A subset W is said to have the mean value property (in short MVP), if for ∀x ∈ M and ∀t0 ¡ t1 , we have that (x; [t0 ; t1 ]) ⊂ W, whenever (x; ti ) ∈ W, for i = 0; 1. Denition 4.3 (Chang and Ghoussoub [5], Gromoll–Meyer pairs). Let g be a C 1 functional on a C 1 -Finsler manifold M , and S be a critical subset of g. Subset pairs (W; W− ) is said to be a Gromoll–Meyer (G–M in short) pair of S associated with a Low generated by a p.g.v.f X of g. If the following conditions hold: (1) W is a closed MVP neighborhood of S satisfying W ∩ K = S, and W ∩ g = ∅ for some . (2) W− is an exit set of W: i.e., ∀x0 ∈ W; ∀t1 ¿ 0, such that (x0 ; t1 ) ∈ W, then ∃ t0 ∈ [0; t1 ] such that (x0 ; [0; t0 ]) ⊂ W and (x0 ; t0 ) ∈ W− . (3) W− is closed and is a union of a "nite number of submanifolds that are transversal to the Low . In general, if g is a C 1 functional satisfying the (PS) condition, S is a dynamically isolated critical set for g, then there are many ways to associate a G–M pair (W; W− ) with S. Indeed, if (O; ; ) is an isolating triplet for S, then one can easily verify that W = O˜ ∩ g−1 [ ; ]:={O} ; W− = W ∩ g−1 ( ) (4.4) form a G–M pair for S (see [5]). Lemma 4.2. Let H1 ; H2 be mutually orthogonal subspace, and (Wi ; Wi− ) be a G–M pair for Si in Hi associated with @ow i (i = 1; 2); then (W1 × W2 ; (W1 × W2− ) ∪ (W2 × W1− ))
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is a G–M pair for S1 × S2 associated with @ow 1 × 2 in H1 × H2 , where 1 × 2 : H1 × H2 × R → H1 × H2 is de>ned by 1 × 2 ((x1 ; x2 ); t) = (1 (x1 ; t); 2 (x2 ; t)). Proof. (1) W1 × W2 has MVP. In fact, for ∀x = (x1 ; x2 ) ∈ H1 × H2 , if 1 × 2 (x; ti ) ∈ W1 × W2 , i = 1; 2. Then 1 (x1 ; ti ) ∈ W1 ; 2 (x2 ; ti ) ∈ W2 , since W1 ; W2 has MVP. Thus 1 (x1 ; [t1 ; t2 ]) ⊂ W1 , 2 (x2 ; [t1 ; t2 ]) ⊂ W2 ; so 1 × 2 (x; [t1 ; t2 ]) ⊂ W1 × W2 . (2) (W1 × W2− ) ∪ (W2 × W1− ) is an exit set. For ∀x0 ∈ W1 × W2 , and ∀t1 ¿ 0 if 1 × 2 (x0 ; t1 ) ∈ W1 × W2 , where x0 = (x01 ; x02 ) ∈ W1 × W2 . Then we have (a) 1 (x01 ; t1 ) ∈ W1 but 2 (x02 ; t1 ) ∈ W2 , (b) 2 (x02 ; t1 ) ∈ W2 but 1 (x01 ; t1 ) ∈ W1 , (c) 1 (x01 ; t1 ) ∈ W1 and 2 (x02 ; t1 ) ∈ W2 . We only prove case (a), the other cases are similar. Case (a): 1 (x01 ; t1 ) ∈ W1 but 2 (x02 ; t1 ) ∈ W2 . Note W1− is the exit of W1 , there exists t0 ∈ [0; t1 ) such that 1 (x01 ; [0; t0 ]) ⊂ W1 and 1 (x01 ; t0 ) ∈ W1− thus, 1 × 2 (x0 ; [0; t0 ]) ⊂ W1 × W2 and 1 × 2 (x1 ; t0 ) ∈ W2 × W1− . (3) By the de"nition of product Low 1 × 2 , it is obvious that 1 × 2 is transversal to (W1 × W2− ) ∪ (W1− × W2 ). Combining (1)–(3), the lemma is proved. Lemma 4.3. Suppose that {f' | ' ∈ [0; 1]} is a family of C 1 functional, and {S' } is a family of isolated critical set corresponding to f' ; and (W' ; W'− ) is a G–M pair for S' associated with @ow ' generated by df' ; for any given '0 ; assume that (O; ; ) is an isolating triplet for S'0 satisfying: for ∀' ∈ [0; 1] cl(O˜ ' ) ∩ Kf'0 ∩ f'−1 [ ; ] = S'0 ; 0 [ ; ] ⊂ int O: S' = cl(O˜ ' ) ∩ Kf' ∩ f'−1 0
(4.5)
Let U be a bounded set which includes all the sets cl(O˜ ' ) ∩ f'−1 [ ; ]. If the map0 1 ping '| → f' is continuous from [0:1] to C (U ); then the groups H ∗ (W' ; W'− ) are independent of '; i.e.; for ∀' ∈ [0; 1] H ∗ (W' ; W'− ) ∼ = H ∗ (W' ; W' ): 0
0−
Proof. For '0 = 0, we have c(O˜ ' )∩Kf0 ∩f0−1 [ ; ] = S0 ; thus, by Chang and Ghoussoub [5, Proposition III:2] W˜ 0 = cl(O˜ ' ) ∩ f0−1 [ ; ]
W˜ 0− = W˜ 0 ∩ f0−1 ( )
(4.6)
form a G–M pair for S0 associated with Low '0 , by Chang [5, Proposition III:3] H ∗ (W0 ; W0− ) ∼ = H ∗ (W˜ 0 ; W˜ 0− ): We say that (W˜ 0 ; W˜ 0− ) is also a G–M pair for S' associated with Low ' , whenever ' is su9ciently approximate to 0. In fact, by the de"nition of G–M pair and (4.6), we need only to show that the restriction of f' on W˜ 0 is a bounded below (whenever ' is su9ciently approximate to 0). Otherwise, there exists 'n → 0 and {x n } ⊂ W˜ 0 ('n ) such that f'n → −∞. Since W˜ 0 ('n ) is a bounded set in "nite-dimensional space. So
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{x n } has a convergence subsequence (denoted still by {x n }). Let x0 be the limit, then x0 ∈ W˜ 0 ('n ). Since f' is continuous about ', let n → ∞, we have f'0 (x0 ) → −∞, this is a contradiction. On the other hand, by Chang and Ghoussoub [5, Proposition III:3] H ∗ (W˜ 0 ; W˜ 0− ) ∼ = ∗ H (W' ; W'− ). So, for ' small enough H ∗ (W0 ; W0− ) ∼ = H ∗ (W' ; W'− ): In the end, by the compactness of [0; 1], we complete the proof of lemma. For the simple case, we have Lemma 4.4. Let E be a >nite-dimensional space, A : E → E be a bounded linear self-adjoint operator, 0 ∈ 3(A). Then S = {} is a dynamically isolated critical set of functional g(x) = 12 Ax; x, associated with negative gradient @ow generated by df(x) = Ax, and (N; L) is a G–M pair for S. Where N := Dr (E + ) × Dr (E − );
L := Dr (E + ) × Sr (E − );
Dr (V ) = {x ∈ V | x ≤ r};
Sr (V ) = {x ∈ V | x = r}:
Proof. Since 0 ∈ 3(A), S ={} is the unique critical point of g, it is obvious that S is a dynamically isolated critical set. By the de"nition of G–M pair, the result is obvious. n Recall, we assume that H n = i=1 Hi , thus H n+1 = H n ⊕ Hn+1 , by the properties of projective operator, we have that P n+1 :=Pn+1 − Pn : H → Hn+1 is a projective operator from H to Hn+1 . In the following, we consider (for n large enough) the following family of vector "elds de"ned by F(x; t) : = tPn+1 (A + K)Pn+1 (x) + (1 − t)(Pn (A + K)Pn (x) +P n+1 (A + P)P n+1 (x)) : H n+1 → H n+1 ;
∀t ∈ [0; 1];
(4.7)
where P is a projective operator from H to the kernel space of operator A. Lemma 4.5. Let (Wt ; Wt− ) be G–M pair for St associated with the @ow t generated by the vector >eld F(x; t), then H ∗ (Wt ; Wt− ) are independent of t. Proof. We de"ne a family of approximation vector "elds G(x; s) as follows: G(x; s) : = (I − Pn+1 )A(I − Pn+1 )(x) +n((n + 1)s − 1)(Pn (A + K)Pn (x) + P n+1 (A + P)P n+1 ) +(1 − n(n + 1)s + n)Pn+1 (A + K)Pn+1 (x) G(x; 0) := A(x) + K(x);
1 for ∀ n+1 ¡ s ≤ 1n ;
for s = 0;
then G(x; s) is the approximation (0) LS vector "elds. By Theorem 3.1, there exists an isolating triplet (O; ; ) for S, such that for n large enough and ∀t ∈ [0; 1]. cl(O˜ (t(n)) ) ∩ K ∩ f−1 [ ; ] = S; cl(O˜ (t(n)) ) ∩ Kt(n) ∩ f−1 [ ; ] ⊂ int O;
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˜ t) = (I − Pn+1 )A(I − Pn+1 )(x) + F(x; t), since where t(n) is the Low generated by F(x; ˜ t) = }, use the same method as Theorem 4.1, we can prove that Kt(n) = {x | F(x; cl((O ∩] H n+1 )t(n) |H n+1 ) ∩ KF(x; t) ∩ f−1 [ ; ] ⊂ int(O ∩ H n+1 ); cl((O ∩] H n+1 )t(n) |H n+1 ) ∩ Kfn+1 ∩ f−1 [ ; ] ⊂ int(O ∩ H n+1 ); that is cl((O ∩] H n+1 )t ) ∩ KF (x; t) ∩ f−1 [ ; ] ⊂ int(O ∩ H n+1 ); cl((O ∩] H n+1 )t ) ∩ Kfn+1 ∩ f−1 [ ; ] = Sn+1 by Theorems 4.2 and 3.1, (O ∩H n+1 ; ; ) is an isolating triplet for Sn+1 , the critical set H n+1 )t ) ∩ of fn+1 in "nite-dimensional space O ∩ H n+1 and for ∀t ∈ [0; 1], cl((O ∩] −1 1 f [ ; ] are included in a bounded set U . Clearly, ft is C continuous as t ∈ [0; 1]. Hence, by Lemma 4.3, H ∗ (Wt ; Wt− ) is indenpendent of t. Here we get the main result of this section. Theorem 4.3. For the considered functional f(x) = 12 Ax; x + G(x), >nite-dimensional subspace H n = ⊕n1 Hi of H and the approximation scheme W = {Pn }∞ 1 w.r.t. A; suppose that (Wn ; Wn− ) is a G–M pair for critical set Sn associated with the @ow generated by dfn = Pn (A + K)Pn : H n → H n ; then for n large enough H ∗+m(Pn (A+P)Pn ) (Wn ; Wn− ) ∼ = H ∗+m(Pn+1 (A+P)Pn+1 ) (Wn+1 ; Wn+1 )
(4.8)
where m(·) denotes the Morse index of operator. Proof. Since P n+1 (A + P)P n+1 : Hn+1 → Hn+1 is invertible, by Lemmas 4.2 and 4.4 (Wn × N; (Wn × L) ∪ (N × Wn− )) is a G–M pair for Sn × {} associated with the Low generated by Pn (A + K)Pn + P n+1 (A + P)P n+1 : Hn+1 → Hn+1 in space H n ⊕ Hn+1 , where − − + + ) × D1 (Hn+1 ) L = D1 (Hn+1 ) × S1 (Hn+1 ): N = D1 (Hn+1
Let (W n+1 (t); W−n+1 (t)) be the G–M pair for St associated with the Low generated by F(x; t) in (4:7), then by Lemma 4.3 H ∗ (W n+1 (0); W−n+1 (0)) ∼ = H ∗ (W n+1 (1); W−n+1 (1)): Hence, for t = 1 and t = 0 H ∗ (Wn × N; (Wn × L) ∪ (Wn− × N )) ∼ = H ∗ (Wn+1 ; W(n+1)− ):
(4.9)
− − ); S1 (Hn+1 )) ,→ (N; L) is homotopic. Hence it Since the immersion map (D1 (Hn+1 induces homotopic map − − )=S1 (Hn+1 )) ∧ (Wn =Wn− ) ,→ (N=L) ∧ (Wn =Wn− ) (D1 (Hn+1
by the property of cohomology modules: H ∗ (S 2(n) ∧ Wn =Wn− ) ∼ = H ∗ (N × Wn ; (N × Wn− ) ∪ (Wn × L));
(4.10)
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thus H ∗ (S 2(n) ∧ Wn =Wn− ) ∼ = H ∗ (Wn+1 ; W(n+1)− );
(4.11)
note that H ∗ (S 2(n) ∧ Wn =Wn− ) ∼ = H ∗−2(n) (Wn =Wn− );
(4.12)
2(n)
is 2(n) dimensional unit ball. 2(n) is the Morse index of P where S as to Hn+1 . Since P n+1 APn + Pn AP n+1 → ;
n+1
(A + P)P n+1
(n → ∞);
hence m(Pn+1 (A + P)Pn+1 ) = m(Pn (A + P)Pn ) + m(P n+1 (A + P)P n+1 ); that is 2(n) = m(P n+1 (A + P)P n+1 ) = m(pn+1 (A + P)Pn+1 ) − m(Pn (A + P)Pn ); combine (4.11) with (4.12), H ∗−(m(Pn+1 (A+P)Pn+1 −m(Pn (A+P)Pn )) (Wn ; Wn− ) ∼ = H ∗ (Wn+1 ; W(n+1)− ) so H ∗+m(Pn (A+P)Pn ) (Wn ; Wn− ) ∼ = H ∗+m(Pn+1 (A+P)Pn+1 ) (Wn+1 ; W(n+1)− ): Now, we can de"ne the critical group for the strongly inde"nite functional on its dynamically isolated set as follows: Denition 4.4. Let f be an inde"nite functional satisfying the hypothesis of (1) – (4), S be a dynamically isolated set for f associated with its negative gradient Low. Assume that Pn ; (Wn ; Wn− ) are same as Theorem 4.3, we de"ne (for n large enough) C∗ (f; S) := H ∗+m(Pn (A+P)Pn ) (Wn ; Wn− ) and call it the qth critical group of S with respect to Low generated by negative gradient vector "eld df. Remark 4.1. De"nition 4.4 is independent of the choice of isolating triplet. Indeed, if (O ; ; ) is another isolating triplet, then for large n; (O ∩ H n ; ; ) is also an isolating triplet for a critical set of restriction functional fn in "nite-dimensional space H n . Assume that (Wn ; Wn− ) is the corresponding G–M pair which is de"ned by (4.6), ) ∼ by Chang and Ghoussoub [5, Proposition III:2 and Theorem III:3]: H ∗ (Wn ; Wn− = ∗ H (Wn ; Wn− ) (for n large enough). Remark 4.2. De"nition 4.4 is independent of the choice of the approximation scheme. ˜ In fact, if W={ P˜ n | n=1; 2 : : :} is another approximation scheme. We take W1 ={Pn | n= 1; 2 : : :} such that Pn = Pn for n = 2k − 1; Pn = P˜ n for n = 2k. Therefore, Theorem 4.3 is true for the approximation scheme W1 , thus for n large enough H ∗+m(Pn (A+P)Pn ) (Wn ; Wn− )∼ ; W(n+1)− ) = H ∗+m(Pn+1 (A+P)Pn+1 ) (Wn+1
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taking n large enough, we have ˜
˜
H ∗+m(Pn (A+P)Pn ) (Wn ; Wn− ) ∼ = H ∗+m(Pn+1 (A+P)Pn+1 ) (W˜ n+1 ; W˜ (n+1)− ): Remark 4.3. For a simple reason, we assume that: ∃a; b ¿ 0 such that |K(x); x| ≤ ax2 + b
∀x ∈ H
in fact, for general case, let (O; ; ) be an isolating triplet for S, and without the loss of generality, we assume O ⊂ B(0; s). De"ne 1 for t ≤ s; C(t) = 1 + s − t for s ¡ t ≤ s + 1; 0 for t ≥ s + 1 and d : H → [0; 1]; d(x) = C(x). Then K1 (x) = d(x)K(x) satis"es the above condition. Hence, we can use K1 (x) to substitute K(x), since the Low generated by A + K is same as the Low generated by A+K1 in bounded set O and for "nite T , so the critical groups de"ned by them are the same. 5. Morse-type inequality We have pointed out that, one of the advantages that we developed the Gromoll– Meyer theory is that a dynamically isolated set has natural Morse decomposition. For a given dynamically isolated critical set S, each of its Morse decomposition set is also a dynamically critical set. The problem is: if there are any relationship between the critical group of S and the critical group of its Morse decomposition sets. The following theorem is a description of this relationship. Suppose that f satis"es the hypothesis in Section 4 and S is a dynamically isolated critical set of f associated with the Low generated by the negative gradient vector "eld, (O; ; ) is its isolating triplet. We denote its corresponding G–M pair by (W; W− ), which is de"ned by W = {O} , W− = W ∩f−1 ( ). Let = d0 ¡d1 ¡d2 ¡ · · · ¡dm−1 ¡ dm = be the regular value of f. Let ˜ ∩ K ∩ f−1 [ ; d1 ]; S1 = cl(O) ··· ˜ ∩ K ∩ f−1 [di−1 ; di ]; Si = cl(O) ······ ˜ ∩ K ∩ f−1 [dm−1 ; ]; Sm = cl(O) then m
i=1
Si = S;
Si ∩ Sj = ∅; i = j
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and Si is a dynamically isolated critical set (it is possible to be ∅), and (O∩f−1 [di−1 ; di ]; di−1 ; di ) is the corresponding isolating triplet. Let M q (W; W− ) =
n
dim Cq (f; Si );
∀q ∈ Z;
i=1
q (W; W− ) = dim Cq (f; S); q
∀q ∈ Z:
q
If M ; are "nite for each q and equal to zero for large |q|. We may de"ne the Morse and Poincare polynomials for (W; W− ) by M (t; W; W− ) =
∞
M q (W; W− )t q ;
q=−∞
P(t; W; W− ) =
∞
q (W; W− )t q :
q=−∞
Theorem 5.1. Assumptions and notations are as above. In addition; if for each q; M q (W; W− ) and q (W; W− ) are >nite ∞and equal to zero for all |q| suGciently large; then there exists a function Q(t) = q=−∞ aq t q ; such that all aq are nonnegative; and M (t; W; W− ) = P(t; W; W− ) + (1 + t)Q(t); moreover; n
P(t; C∗ (f; Si )) = P(t; C∗ (f; S)) + (1 + t)Q(t):
i=1
Proof. Let Z ⊂ Y ⊂ X be closed subsets of H: Then we have long exact sequence (see [8,11]). j∗
i∗
,∗
· · · → H q (X; Z) → H q (Y; Z) → H q+1 (X; Y ) → H q+1 (X; Z) → · · · : We denote the rank of a mapping by R and dim H (·) by q (·). Since q are "nite for all q and zero for all |q| large. It follows from the exactness, we have q (X; Z) = dim R(j q ) + dim R(iq ); q (Y; Z) = dim R(iq ) + dim R(,q ); q (X; Y ) = dim R(,q−1 ) + dim R(j q ); thus q (X; Y ) + q (Y; Z) = q (X; Z) + dim R(,q−1 ) + dim R(,q ): Denote the Poincare polynomial of (X; Y ) by P(t; X; Y ) and set q(t; X; Y; Z) =
∞
(dim R(,q ))t q ;
q=−∞
(5.1)
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since dim R(,q ) = 0, whenever |q| is large. It follows by multiplying t q to (5.1) and summing over q that P(t; X; Y ) + P(t; Y; Z) = P(t; X; Z) + (1 + t)q(t; X; Y; Z):
(5.2)
Let Wi = (W ∩ fdi ) ∪ W− ; 1 ≤ i ≤ m then Wm = W ∩ f ∪ W− = W;
W 0 = W ∩ f ∪ W− = W −
and W0 ⊂ W1 ⊂ W2 · · · ⊂ Wm are closed sets, for j ≥ 2, apply (5.2) to W0 ⊂ Wj−1 ⊂ Wj , and then summing from 2 to m, we have m
P(t; Wj ; Wj−1 ) +
j=2
m
P(t; Wj−1 ; W0 )
j=2
=
m
P(t; Wj ; W0 ) + (1 + t)
j=2
m
q(t; Wj ; Wj−1 ; W0 );
j=2
thus m
P(t; Wj ; Wj−1 ) = P(t; Wm ; W0 ) + (1 + t)Q(t);
j=1
where Q(t) =
m
q(t; Wj ; Wj−1 ; W0 );
j=2
that is m
P(t; Wj ; Wj−1 ) = P(t; W; W− ) + (1 + t)Q(t);
(5.3)
j=1
set W˜ i = {x ∈ W | di−1 ≤ f(x) ≤ di }; W˜ i− = W˜ i ∩ (W− ∪ f−1 (di−1 )); then it is easy to see that (W˜ i ; W˜ i− ) is a G–M pair for Si (0 ≤ i ≤ m). By excision, it follows that H ∗ (Wi ; Wi−1 ) ∼ = H ∗ (W˜ i ; W˜ i− ); de"ne P(t; C∗ (f; Si )) = P(t; W˜ i ; W˜ i− ); then (5.3) implies n
P(t; C∗ (f; Si )) = P(t; C∗ (f; S)) + (1 + t)Q(t);
i=1
we complete the proof of the theorem.
Y.X. Guo, J.Q. Liu / Nonlinear Analysis 48 (2002) 831 – 851
851
References [1] V. Benci, The direct method in the study of periodic solutions of Hamiltonian systems with prescribed period, in: P. Aubin, A. Bensoussan, I. Ekeland (Eds.), Advances in Hamiltonian Systems, Birkhauser, Basel, 1983. [2] D.V. Benci, P.H. Rabinowitz, Critical point theorems for inde"nite functionals, Invent. Math. 52 (1979) 241–273. [3] K.C. Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure. Appl. Math. 34 (1981) 691–712. [4] K.C. Chang, In"nite dimensional Morse theory and its applications, CRM, Universite’ de Montreal, SMS, 97 (1985). [5] K.C. Chang, N. Ghoussoub, The Conley index and the critical groups via an extension of Gromoll– Meyer theory, Topology Method Nonlinear Anal. 7 (1996) 77–93. [6] K.C. Chang, In"nite Dimensional Morse Theory and Multiple Solution Problems, Birkhauser, Boston, 1993. [7] C.C. Conley, in: Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series, Vol. 38, American Mathematical Society, Providence, RI, 1978. [8] A. Dold, Lectures on Algebraic Topology, Springer, Berlin, 1980. [9] I. Ekeland, J.M. Lasry, On the number of periodic trajectories for a Hamiltonian Low on a convex energy surface, Ann. Math. (2)112 (1980) 283–319. [10] K. Geba, M. Izydorek, A. Pruszko, The Conley index in Hilbert spaces and its applications, preprint, 1997. [11] M.J. Greeberg, Lectures on Algebraic Topology, Benjamin, New York, 1967. [12] H. Hofer, On strongly inde"nite functionals with applications, J. AMS 275 (1) (1983) 68–79. [13] Y.M. Long, Conley Index Theory and Hamiltonian System, Science Publication, Beijing, 1993 (in Chinese). [14] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Berlin, Heidelberg, 1989. [15] P.H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978) 157–184. [16] A. Szulkin, Cohomology and Morse theory for strongly inde"nite functionals, Math. Z. 209 (1992) 375–418.
www.elsevier.com/locate/na
Morse theory for strongly inde"nite functional Y.X. Guoa; ∗ , J.Q. Liub a Institute
of Systems Science, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China b Institute of Mathematics, Peking University, Beijing 100871, People’s Republic of China Received 12 October 1999
Keywords: Morse theory; Inde"nite functional; Critical groups; Morse inequalities; Dynamically isolated set
1. Introduction The classical Morse theory has proved to be very useful in nonlinear analysis, in proving the existence of solutions and estimating the number of solutions to the differential equations. However, because only nondegenerate functional, or nondegenerate critical points, were well studied, and the nondegeneracy of a solution to a partial differential equation occurring in practice is di9cult to verify in advance, sometimes it is even not true. So, the applications of the classical Morse theory to partial di;erential equations were very limited. To extend the applications of classical Morse theory, there are two theories in front of us: one is Gromoll–Meyer theory on isolated critical point which may be degenerate (see [4,6]); the other is Conley’s Morse index theory on isolated invariant sets (see [7]). However, since the Conley index is homotopy class, it is di9cult to "gure out. In this sense, Gromoll–Meyer theory is easier than Conley index theory, and among other things, Gromoll–Meyer theory is su9cient for many applications (see [3]). Recently, in their paper [5], K.C. Chang and N. Ghoussoub, in a variational setting, investigated the connection between the two theories, more precisely, they studied the relationship between the Gromoll–Meyer pair of a dynamically isolated critical set and the Conley index pair of its isolated invariant neighborhoods, and showed that ∗
Corresponding author. E-mail address: [email protected] (Y.X. Guo).
0362-546X/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 0 ) 0 0 2 1 8 - 2
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Y.X. Guo, J.Q. Liu / Nonlinear Analysis 48 (2002) 831 – 851
the Gromoll–Meyer pairs are Conley index pairs; and Conley index coincides with the critical groups. That is the information given by the critical groups of such a dynamically isolated critical set is equivalent to which given by the Conley index of its isolated invariant neighborhoods. Thus, in many cases, we can use Gromoll–Meyer theory only to resolve problem, and the proof is also relatively easier. However, we consider the following functional G(x) = 12 x+ 2 − 12 x− 2 (functional of such type is called strongly inde"nite) de"ned on a Hilbert space E = E + + E − , where E + ; E − are orthogonal decomposition of E, and both of them are in"nite-dimensional subspace. Clearly, is the only critical point of G. According to Morse theory Cq (G; ) = ; q = 0; 1; 2; : : : : This example explains why the previous de"nition for critical group is of little use for strongly inde"nite functional. The purpose of our paper is to construct a Morse theory for inde"nite functional. Here a strongly inde"nite functional is a C 1 map which is neither bounded from above nor from below, not even model a "nite-dimensional subspace. In the study of existence theory for systems of elliptic equations and of periodic solutions of Hamiltonian systems of ordinary di;erential equations, the common feature is that these equations can be derived from variational problems which are inde"nite in a very strong sense, in particular the corresponding functional are strongly inde"nite. To "nd the critical points of strongly inde"nite functional, there are at least two sets of methods that have been developed: (i) Morse theory and its generalizations (Gromoll–Meyer theory, Conley index theory). (ii) Minimax theory. And there have been many papers based on minimax theory (see [1,2,9,12,13,15]), however, all these methods have some disadvantages, because one needs strong hypotheses (such as uniformly bounded, convexity, etc.); in 1992, A. Szukin (see [16]) constructed a di;erent theory for an isolated critical point which turned out to be useful for strongly inde"nite functional; on the other hand, in 1997, Geba [10] presented a new generalization of classical Conley index theory, and gave the de"nition of homotopy Conley index about an isolated neighborhood of an isolated invariant set. Motivated by the ideas of papers [5,10], based on the Gromoll–Meyer theory, we construct a di;erent Morse theory for strongly inde"nite functional on its dynamically isolated set. Our paper is organized as follows: In Section 2, we investigate the properties of Low and vector "eld which we need in developing our theory. In Sections 3–5 we construct a Morse theory for strongly inde"nite functional, including the de"nition of critical groups, Morse-type equalities.
2. Flow and vector eld Let H be a real Hilbert space with an inner product ·; · and norm ·. Let f ∈ C 1 (H; R), ∀x ∈ H , we denote the di;erential of f at x by df(x), x is said to be a critical point of f if df(x) = ; ∀a ∈ R1 , fa = {x ∈ H | f(x) ≤ a} is called the level set; K = {x ∈ H | df(x) = } is called the critical set of f; ∀c ∈ R1 , Kc = {x ∈ K | f(x) = c}, if Kc = ∅, then we say c is a critical value of f; otherwisze c is said to be a regular value. Function f is said to satisfy the Palais–Smale
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833
(PS for abbreviation) condition, if for any sequence {x n } such that f(x n ) is bounded and df(x n ) → (n → ∞) possesses {x n } has a convergent subsequence. Denition 2.1 (Chang and Ghoussoub [5], Flow). Let M be a metric space, a continuous map : R × M → M is said to be a Low, if it satis"es the following properties: (i) (0; x) = x; ∀x ∈ M (ii) (t1 ; (t2 ; x)) = (t1 + t2 ; x); ∀t1 ; t2 ∈ R; x ∈ M . We say that Low is bounded if it is a bounded continuous map. Recall that the integral curve starting at u of a Locally Lipschitz continuous vector "eld g : X → X on some Banach space X is the maximal C 1 -curve (t; u) de"ned on (t − (u); t + (u)) and satisfying the cauchy problem (t; ˙ u) = g((t; u)); (2.1) (0; u) = u: The existence of such integral curves is well known. If we set D() = {(t; u) ∈ R × X; t − (u) ¡ t ¡ t + (u)}, then D() is open and : D() → X is continuous. The map is called the local Low generated by g. It is natural to ask, when the local Low generated by g is global, that is the Low is well de"ned on R × X . Remark 2.1. According to [14, Chapter 8, Proposition 8:1], if t + (u) ¡ ∞; ([0; t + (u)]; u) is unbounded. In addition, if g satis"es |g(u); u| ≤ 12 cu2 + d;
∀u ∈ X;
(2.2)
for some positive constants c; d ¿ 0, then Low generated by g is well de"ned and bounded on R × X . In this paper, what we are interested is the following vector "eld and Low with special form. Denition 2.2 (Geba et al. [10]). We say that F:H →H is a LS vector "eld, if there exists a completely continuous and locally lipschitz continuous map K : H → H such that F(x) = Ax + K(x);
(2.3)
where A : H → H is a linear bounded self-adjoint operator. Denition 2.3 (Geba et al. [10]). We say that a Low : R × H → H is a LS Low, if (t; x) = etA x + U (t; x);
(2.4)
where U : R × H → H is completely continuous. Theorem 2.1. Suppose that F(x) = Ax + K(x) is a LS vector >eld; and K satis>es condition (2:2). Then the @ow generated by F is a LS @ow.
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Y.X. Guo, J.Q. Liu / Nonlinear Analysis 48 (2002) 831 – 851
Proof. Let be the Low generated by F; then d − A = K(); dt this means that satis"es an inhomogeneous linear equation and therefore it can be t represented as (t; u) = exp(tA)u + W (t; u), where W (t; u) = etA 0 e− A K(( ; u)) d . To see that W is compact, suppose [ ; ] × B ⊂ R × H is bounded. Since is bounded, without loss of generality, we may assume that ([ ; ] × B) ⊂ BR for some R ¿ 0. Where BR denotes a ball with radius R. Therefore, K(([ ; ] × B)) ⊂ K(BR ) ⊂ cl(K(BR )); note that K is compact, it follows that cl(K(BR )) is a compact set. Now let Y = {e−sA z | s ∈ [ ; ]; z ∈ cl(K(BR ))} since the map (s; z) → e−sA z is continuous on compact set [ ; ] × cl(K(BR ), its range Y is compact. Therefore, the closed convex hull Yˆ of Y is also compact. So for any T ¿ 0, T (Yˆ ∪ ) is also compact. Note that z(s) = e−sA K((s; u)) ∈ Y; ∀s ∈ [ ; ]; u ∈ B: t Hence for t ∈ [0; T ]; (1=T ) 0 z(s) ds ∈ Yˆ ; and t S Y= z(s) ds | t ∈ [ ; ]; z(s) ∈ Y ⊂ T (Yˆ ∪ ): 0
It follows that W : R × H → H is compact. We say that continuous map ' : D(' ) → H is a family of local Lows on H indexed by ', if D(' ) is an open subset of R × H × T and ' (where ' (t; x) = (t; x; ')) is a local Low for every ' ∈ T. If D(' ) = R × H × T, then we say that is a family of Lows on H indexed by T. Denition 2.4 (Geba et al. [10]). Let T be a compact metric space. We say that F : H × T → H is a family of LS vector "elds, if there exists a completely continuous and local lipschitz continuous map K : H × T → H such that F(x; ') = Ax + K(x; ');
∀(x; ') ∈ H × T;
(2.5)
where A is a bounded linear self-adjoint operator. Denition 2.5 (Geba et al. [10]). We say that a family of Lows : R × H × T → H is a family of LS Lows, if (t; x; ') = etA x + U (t; x; '); U : R × H × T → H is completely continuous. Similar to De"nition 2.4 and De"nition 2.5, we de"ne
(2.6)
Y.X. Guo, J.Q. Liu / Nonlinear Analysis 48 (2002) 831 – 851
835
Denition 2.4 . For any given '0 ∈ T, we say that F˜ : H × ' → H is a family of approximation ('0 ) LS vector "elds, if ˜ ') = Ax + K(x; ') + !(x; ') F(x; satisfying (i) A is a bounded linear self-adjoint operator. (ii) K : H × T → H is a compact and Lipschitz continuous map. (iii) !(x; ') → 0 as ' → '0 , uniformly in x belonging to a bounded set. Denition 2.5 . T; '0 are same as in De"nition 2:4. We say that ˜ : R × H × T → H is a family of approximation ('0 ) LS Lows, if (t; ˜ x; ') = etA x + U (t; x; ') + !(t; x; ') satisfying (i) A is a bounded linear self-adjoint operator. (ii) U : R × H × T → H is compact. (iii) !(t; x; ') → 0 as ' → '0 , uniformly in x belonging to a bounded set and t ¡ T for some T ¿ 0. Use the same arguments as in Theorem 2.1, we can prove Theorem 2.2. Assume that F(x; ') = Ax + K(x; ') is a family of LS vector >elds; and there exist a; b ¿ 0 such that |K(x; '); x| ≤ a|x|2 + b;
∀' ∈ T; x ∈ H
then the @ows generated by F(x; ') is a family of LS @ows. Proof. Similar to Theorem 2.1, Lows ' generated by F' (x) := F(x; ') can be repret sented as (t; u; ') = exp(tA)u + W (t; u; '); where W (t; u; ') = etA 0 e− A K(( ; u); ') d . To see that W is compact, suppose [ ; ] × B × T1 ⊂ R × H × T is bounded. Since ' is bounded for any ' ∈ T, hence ([ ; ] × B × T1 ) ⊂ BR × T2 (for some R ¿ 0), so K(([ ; ] × B × T1 )) ⊂ K(BR × T2 ) ⊂ cl(K(BR × T2 )): Since K is compact, so is cl(K(BR × T2 )). Set Y = {e−sA z | s ∈ [ ; ]; z ∈ cl(K(BR × T2 ))} note that (s; z) → e−sA z is continuous on the compact set [ ; ] × cl(K(BR × T2 )), Y and its closed convex hull Yˆ are also compact. Using the same arguments as in Theorem 2.1, we complete the proof of Theorem 2.2. Theorem 2.3. Let F' be a family of approximation ('0 ) LS vector >elds; then a family of @ows ' generated by F(x; ') is a family of approximation ('0 ) LS @ows. Proof. Suppose that ' is the Low generated by F' , by Theorem 2.2 (t; u; ') = exp(tA)u + w(t; u; ') + !(t; u; ');
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Y.X. Guo, J.Q. Liu / Nonlinear Analysis 48 (2002) 831 – 851
where w(t; u; ') = etA !(t; u; ') = etA
t
0
0
t
e− A K(( ; u); ') d ; e− A !(( ; u); ') d
(2.7)
in the Low, we need only to show !(t; u; ') → 0 as ' → '0 uniformly in u bounded and t ¡ T for some T ¿ 0. In fact, from De"nition 2:5 (iii), this is obviously true. 3. Dynamically isolated critical set and isolating triplet Let H be a Hilbert space, B ⊂ H be a subset, denote by N, (B) the , closed neighborhood of B, and denote by cl(B) the closed closer of B; let be a Low, and f be + a functional, B˜ = t∈R (t; B); B˜ = t∈R+ (t; B) and f−1 [ ; ] = {x | ≤ f(x) ≤ }. We now recall the following key concept due to [5]: Denition 3.1 (Dynamically isolated critical set). A subset S of the critical set K is said to be a dynamically isolated critical set, if there exists a closed neighborhood O of S and regular values ; of functional f, such that ˜ ∩ K ∩ f−1 [ ; ] = S; O ⊂ f−1 [ ; ] cl(O) where O˜ = t∈R (t; O) and is the Low associated with functional f, we shall then say that (O; ; ) is an isolating triplet for S. Obviously, the isolated property of a dynamically isolated critical set is dependent on the Low, we also say it is an isolated critical set associated with some Low. In the following, we will investigate the properties of a dynamically isolated critical set about its nearby Lows. Let S be a dynamically isolated critical set associated with Low '0 , here, the so-called nearby Lows mean that the Lows ' , as ' is su9ciently near to '0 . Lemma 3.1. Suppose f ∈ C 2 (H; R) satisfying the (PS) condition; T is a compact metric space; ' (' ∈ T) is a family of LS @ows generated by F' = A + K' ; S is a dynamically isolated critical set of f; (O; ; ) is one of its isolating triplet. If −df ≡ F'0 ; K'0 satis>es global Lipschitz continuative on any bounded set; and ˜ F' (x) − F'0 (x) → 0 uniformly as to bounded x. Then ∃, ¿ 0; such that cl(O' ) ∩ f−1 [ ; ] (uniformly in ') is bounded; whenever |'−'0 | ¡ ,; where O˜ ' = t∈R ' (t; O). Proof. (1) We will show for ∀0 ¿ 0; ∃, ¿ 0 such that ' (x; t) − '0 (x; t) ¡ 0
whenever |' − '0 | ¡ ,
uniformly holds for any bounded x and t.
Y.X. Guo, J.Q. Liu / Nonlinear Analysis 48 (2002) 831 – 851
837
In fact, d' = A' + K' (' ); dt
so
d'0 = A'0 + K'0 ('0 ) dt
d(' − '0 ) = A(' − '0 ) + K' (' ) − K'0 ('0 ) dt = A(' − '0 ) + K' (' ) − K'0 (' ) + K'0 (' ) − K'0 ('0 ) since ' is bounded. Therefore, ∀0 ¿ 0; ∃,1 ¿ 0 such that K' (' ) − K'0 (' ) ¡ 0
for ∀' ∈ T; |' − '0 | ¡ ,1
hence, d(' − '0 ) ≤ c|' − ' | + 0 0 dt and |' − '0 | ≤ ect
0
t
0e−cs ds
noting that, c is bounded and 0 is small enough, (1) is proved. (2) Let S1 = K ∩ f−1 [ ; ], we take S2 as the bounded closed neighborhood of S1 such that O ⊂ S2 . Since f satis"es (PS), there exists T1 ¿ 0, such that cl(O˜ '0 ) ∩ f−1 [ ; ] ⊂ '0 (S2 ; [ − T1 ; T1 ]): Now, we choose f−1 [ ; ] ⊃ S3 ⊃ S˜2 = '0 (S2 ; [ − T1 ; T1 ]), such that d(S˜2 ; @S3 ) ¿ 0. By (PS) condition, there exists 2 ¿ 0 such that F'0 (x) ≥ 22 for ∀x ∈ S3 − S˜2 : Since F' (x) − F'0 (x) → 0
uniformly for bounded x;
there exists ,2 ¿ 0 such that for ∀'; |' − '0 | ¡ ,2 , F' (x) − F'0 (x) ¡ 2; thus F' (x) ≥ 2 for ∀'; |' − '0 | ¡ ,2
and
∀x ∈ S3 − S˜2 :
Now, if ' (x; [0; t]) ⊂ S3 − S˜2 , then t ¡ T2 for some T2 ¿ 0. Hence, there exist , = min(,1 ; ,2 ) ¿ 0, and T = max(T1 ; T2 ), such that ∀' ∈ T, |' − '0 | ¡ ,, cl(O˜ ' ) ∩ f−1 [ ; ] ⊂ ' (S˜2 ; [ − T; T ]). The later is a bounded set. The lemma is proved. Here is the main result of this section. Theorem 3.1. Suppose that f ∈ C 2 (H; R) satis>es (PS); ' (t; x) = (t; x; ') = etA x + U (t; x; ') is a family of LS @ows generated by F' ; the zero eigenvalue of A is isolated in 3(A): Assume '0 is the negative gradient @ow generated by df = A + K'0 and S is a dynamically isolated critical set of f associated with @ow '0 . If K'0 is global Lipschitz continuous on any bounded set and F' (x) − df(x) → 0; uniformly in
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Y.X. Guo, J.Q. Liu / Nonlinear Analysis 48 (2002) 831 – 851
bounded x. Then there exists an isolating triplet (O; ; ) of S and a positive number , ¿ 0 such that ∀' ∈ T; |' − '0 | ¡ , we have cl(O˜ ' ) ∩ K ∩ f−1 [ ; ] = S;
(3.1)
cl(O˜ ' ) ∩ K' ∩ f−1 [ ; ] ⊂ int O;
(3.2)
where K' = {x | F' (x) = 0}. Proof. (1) Let (G; ; ) be one of the isolating triplet of S; then S = cl(G˜ '0 ) ∩ K ∩ f−1 [ ; ]; (3.3) where G˜ '0 = t∈R '0 (t; G). We will show: ∃, ¿ 0 (small enough) and a closed neighborhood G, of S including G such that cl((G˜ , )' ) ∩ K ∩ f−1 [ ; ] = S
for ∀' ∈ T; |' − '0 | ¡ ,:
{Gn }∞ 1
If it is not true, then there exist 1 ⊂ G; Gn = x | d(x; S) ≤ n
and
{'n }∞ 1
(3.4)
satisfying
'n → '0 (n → ∞);
cl((G˜ , )'n ) ∩ K ∩ f−1 [ ; ] = S; that is ∀n ∈ N;
∃yn ∈ cl((G˜ n )'n ) ∩ K ∩ f−1 [ ; ];
and
yn ∈ S;
so yn ∈ Y = cl(G˜ '0 ) ∩ f−1 [ ; ]; now we take a sequence {zn } ∈ (G˜ n )'n such that d(zn ; yn ) → 0 (n → ∞); zn ∈ Y; ∀n. By (PS) condition, {yn } ⊂ K ∩ f−1 [ ; ] has a convergence subsequence denoted by {yn } still. Suppose yn → y (n → ∞), then y ∈ Y . In fact, if y ∈ Y; then y ∈ S. Since G is the closed neighborhood of S, we have {yn } ⊂ G (for n large enough), this is a contradiction. (2) Since 'n is LS Low, which can be written as 'n (x; t) = etA x + U (t; x; 'n ); without loss of generalization, assume zn = 'n (x n ; tn ); tn ¿ 0; x n ∈ Gn and x n → x ∈ S: Note that x n ∈ Y , and zn ∈ Y , there must exist a point on the integral curve starting at x n and ending at zn which belongs to @Y . We suppose wn is the last point in @Y; zn = 'n (wn ; 3n ); x n = 'n (wn ; −sn ), then we claim that 3n ; sn → +∞: If not, 3n is bounded, without loss of generality, we assume that 3n → 30 (n → ∞); thus wn = 'n (zn ; −3n ) → '0 (y; −30 ) = y (note that y ∈ S is a critical point). So, there exists a neighborhood V (y) of y such that {wn } ⊂ V (y) (for n large enough), this is in contradiction with wn ∈ @Y; ∀n. (3) We denote the trajectories starting from x n to zn by 'n (x n zn ) and denote the set of all these trajectories by D. Then Lemma 3.1 implies that D is bounded. (4) We prove that {wn } is compact. By (2), zn = 'n (wn ; 3n ) and x n = 'n (wn ; −sn ) convergence to a critical point in f−1 [ ; ]; moreover 3n ; sn → +∞. Thus for ∀T ¿ 0,
Y.X. Guo, J.Q. Liu / Nonlinear Analysis 48 (2002) 831 – 851
839
−1 TA −T ≤t≤T 'n (wn ; 3n ) ⊂ f [ ; ], for n large enough. Therefore un =e wn +U (wn ; T; 'n ) is bounded, let M be the boundary, H =H+ ⊕H0 ⊕H− be the orthogonal decomposition of H according to the positive, negative, zero eigenvalue subspace of A. Since the zero eigenvalue of A is isolated in the spectra set 3(A). Thus for ∀8 ¿ 0, there exists T ¿ 0 such that
etA ≥ 8x;
∀t ≥ T; x ∈ H+ ;
(3.5)
etA ≥ 8x;
∀t ≤ −T; x ∈ H− :
(3.6)
Assume wn = wn+ + wn0 + wn− ∈ H+ ⊕ H0 ⊕ H− ; ∀n. Since H0 is a "nite-dimensional space, so {wn0 } has a convergence subsequence. In the following, without loss of generality, we assume that {wn+ } has no convergence subsequence, then ∃0 ¿ 0
such that wi+ − wj+ ≥ 0;
for i = j;
(3.7)
thus for 8 = 3M=0; ∃T0 ¿ 0, such that (3.5) is true for ∀t ≥ T0 . Therefore, eT0 A wi − eT0 A wj = eT0 A (wi − wj ) ≥ eT0 A (wi+ − wj+ ) ≥
3M wi+ − wj+ ≥ 3M; 0
on the other hand, eT0 A wi − et0 A wj = eT0 A wi + U (T0 ; wi ; 'i ) + U (T0 ; wi ; 'i ) − U (T0 ; wj ; 'j ) + eT0 A wj + U (To ; wj ; 'j ) ≤ 2M + U (To ; wi ; 'i ) − U (T0 ; wj ; 'j ); hence U (T0 ; wi ; 'i ) − U (T0 ; wj ; 'j ) ¿ M;
for i = j
this is a contradiction with the fact that U is compact. Let w be the limit of the convergence subsequence, then w ∈ @Y: At last, we will deal with a contradiction. Since w ∈ @Y; then we have w = '0 (y; t); for ∀y ∈ G; ∀t ∈ R. Otherwise, by the continuous Low, there exists a neighborhood Br (y) ⊂ G of y such that t∈R '0 (Br (y); t) ⊂ Y is the neighborhood of w, this contradicts with w ∈ @Y . So, for 0 ¿ 0 small enough, ∃h ¿ 0 such that f('0 (w; h)) = + 0. Note ∀T ¿ 0, and n large enough, we have f('n (wn ; T )) ≤ ; hence, for T = h; f('n (wn ; h)) ≤ ; let n → ∞, we obtain f('0 (w; h)) ≤ this is a contradiction. Eq. (3.1) is proved. Now we prove (3.2). In fact, by (PS) condition, ∃2 ¿ 0, such that df(x) ≥ 2
∀x ∈ f−1 [ ; ] \ O
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Y.X. Guo, J.Q. Liu / Nonlinear Analysis 48 (2002) 831 – 851
by Lemma 3.1, ∃,1 ¿ 0, such that for all ', |' − '0 | ¡ ,1 , cl(O˜ ' ) ∩ f−1 [ ; ] are included in a bounded set G. So F' (x) − df(x) → 0 uniformly as to x ∈ G: Therefore, ∃,2 ¿ 0, such that 2 F' (x) − df(x) ¡ ; 2
∀x ∈ G
∀'; |' − '0 | ¡ ,2
thus, if |' − '0 | ¡ , = min{,1 ; ,2 }; y' ∈ cl(O˜ ' ) ∩ K' ∩ f−1 [ ; ], and y' ∈ int O, then 2 ≤ df(y' ) ≤ F' (y' ) − df(y' ) ¡ 2=2 this implies that, ∃, ¿ 0, such that cl(O˜ ' ) ∩ K' ∩ f−1 [ ; ] ⊂ int O
for ∀' ∈ T |' − '0 | ¡ ,:
The theorem is proved. Theorem 3.1 shows if S is a dynamically isolated critical set associated with Low '0 (generated by df), then it is also dynamically isolated about to its nearby Lows ' (generated by F' ). Moreover, the zero point set of the vector "eld F' , which includes cl(O˜ ' ) ∩ f−1 [ ; ] will all be in the interior of O, while ' is su9ciently approximating '0 . As to the approximation ('0 )LS, we have Theorem 3.2. Suppose that f satis>es the assumes in Theorem 3:1; '0 is negative LS gradient @ow of f; S is a dynamically isolated critical set of f associated with @ow '0 . ' (' ∈ T) is a family of approximation ('0 ) LS @ows; then there exists an isolate triplet (O; ; ) of S and a real number , ¿ 0 such that cl(O˜ ' ) ∩ K ∩ f−1 [ ; ] = S;
(3.8)
cl(O˜ ' ) ∩ K' ∩ f−1 [ ; ] ⊂ int O for ∀' ∈ T; |' − '0 | ¡ ,:
(3.9)
Proof. We use the same notation as in Theorem 3.1. The proof of Theorem 3.2 is similar to Theorem 3.1 but a little di;erence in step (4), we give the sketch. Indeed, to prove the compactness of {wn }, we denote the boundary of un =eTA wn +U (wn ; T; 'n )+ !(!n ; T; 'n ) by M , since !(!n ; T; 'n ) → 0 ('n → 0) uniformly in bounded !n , we denote its boundary by m. Similar to Theorem 3.1, for ∀8 ¿ 0, there exists T ¿ 0 such that etA ≥ 8x;
∀t ≥ T; x ∈ H+ ;
(3.10)
etA ≥ 8x;
∀t ≤ −T; x ∈ H− ;
(3.11)
moreover, without loss of generality, again we assume that {wn+ } has no convergence subsequence, then ∃0 ¿ 0;
such that wi+ − wj+ ≥ 0;
for i = j;
(3.12)
thus for 8 = 3(M + m)=0; ∃T0 ¿ 0, such that (3.10) is true for ∀t ≥ T0 . Therefore, eT0 A wi − eT0 A wj = eT0 A (wi − wj )
Y.X. Guo, J.Q. Liu / Nonlinear Analysis 48 (2002) 831 – 851
≥
841
3(M + m) wi+ − wj+ 0
≥ 3(M + m): On the other hand, eT0 A wi − et0 A wj = eT0 A wi + U (T0 ; wi ; 'i ) + !(T0 ; !i ; 'i ) + U (T0 ; wi ; 'i ) − U (T0 ; wj ; 'j )) + !(T0 ; !i ; 'i ) − !(T0 ; !j ; 'j ) + eT0 A wj + U (To ; wj ; 'j ) + !(T0 ; !j ; 'j ) ≤ 2(M + m) + U (To ; wi ; 'i ) − U (T0 ; wj ; 'j ): Thus, U (T0 ; wi ; 'i ) − U (T0 ; wj ; 'j ) ¿ M + m;
for any i = j;
this contradicts with the fact that U is compact. The remainder is same as Theorem 3.1. We omit it. The proof of (3.9) is same as (3.2). 4. Critical group Let be a real separable Hilbert space with inner product ·; · and norm · ; and H ∞ H = i=1 Hi with all subspace Hi mutually orthogonal and "nite dimension. Henceforth, we assume that (1) f ∈ C 2 (H; R) with the form f(x) = 12 Ax; x + G(x). (2) A : H → H is a linear, self-adjoint and bounded operator with a "nite-dimensional kernel N , and the zero eigenvalue of A is isolated in 3(A). (3) ∇G:=K(x) is compact, and global Lipschitz continuous on any bounded set. For a simple reason, we assume that (4) There exist a; b ¿ 0 such that |K(x); x| ≤ ax2 + b ∀x ∈ H . Then it is easy to know, df = A + K is a LS vector "eld. Denition 4.1. Let W = {Pn | n = 1; 2; : : :} be a sequence of orthogonal projections. We said W is an approximation scheme with respect to A, if the following properties hold: (1) Pn H = H n is "nite dimensional for ∀n, (2) Pn → I; (n → ∞) (strongly), (3) [Pn ; A] = Pn A − APn → 0 (n → ∞) (in the operator norm). Theorem 4.1. Suppose that S is a dynamically isolated critical set associated with negative gradient @ow generated by df(x)=Ax+K(x), then there exists an isolating triplet (O; ; ) for S and n0 ∈ N such that for all n ≥ n0 (4.1) cl(O˜ n ) ∩ K ∩ f−1 [ ; ] = S;
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cl(O˜ n ) ∩ Kn ∩ f−1 [ ; ] ⊂ int O; (4.2) where O˜ n = t∈R n (O; t); n is the negative gradient @ow generated by Fn = (I − Pn )A(I − Pn ) + Pn (A + K)Pn ; and Kn denotes the zero points set of vector >eld Fn . Proof. We de"ne a family of approximation (0) LS vector "eld F : H × [0; 1] → H as follows: F(x; s) := (I − Pn )A(I − Pn )x + Pn APn (x) + Pn KPn (x);
1 ∀ n+1 ¡ s ≤ 1n ;
F(x; 0) = Ax + K(x); since F(x; s) = Ax + Pn KPn (x) + (I − Pn )A(I − Pn )(x) + Pn APn (x) − Ax and (Pn A − APn )x → 0 (n → ∞) uniformly w.r.t. bounded x, so [(I − Pn )A(I − Pn ) + Pn APn − A](x) → 0
(n → ∞) uniformly in x
hence, for large n; F(x; s) is a family of approximation (0) LS vector "elds satisfying (2.2). Thus, according to Theorem 2.3, the Lows generated by them is a family of approximation (0) LS Lows, denoted by ˜s : R × H × T → H and ˜0 ≡ . Therefore, by Theorem 3.2, there exists an isolating triplet (O; ; ) for S and s0 ∈ [0; 1] such that for all s ∈ [0; s0 ] cl(O˜ s ) ∩ K ∩ f−1 [ ; ] = S; cl(O˜ s ) ∩ Ks ∩ f−1 [ ; ] ⊂ int O; take n0 ¡ 1=s0 , then for all n ≥ n0 , we have cl(O˜ 1=n ) ∩ K ∩ f−1 [ ; ] = S; cl(O˜ 1=n ) ∩ K1=n ∩ f−1 [ ; ] ⊂ int O; note that ˜1=n = n , the theorem is proved. By the de"nition of the vector "eld Fn and F˜ n , the following lemma is obvious. Lemma 4.1. Let ˜n be the @ow generated by F˜ n (x) = Pn (A + K)Pn (x) : H n → H n , n is the @ow generated by Fn (x) = (I − Pn )A(I − Pn )(x) + Pn (A + K)Pn (x) : H → H; then n |H n = ˜n . Here n |H n is the restriction of n on H n . Theorem 4.2. Suppose that S is a dynamically isolated critical set associated with negative gradient @ow generated by the gradient >eld df(x) = Ax+K(x). If (O; ; ) is an isolating triplet for S associated with @ow n (generated by Fn (x)) satisfying Theorem 4:1 for n large enough. Then (O ∩ H n ; ; ) is an isolating triplet for the critical set Sn = O ∩H n ∩Kfn associated with the @ow generated by dfn (x) = Fn (x)|H n , where fn = f|H n ; Kfn is the critical set of fn .
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843
Proof. By Lemma 4.1, the Low generated by dfn (x) in H n is exactly the restriction of the Low n on H n , which is generated by Fn (x), that is n |H n . Thus, we need only to show cl((O ] ∩ H n )n |H n ) ∩ Kfn ∩ fn−1 [ ; ] ⊂ int (O ∩ H n ) (4.3) since cl(O˜ n )∩K1=n ∩f−1 [ ; ] ⊂ int O and K1=n = {x | (I −Pn )A(I −Pn )(x)+dfn (x) = 0}, if fn has critical point in cl(O˜ n ) ∩ f−1 [ ; ] then it belongs to int(O ∩ H n ). Therefore, for ∀y ∈ cl((O ] ∩ H n )n | n ) ∩ Kfn ∩ fn−1 [ ; ]; H
since cl((O ] ∩ H n )n |H n ) ∩ fn−1 [ ; ] ⊂ cl(O˜ n ) ∩ f−1 [ ; ]; so y ∈ cl(O˜ n ) ∩ f−1 [ ; ] thus y ∈ int(O ∩ H n ). We complete the proof of the theorem. In the following, we need the following de"nitions which will be very useful to what follows: Denition 4.2 (Chang and Ghoussoub [5], Mean value property). Let : R × M → M be a Low, M be a metric space. A subset W is said to have the mean value property (in short MVP), if for ∀x ∈ M and ∀t0 ¡ t1 , we have that (x; [t0 ; t1 ]) ⊂ W, whenever (x; ti ) ∈ W, for i = 0; 1. Denition 4.3 (Chang and Ghoussoub [5], Gromoll–Meyer pairs). Let g be a C 1 functional on a C 1 -Finsler manifold M , and S be a critical subset of g. Subset pairs (W; W− ) is said to be a Gromoll–Meyer (G–M in short) pair of S associated with a Low generated by a p.g.v.f X of g. If the following conditions hold: (1) W is a closed MVP neighborhood of S satisfying W ∩ K = S, and W ∩ g = ∅ for some . (2) W− is an exit set of W: i.e., ∀x0 ∈ W; ∀t1 ¿ 0, such that (x0 ; t1 ) ∈ W, then ∃ t0 ∈ [0; t1 ] such that (x0 ; [0; t0 ]) ⊂ W and (x0 ; t0 ) ∈ W− . (3) W− is closed and is a union of a "nite number of submanifolds that are transversal to the Low . In general, if g is a C 1 functional satisfying the (PS) condition, S is a dynamically isolated critical set for g, then there are many ways to associate a G–M pair (W; W− ) with S. Indeed, if (O; ; ) is an isolating triplet for S, then one can easily verify that W = O˜ ∩ g−1 [ ; ]:={O} ; W− = W ∩ g−1 ( ) (4.4) form a G–M pair for S (see [5]). Lemma 4.2. Let H1 ; H2 be mutually orthogonal subspace, and (Wi ; Wi− ) be a G–M pair for Si in Hi associated with @ow i (i = 1; 2); then (W1 × W2 ; (W1 × W2− ) ∪ (W2 × W1− ))
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is a G–M pair for S1 × S2 associated with @ow 1 × 2 in H1 × H2 , where 1 × 2 : H1 × H2 × R → H1 × H2 is de>ned by 1 × 2 ((x1 ; x2 ); t) = (1 (x1 ; t); 2 (x2 ; t)). Proof. (1) W1 × W2 has MVP. In fact, for ∀x = (x1 ; x2 ) ∈ H1 × H2 , if 1 × 2 (x; ti ) ∈ W1 × W2 , i = 1; 2. Then 1 (x1 ; ti ) ∈ W1 ; 2 (x2 ; ti ) ∈ W2 , since W1 ; W2 has MVP. Thus 1 (x1 ; [t1 ; t2 ]) ⊂ W1 , 2 (x2 ; [t1 ; t2 ]) ⊂ W2 ; so 1 × 2 (x; [t1 ; t2 ]) ⊂ W1 × W2 . (2) (W1 × W2− ) ∪ (W2 × W1− ) is an exit set. For ∀x0 ∈ W1 × W2 , and ∀t1 ¿ 0 if 1 × 2 (x0 ; t1 ) ∈ W1 × W2 , where x0 = (x01 ; x02 ) ∈ W1 × W2 . Then we have (a) 1 (x01 ; t1 ) ∈ W1 but 2 (x02 ; t1 ) ∈ W2 , (b) 2 (x02 ; t1 ) ∈ W2 but 1 (x01 ; t1 ) ∈ W1 , (c) 1 (x01 ; t1 ) ∈ W1 and 2 (x02 ; t1 ) ∈ W2 . We only prove case (a), the other cases are similar. Case (a): 1 (x01 ; t1 ) ∈ W1 but 2 (x02 ; t1 ) ∈ W2 . Note W1− is the exit of W1 , there exists t0 ∈ [0; t1 ) such that 1 (x01 ; [0; t0 ]) ⊂ W1 and 1 (x01 ; t0 ) ∈ W1− thus, 1 × 2 (x0 ; [0; t0 ]) ⊂ W1 × W2 and 1 × 2 (x1 ; t0 ) ∈ W2 × W1− . (3) By the de"nition of product Low 1 × 2 , it is obvious that 1 × 2 is transversal to (W1 × W2− ) ∪ (W1− × W2 ). Combining (1)–(3), the lemma is proved. Lemma 4.3. Suppose that {f' | ' ∈ [0; 1]} is a family of C 1 functional, and {S' } is a family of isolated critical set corresponding to f' ; and (W' ; W'− ) is a G–M pair for S' associated with @ow ' generated by df' ; for any given '0 ; assume that (O; ; ) is an isolating triplet for S'0 satisfying: for ∀' ∈ [0; 1] cl(O˜ ' ) ∩ Kf'0 ∩ f'−1 [ ; ] = S'0 ; 0 [ ; ] ⊂ int O: S' = cl(O˜ ' ) ∩ Kf' ∩ f'−1 0
(4.5)
Let U be a bounded set which includes all the sets cl(O˜ ' ) ∩ f'−1 [ ; ]. If the map0 1 ping '| → f' is continuous from [0:1] to C (U ); then the groups H ∗ (W' ; W'− ) are independent of '; i.e.; for ∀' ∈ [0; 1] H ∗ (W' ; W'− ) ∼ = H ∗ (W' ; W' ): 0
0−
Proof. For '0 = 0, we have c(O˜ ' )∩Kf0 ∩f0−1 [ ; ] = S0 ; thus, by Chang and Ghoussoub [5, Proposition III:2] W˜ 0 = cl(O˜ ' ) ∩ f0−1 [ ; ]
W˜ 0− = W˜ 0 ∩ f0−1 ( )
(4.6)
form a G–M pair for S0 associated with Low '0 , by Chang [5, Proposition III:3] H ∗ (W0 ; W0− ) ∼ = H ∗ (W˜ 0 ; W˜ 0− ): We say that (W˜ 0 ; W˜ 0− ) is also a G–M pair for S' associated with Low ' , whenever ' is su9ciently approximate to 0. In fact, by the de"nition of G–M pair and (4.6), we need only to show that the restriction of f' on W˜ 0 is a bounded below (whenever ' is su9ciently approximate to 0). Otherwise, there exists 'n → 0 and {x n } ⊂ W˜ 0 ('n ) such that f'n → −∞. Since W˜ 0 ('n ) is a bounded set in "nite-dimensional space. So
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{x n } has a convergence subsequence (denoted still by {x n }). Let x0 be the limit, then x0 ∈ W˜ 0 ('n ). Since f' is continuous about ', let n → ∞, we have f'0 (x0 ) → −∞, this is a contradiction. On the other hand, by Chang and Ghoussoub [5, Proposition III:3] H ∗ (W˜ 0 ; W˜ 0− ) ∼ = ∗ H (W' ; W'− ). So, for ' small enough H ∗ (W0 ; W0− ) ∼ = H ∗ (W' ; W'− ): In the end, by the compactness of [0; 1], we complete the proof of lemma. For the simple case, we have Lemma 4.4. Let E be a >nite-dimensional space, A : E → E be a bounded linear self-adjoint operator, 0 ∈ 3(A). Then S = {} is a dynamically isolated critical set of functional g(x) = 12 Ax; x, associated with negative gradient @ow generated by df(x) = Ax, and (N; L) is a G–M pair for S. Where N := Dr (E + ) × Dr (E − );
L := Dr (E + ) × Sr (E − );
Dr (V ) = {x ∈ V | x ≤ r};
Sr (V ) = {x ∈ V | x = r}:
Proof. Since 0 ∈ 3(A), S ={} is the unique critical point of g, it is obvious that S is a dynamically isolated critical set. By the de"nition of G–M pair, the result is obvious. n Recall, we assume that H n = i=1 Hi , thus H n+1 = H n ⊕ Hn+1 , by the properties of projective operator, we have that P n+1 :=Pn+1 − Pn : H → Hn+1 is a projective operator from H to Hn+1 . In the following, we consider (for n large enough) the following family of vector "elds de"ned by F(x; t) : = tPn+1 (A + K)Pn+1 (x) + (1 − t)(Pn (A + K)Pn (x) +P n+1 (A + P)P n+1 (x)) : H n+1 → H n+1 ;
∀t ∈ [0; 1];
(4.7)
where P is a projective operator from H to the kernel space of operator A. Lemma 4.5. Let (Wt ; Wt− ) be G–M pair for St associated with the @ow t generated by the vector >eld F(x; t), then H ∗ (Wt ; Wt− ) are independent of t. Proof. We de"ne a family of approximation vector "elds G(x; s) as follows: G(x; s) : = (I − Pn+1 )A(I − Pn+1 )(x) +n((n + 1)s − 1)(Pn (A + K)Pn (x) + P n+1 (A + P)P n+1 ) +(1 − n(n + 1)s + n)Pn+1 (A + K)Pn+1 (x) G(x; 0) := A(x) + K(x);
1 for ∀ n+1 ¡ s ≤ 1n ;
for s = 0;
then G(x; s) is the approximation (0) LS vector "elds. By Theorem 3.1, there exists an isolating triplet (O; ; ) for S, such that for n large enough and ∀t ∈ [0; 1]. cl(O˜ (t(n)) ) ∩ K ∩ f−1 [ ; ] = S; cl(O˜ (t(n)) ) ∩ Kt(n) ∩ f−1 [ ; ] ⊂ int O;
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˜ t) = (I − Pn+1 )A(I − Pn+1 )(x) + F(x; t), since where t(n) is the Low generated by F(x; ˜ t) = }, use the same method as Theorem 4.1, we can prove that Kt(n) = {x | F(x; cl((O ∩] H n+1 )t(n) |H n+1 ) ∩ KF(x; t) ∩ f−1 [ ; ] ⊂ int(O ∩ H n+1 ); cl((O ∩] H n+1 )t(n) |H n+1 ) ∩ Kfn+1 ∩ f−1 [ ; ] ⊂ int(O ∩ H n+1 ); that is cl((O ∩] H n+1 )t ) ∩ KF (x; t) ∩ f−1 [ ; ] ⊂ int(O ∩ H n+1 ); cl((O ∩] H n+1 )t ) ∩ Kfn+1 ∩ f−1 [ ; ] = Sn+1 by Theorems 4.2 and 3.1, (O ∩H n+1 ; ; ) is an isolating triplet for Sn+1 , the critical set H n+1 )t ) ∩ of fn+1 in "nite-dimensional space O ∩ H n+1 and for ∀t ∈ [0; 1], cl((O ∩] −1 1 f [ ; ] are included in a bounded set U . Clearly, ft is C continuous as t ∈ [0; 1]. Hence, by Lemma 4.3, H ∗ (Wt ; Wt− ) is indenpendent of t. Here we get the main result of this section. Theorem 4.3. For the considered functional f(x) = 12 Ax; x + G(x), >nite-dimensional subspace H n = ⊕n1 Hi of H and the approximation scheme W = {Pn }∞ 1 w.r.t. A; suppose that (Wn ; Wn− ) is a G–M pair for critical set Sn associated with the @ow generated by dfn = Pn (A + K)Pn : H n → H n ; then for n large enough H ∗+m(Pn (A+P)Pn ) (Wn ; Wn− ) ∼ = H ∗+m(Pn+1 (A+P)Pn+1 ) (Wn+1 ; Wn+1 )
(4.8)
where m(·) denotes the Morse index of operator. Proof. Since P n+1 (A + P)P n+1 : Hn+1 → Hn+1 is invertible, by Lemmas 4.2 and 4.4 (Wn × N; (Wn × L) ∪ (N × Wn− )) is a G–M pair for Sn × {} associated with the Low generated by Pn (A + K)Pn + P n+1 (A + P)P n+1 : Hn+1 → Hn+1 in space H n ⊕ Hn+1 , where − − + + ) × D1 (Hn+1 ) L = D1 (Hn+1 ) × S1 (Hn+1 ): N = D1 (Hn+1
Let (W n+1 (t); W−n+1 (t)) be the G–M pair for St associated with the Low generated by F(x; t) in (4:7), then by Lemma 4.3 H ∗ (W n+1 (0); W−n+1 (0)) ∼ = H ∗ (W n+1 (1); W−n+1 (1)): Hence, for t = 1 and t = 0 H ∗ (Wn × N; (Wn × L) ∪ (Wn− × N )) ∼ = H ∗ (Wn+1 ; W(n+1)− ):
(4.9)
− − ); S1 (Hn+1 )) ,→ (N; L) is homotopic. Hence it Since the immersion map (D1 (Hn+1 induces homotopic map − − )=S1 (Hn+1 )) ∧ (Wn =Wn− ) ,→ (N=L) ∧ (Wn =Wn− ) (D1 (Hn+1
by the property of cohomology modules: H ∗ (S 2(n) ∧ Wn =Wn− ) ∼ = H ∗ (N × Wn ; (N × Wn− ) ∪ (Wn × L));
(4.10)
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847
thus H ∗ (S 2(n) ∧ Wn =Wn− ) ∼ = H ∗ (Wn+1 ; W(n+1)− );
(4.11)
note that H ∗ (S 2(n) ∧ Wn =Wn− ) ∼ = H ∗−2(n) (Wn =Wn− );
(4.12)
2(n)
is 2(n) dimensional unit ball. 2(n) is the Morse index of P where S as to Hn+1 . Since P n+1 APn + Pn AP n+1 → ;
n+1
(A + P)P n+1
(n → ∞);
hence m(Pn+1 (A + P)Pn+1 ) = m(Pn (A + P)Pn ) + m(P n+1 (A + P)P n+1 ); that is 2(n) = m(P n+1 (A + P)P n+1 ) = m(pn+1 (A + P)Pn+1 ) − m(Pn (A + P)Pn ); combine (4.11) with (4.12), H ∗−(m(Pn+1 (A+P)Pn+1 −m(Pn (A+P)Pn )) (Wn ; Wn− ) ∼ = H ∗ (Wn+1 ; W(n+1)− ) so H ∗+m(Pn (A+P)Pn ) (Wn ; Wn− ) ∼ = H ∗+m(Pn+1 (A+P)Pn+1 ) (Wn+1 ; W(n+1)− ): Now, we can de"ne the critical group for the strongly inde"nite functional on its dynamically isolated set as follows: Denition 4.4. Let f be an inde"nite functional satisfying the hypothesis of (1) – (4), S be a dynamically isolated set for f associated with its negative gradient Low. Assume that Pn ; (Wn ; Wn− ) are same as Theorem 4.3, we de"ne (for n large enough) C∗ (f; S) := H ∗+m(Pn (A+P)Pn ) (Wn ; Wn− ) and call it the qth critical group of S with respect to Low generated by negative gradient vector "eld df. Remark 4.1. De"nition 4.4 is independent of the choice of isolating triplet. Indeed, if (O ; ; ) is another isolating triplet, then for large n; (O ∩ H n ; ; ) is also an isolating triplet for a critical set of restriction functional fn in "nite-dimensional space H n . Assume that (Wn ; Wn− ) is the corresponding G–M pair which is de"ned by (4.6), ) ∼ by Chang and Ghoussoub [5, Proposition III:2 and Theorem III:3]: H ∗ (Wn ; Wn− = ∗ H (Wn ; Wn− ) (for n large enough). Remark 4.2. De"nition 4.4 is independent of the choice of the approximation scheme. ˜ In fact, if W={ P˜ n | n=1; 2 : : :} is another approximation scheme. We take W1 ={Pn | n= 1; 2 : : :} such that Pn = Pn for n = 2k − 1; Pn = P˜ n for n = 2k. Therefore, Theorem 4.3 is true for the approximation scheme W1 , thus for n large enough H ∗+m(Pn (A+P)Pn ) (Wn ; Wn− )∼ ; W(n+1)− ) = H ∗+m(Pn+1 (A+P)Pn+1 ) (Wn+1
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taking n large enough, we have ˜
˜
H ∗+m(Pn (A+P)Pn ) (Wn ; Wn− ) ∼ = H ∗+m(Pn+1 (A+P)Pn+1 ) (W˜ n+1 ; W˜ (n+1)− ): Remark 4.3. For a simple reason, we assume that: ∃a; b ¿ 0 such that |K(x); x| ≤ ax2 + b
∀x ∈ H
in fact, for general case, let (O; ; ) be an isolating triplet for S, and without the loss of generality, we assume O ⊂ B(0; s). De"ne 1 for t ≤ s; C(t) = 1 + s − t for s ¡ t ≤ s + 1; 0 for t ≥ s + 1 and d : H → [0; 1]; d(x) = C(x). Then K1 (x) = d(x)K(x) satis"es the above condition. Hence, we can use K1 (x) to substitute K(x), since the Low generated by A + K is same as the Low generated by A+K1 in bounded set O and for "nite T , so the critical groups de"ned by them are the same. 5. Morse-type inequality We have pointed out that, one of the advantages that we developed the Gromoll– Meyer theory is that a dynamically isolated set has natural Morse decomposition. For a given dynamically isolated critical set S, each of its Morse decomposition set is also a dynamically critical set. The problem is: if there are any relationship between the critical group of S and the critical group of its Morse decomposition sets. The following theorem is a description of this relationship. Suppose that f satis"es the hypothesis in Section 4 and S is a dynamically isolated critical set of f associated with the Low generated by the negative gradient vector "eld, (O; ; ) is its isolating triplet. We denote its corresponding G–M pair by (W; W− ), which is de"ned by W = {O} , W− = W ∩f−1 ( ). Let = d0 ¡d1 ¡d2 ¡ · · · ¡dm−1 ¡ dm = be the regular value of f. Let ˜ ∩ K ∩ f−1 [ ; d1 ]; S1 = cl(O) ··· ˜ ∩ K ∩ f−1 [di−1 ; di ]; Si = cl(O) ······ ˜ ∩ K ∩ f−1 [dm−1 ; ]; Sm = cl(O) then m
i=1
Si = S;
Si ∩ Sj = ∅; i = j
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and Si is a dynamically isolated critical set (it is possible to be ∅), and (O∩f−1 [di−1 ; di ]; di−1 ; di ) is the corresponding isolating triplet. Let M q (W; W− ) =
n
dim Cq (f; Si );
∀q ∈ Z;
i=1
q (W; W− ) = dim Cq (f; S); q
∀q ∈ Z:
q
If M ; are "nite for each q and equal to zero for large |q|. We may de"ne the Morse and Poincare polynomials for (W; W− ) by M (t; W; W− ) =
∞
M q (W; W− )t q ;
q=−∞
P(t; W; W− ) =
∞
q (W; W− )t q :
q=−∞
Theorem 5.1. Assumptions and notations are as above. In addition; if for each q; M q (W; W− ) and q (W; W− ) are >nite ∞and equal to zero for all |q| suGciently large; then there exists a function Q(t) = q=−∞ aq t q ; such that all aq are nonnegative; and M (t; W; W− ) = P(t; W; W− ) + (1 + t)Q(t); moreover; n
P(t; C∗ (f; Si )) = P(t; C∗ (f; S)) + (1 + t)Q(t):
i=1
Proof. Let Z ⊂ Y ⊂ X be closed subsets of H: Then we have long exact sequence (see [8,11]). j∗
i∗
,∗
· · · → H q (X; Z) → H q (Y; Z) → H q+1 (X; Y ) → H q+1 (X; Z) → · · · : We denote the rank of a mapping by R and dim H (·) by q (·). Since q are "nite for all q and zero for all |q| large. It follows from the exactness, we have q (X; Z) = dim R(j q ) + dim R(iq ); q (Y; Z) = dim R(iq ) + dim R(,q ); q (X; Y ) = dim R(,q−1 ) + dim R(j q ); thus q (X; Y ) + q (Y; Z) = q (X; Z) + dim R(,q−1 ) + dim R(,q ): Denote the Poincare polynomial of (X; Y ) by P(t; X; Y ) and set q(t; X; Y; Z) =
∞
(dim R(,q ))t q ;
q=−∞
(5.1)
850
Y.X. Guo, J.Q. Liu / Nonlinear Analysis 48 (2002) 831 – 851
since dim R(,q ) = 0, whenever |q| is large. It follows by multiplying t q to (5.1) and summing over q that P(t; X; Y ) + P(t; Y; Z) = P(t; X; Z) + (1 + t)q(t; X; Y; Z):
(5.2)
Let Wi = (W ∩ fdi ) ∪ W− ; 1 ≤ i ≤ m then Wm = W ∩ f ∪ W− = W;
W 0 = W ∩ f ∪ W− = W −
and W0 ⊂ W1 ⊂ W2 · · · ⊂ Wm are closed sets, for j ≥ 2, apply (5.2) to W0 ⊂ Wj−1 ⊂ Wj , and then summing from 2 to m, we have m
P(t; Wj ; Wj−1 ) +
j=2
m
P(t; Wj−1 ; W0 )
j=2
=
m
P(t; Wj ; W0 ) + (1 + t)
j=2
m
q(t; Wj ; Wj−1 ; W0 );
j=2
thus m
P(t; Wj ; Wj−1 ) = P(t; Wm ; W0 ) + (1 + t)Q(t);
j=1
where Q(t) =
m
q(t; Wj ; Wj−1 ; W0 );
j=2
that is m
P(t; Wj ; Wj−1 ) = P(t; W; W− ) + (1 + t)Q(t);
(5.3)
j=1
set W˜ i = {x ∈ W | di−1 ≤ f(x) ≤ di }; W˜ i− = W˜ i ∩ (W− ∪ f−1 (di−1 )); then it is easy to see that (W˜ i ; W˜ i− ) is a G–M pair for Si (0 ≤ i ≤ m). By excision, it follows that H ∗ (Wi ; Wi−1 ) ∼ = H ∗ (W˜ i ; W˜ i− ); de"ne P(t; C∗ (f; Si )) = P(t; W˜ i ; W˜ i− ); then (5.3) implies n
P(t; C∗ (f; Si )) = P(t; C∗ (f; S)) + (1 + t)Q(t);
i=1
we complete the proof of the theorem.
Y.X. Guo, J.Q. Liu / Nonlinear Analysis 48 (2002) 831 – 851
851
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