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Abstract and Applied Analysis Homological local linking
Homological local linking
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卷:
3
年:
1998
语言:
english
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Abstract and Applied Analysis
DOI:
10.1155/s1085337598000505
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HOMOLOGICAL LOCAL LINKING KANISHKA PERERA Abstract. We generalize the notion of local linking to include certain cases where the functional does not have a local splitting near the origin. Applications to secondorder Hamiltonian systems are given. 1. Introduction The notion of local linking introduced by Li and Liu [4] plays a useful role in a wide variety of problems in the Calculus of Variations. Let F be a real C 1 function deﬁned on a Banach space X. We say that F has a local linking near the origin if X has a direct sum decomposition X = X1 ⊕ X2 with j = dim X1 < ∞, F (0) = 0, and, for some r > 0, (1) F (u) ≤ 0 for u ∈ X1 , u ≤ r, F (u) > 0 for u ∈ X2 , 0 < u ≤ r. Then it is clear that 0 is a critical point of F . In this paper we give the following more general deﬁnition of “homological” local linking: Deﬁnition 1.1. Assume that 0 is an isolated critical point of F with F (0) = 0 and let q, β be positive integers. We say that F has a local (q, β)linking near the origin if there exist a neighborhood U of 0 and subsets A, S, B of U with A ∩ S = ∅, 0 ∈ A, A ⊂ B such that 1. 0 is the only critical point of F in F0 ∩ U where F0 is the sublevel set {u ∈ X : F (u) ≤ 0}, 2. denoting by i1∗ : Hq−1 (A) −→ Hq−1 (U \ S) and i2∗ : Hq−1 (A) −→ Hq−1 (B) the embeddings of the singular homology groups induced by inclusions, rank i1∗ − rank i2∗ ≥ β, 1991 Mathematics Subject Classiﬁcation. Primary: 58E05. Key words and phrases. Morse theory, critical groups, local linking. Received: March 10, 1998. c 1996 Mancorp Publishing, Inc. 181 182 K. PERERA 3. F ≤ 0 on B, 4. F > 0 on S \{0}. If F satisﬁes the condition (1) with j ≥ 1 and 0 is an isolated critical point of F , taking U to be a suﬃciently small closed ball Bρ centered at the origin, A = ∂Bρ ∩ X1 , S = Bρ ∩ X2 , and B = Bρ ∩ X1 , we see that F has a local (j, 1)linking near the origin. The following example in R shows that our deﬁnition is, in fact, weaker than (1): Example 1.2 (Monkey saddle). F (x, y) = x3 − 3xy 2 has a loc; al (1, 2)linking near the origin; see Proposition 2.1. Note that the critical groups of F at 0 are given by C∗ (F, 0) = H∗ (F0 ∩ U, (F0 ∩ U )\{0}) (see Chang [2] or Mawhin and Willem [8]). It was proved in Liu [7] that if F satisﬁes (1) and 0 is an isolated critical point of F , then Cj (F, 0) = 0. This fact was used in Perera [9] to obtain a nontrivial critical point u with either Cj+1 (F, u) = 0 or Cj−1 (F, u) = 0, under an additional assumption on the behavior of F at inﬁnity. Here we extend these results to the case where F satisﬁes the weaker conditions given in Deﬁnition 1.1 near the origin. As an application we prove the existence of nontrivial timeperiodic solutions of a system of ordinary diﬀerential equations, under diﬀerent hypotheses on the behavior of the nonlinearity at inﬁnity. For the existence of nontrivial critical points under the usual deﬁnition of local linking and various assumptions at inﬁnity see Li and Liu [4], Li and Liu [5], Liu [7], Silva [10], Brézis and Nirenberg [1], Li and Willem [6], and their references. 2. An Example As an example of homological local linking, we prove the following proposition: Proposition 2.1. Assume that 0 is an isolated critical point of F and F (u) = P (u) + o(us ) at u = 0 where P is homogeneous of degree s > 1, i.e., P (λ u) = λs P (u) ∀λ ≥ 0, ∀u ∈ X. Assume also that there are disjoint subsets Ã, S̃ of S ∞ , the unit sphere in X, such that 1. the rank of the embedding Hq−1 (Ã) −→ Hq−1 (S ∞\S̃) is at least β + δq1 , 2. supÃ P < 0, 3. inf S̃ P > 0. Then F has a local (q, β)linking near the origin. HOMOLOGICAL LOCAL LINKING 183 Proof. Fix > 0 so that supÃ P + < 0 < inf S̃ P − and take ρ > 0 suﬃciently small such that 0 is the only critical point of F in U = Bρ = {u ∈ X : u ≤ ρ} and F (u) − P (u) ≤ us for u ≤ ρ. Then take A = B = S = ρ u : u ∈ Ã , λ u : u ∈ Ã, 0 ≤ λ ≤ ρ , λ u : u ∈ S̃, 0 ≤ λ ≤ ρ . On B, s s F (λ u) ≤ P (λ u) + λ u = λ (P (u) + ) ≤ λ s s F (λ u) ≥ P (λ u) − λ u = λ (P (u) − ) ≥ λ sup P + Ã and on S \{0}, s s ≤ 0, inf P − > 0. S̃ Nowwe verify the condition 2 of Deﬁnition 1.1. Set Sρ = ∂Bρ and S̃ρ = ρ u : u ∈ S̃ . By the assumption 1, the rank of the embedding i∗ : Hq−1 (A) −→ Hq−1 (Sρ \ S̃ρ ) is at least β + δq1 . The map (t, λ u) −→ [(1 − t)λ + t] u 0 ≤ t ≤ 1, u ∈ S ∞ \ S̃, 0 < λ ≤ ρ is a strong deformation retraction of Bρ \S onto Sρ \ S̃ρ and hence the embedding Hq−1 (Sρ \ S̃ρ ) −→ Hq−1 (Bρ \S) is an isomorphism. It follows that rank i1∗ = rank i∗ ≥ β + δq1 . On the other hand, B is contractible to 0 via (t, λ u) −→ (1 − t)λ u 0 ≤ t ≤ 1, u ∈ Ã, 0 ≤ λ ≤ ρ and hence rank i2∗ = δq1 . 3. Critical Groups of the Origin and Nontrivial Critical Points The following theorem extends Theorem 2.1 of Liu [7]: Theorem 3.1. If F has a local (q, β)linking near the origin, then rank Cq (F, 0) ≥ β. 184 K. PERERA Proof. Consider the following portion of the exact sequence of the pair (F0 ∩ U, F0 ∩ U \{0}): Cq (F, 0) ∂∗ ✲ Hq−1 (F0 ∩ U \{0}) i∗✲ Hq−1 (F0 ∩ U ) We have rank Cq (F, 0) ≥ rank ∂∗ = nullity i∗ . Consider the following commutative diagram induced by inclusions: Hq−1 (A) i2∗ ✲ Hq−1 (B) i1∗ ✠ Hq−1 (U \S) i3∗ ❅ I ❅ ❅ ❄ Hq−1 (F0 ∩ U \{0}) ❄ ✲ Hq−1 (F0 ∩ U ) i∗ We have nullity i∗ ≥ rank i3∗ − rank i2∗ ≥ rank i1∗ − rank i2∗ ≥ β. Now we assume that F satisﬁes the PalaisSmale compactness condition (PS) and has only isolated critical values, with each critical value corresponding to a ﬁnite number of critical points, and set Kc = set of critical points of F where F = c, Kab = set of critical points of F where a < F < b. The main result of this section is the following: Theorem 3.2. Suppose that F has a local (q, β)linking near the origin and assume that there are regular values a, b of F such that a < 0 < b and rank Hq (Fb , Fa ) < β. Then rank Cq−1 (F, u) + u∈Ka0 rank Cq+1 (F, u) ≥ β − rank Hq (Fb , Fa ). u∈K0b In particular, F has a (nontrivial) critical point u with either a < F (u) < 0 and Cq−1 (F, u) = 0, or 0 < F (u) < b and Cq+1 (F, u) = 0. Theorem 3.2 follows from Lemma 3.3 below and Theorem 3.1. HOMOLOGICAL LOCAL LINKING 185 Lemma 3.3. If a < b are regular values of F , c ∈ (a, b), and q ∈ Z, then u∈Kac ≥ rank Cq−1 (F, u) + rank Cq+1 (F, u) u∈Kcb rank Cq (F, u) − rank Hq (Fb , Fa ). u∈Kc The proof of Lemma 3.3 makes use of the following topological lemma: Lemma 3.4. If B ⊂ B ⊂ A ⊂ A are topological spaces and q ∈ Z, then rank Hq−1 (B, B ) + rank Hq+1 (A , A) ≥ rank Hq (A, B) − rank Hq (A , B ). Proof. Consider the following portions of the exact sequences of the triples (A, B, B ) and (A , A, B ), respectively: Hq (A, B ) Hq+1 (A , A) j∗ ✲ Hq (A, B) ∂∗✲ Hq (A, B ) ∂∗✲ Hq−1 (B, B ), i∗ ✲ Hq (A , B ) We have rank Hq−1 (B, B ) + rank Hq (A, B ) ≥ rank Hq (A, B), rank Hq (A , B ) + rank Hq+1 (A , A) ≥ rank Hq (A, B ). Proof of Lemma 3.3. Take > 0 such that a < c − < c + < b and c is the only critical value of F in [c − , c + ]. Applying Lemma 3.4 to Fa ⊂ Fc− ⊂ Fc+ ⊂ Fb , rank Hq−1 (Fc− , Fa ) + rank Hq+1 (Fb , Fc+ ) ≥ rank Hq (Fc+ , Fc− ) − rank Hq (Fb , Fa ). But, by Chapter I, Theorem 4.3, Corollary 4.1, and Theorem 4.2 of Chang [2], rank Hq−1 (Fc− , Fa ) ≤ u∈Kac rank Hq+1 (Fb , Fc+ ) ≤ rank Cq−1 (F, u), rank Cq+1 (F, u), u∈Kcb rank Hq (Fc+ , Fc− ) = rank Cq (F, u). u∈Kc The following corollary generalizes Theorem 2.2 of Liu [7] and Theorem 5 of Brézis and Nirenberg [1]. See Remark 2.3 of Liu [7] and the remarks following Theorem 4 and proof of Theorem 5 of Brézis and Nirenberg [1] for the history of this result: 186 K. PERERA Corollary 3.5 (Three Critical Point Theorem). Suppose that F has a local (q, β)linking near the origin and assume that F is bounded below. If q = 1 and β ≥ 2, or q ≥ 2, then F has at least two nontrivial critical points. If q ≥ 2, then F has a nontrivial critical point which is not a local minimizer. Proof. F achieves its minimum at some point u0 with F (u0 ) ≤ 0 and rank Cj (F, u0 ) = δj0 (for the critical groups of an isolated local minimum point see Example 1 in Chapter I, Section 4 of Chang [2]). By Theorem 3.1, Cq (F, 0) = 0 and hence u0 = 0. Supposing 0 and u0 to be the only critical points and taking a < inf X F and b = +∞ in Theorem 3.2, we have β ≤ δq1 . If q ≥ 2, the critical point u = 0 with either Cq−1 (F, u) = 0 or Cq+1 (F, u) = 0 obtained in Theorem 3.2 is not a local minimizer. 4. Secondorder Hamiltonian systems Consider the secondorder nonautonomous system (2) ẍ = ∇ V (t, x) where V ∈ C 1 (R × R , R) is 2πperiodic in t and satisﬁes (V1 ): there are constants µ > 2 and R > 0 such that 0 < µ V (t, x) ≤ x · ∇ V (t, x) for x ≥ R, ∀t, (V2 ): there is a homogeneous function P ∈ C 1 (R , R) of degree s > 2 such that V (t, x) = P (x) + o(xs ) at x = 0, uniformly in t. Theorem 4.1. Assume (V1 ), (V2 ), and the following condition on P : (P ): there are disjoint subsets Ã, S̃ of S n−1 such that 1. for some positive integers q ≤ n and β, the rank of the embedding Hq−1 (Ã) −→ Hq−1 (S n−1 \ S̃) is at least β + δq1 , 2. P < 0 on Ã, 3. P > 0 on S̃. Then (2) has at least one nonzero 2πperiodic solution. Remark 4.2. Our assumption (P ) generalizes the condition (P4 ) of Felmer and Silva [3]. See Theorem 7 and the remark following it in Li and Willem [6] for related results. HOMOLOGICAL LOCAL LINKING 187 Proof of Theorem 4.1. We seek solutions of (2) as critical points of the functional 2π 1 F (x) = ẋ2 + V (t, x(t)) dt 2 0 deﬁned on the Hilbert space X of vector functions x(t) having period 2π and belonging to H 1 on [0, 2π], with the norm 1 x = √ 2π 2π 0 2 1/2 2 ẋ + x . It is wellknown that F satisﬁes (PS). As in the proof of Lemma 3.2 of Wang [11], (V1 ) also implies that Hj (X, Fa ) = 0 ∀j ∈ Z for a < 0 and a suﬃciently large. The conclusion follows from Theorem 3.2 and the Lemma 4.3 below. Lemma 4.3. If V satisﬁes (V2 ) with P as in Theorem 4.1, then F has a local (q, β)linking near the origin. Proof. We have the splitting X = X1 ⊕ X2 where X1 is the space of constant functions, identiﬁed with R , and X2 is the space of functions in X whose integral is zero. We take U = A = B = S = λ x1 + x2 : x1 ∈ S n−1 , x2 ∈ X2 , 0 ≤ λ ≤ ρ, x2 ≤ ρ , ρ x1 : x1 ∈ Ã , λ x1 : x1 ∈ Ã, 0 ≤ λ ≤ ρ , λ x1 + x2 : x1 ∈ S̃, x2 ∈ X2 , 0 ≤ λ ≤ ρ, x2 ≤ ρ . As in the proof of Proposition 2.1, F ≤ 0 on B for ρ > 0 suﬃciently small. On S \{0}, 1 ẋ2 2 + P (λ x1 + x2 ) + o(1) λ x1 + x2 s as ρ → 0. 2 By the mean value theorem and the Young’s inequality, F (λ x1 + x2 ) = P (λ x1 + x2 ) = P (λ x1 ) + ∇P (λ x1 + θ x2 ) · x2 for some θ ∈ [0, 1] ≥ λs P (x1 ) − C λ x1 + θ x2 s−1 x2  ≥ λs min P − C λ(s−1)p + x2 p + x2 s S̃ where p, p are conjugate exponents with 2 < p < s. It follows that F (λ x1 + x2 ) ≥ λ s + on S for small ρ. 2π min P − C ρ(s/p −1)p + o(1) S̃ 1 ẋ2 2 − C x2 p + x2 s > 0 2 188 K. PERERA Set U1 = {x1 ∈ X1 : x1  ≤ ρ} and S1 = λ x1 : x1 ∈ S̃, 0 ≤ λ ≤ ρ . As in the proof of Proposition 2.1, the rank of the embedding Hq−1 (A) −→ Hq−1 (U1 \S1 ) is at least β + δq1 . The map (t, λ x1 + x2 ) −→ λ x1 + (1 − t) x2 0 ≤ t ≤ 1, x1 ∈ S n−1 \ S̃, x2 ∈ X2 , 0 < λ ≤ ρ, x2 ≤ ρ is a strong deformation retraction of U \S onto U1 \S1 , and hence the embedding Hq−1 (U1 \S1 ) −→ Hq−1 (U \S) is an isomorphism. Remark 4.4. Note that Theorem 3.2 gives a critical point x with either F (x) < 0 and Cq−1 (F, x) = 0, or F (x) > 0 and Cq+1 (F, x) = 0, yielding Morse index estimates for x via the Shifting theorem when V , and hence F , is C 2 (see Chapter I, Theorem 5.4 of Chang [2]): either F (x) < 0 and m(x) ≤ q − 1 ≤ m∗ (x), or F (x) > 0 and m(x) ≤ q + 1 ≤ m∗ (x) where m(x) and m∗ (x) = m(x) + dim ker d2 F (x) denote the Morse index and the large Morse index of x, respectively. This additional information can sometimes be used to distinguish x from the constant solutions, when they exist. Now we replace (V1 ) by the condition (V1 ) : V (t, x) → +∞ as x → ∞ uniformly in t, which implies that F satisﬁes (PS) and is bounded below. Then Corollary 3.5 yields Theorem 4.5. Assume (V1 ) , (V2 ), and (P ). If q = 1 and β ≥ 2, or q ≥ 2, then (2) has at least two nonzero 2πperiodic solutions. Remark 4.6. See Theorems 7 and 7’ in Brézis and Nirenberg [1] for related results. References 1. H. Brézis and L. Nirenberg, Remarks on ﬁnding critical points, Comm. Pure Appl. Math. XLIV (1991), 939–963. 2. K.C. Chang, Inﬁnitedimensional Morse theory and multiple solution problems, Progress in Nonlinear Diﬀerential Equations and their Applications, #6, Birkhäuser, Boston, 1993. 3. P. L. Felmer and E.A. de B. e Silva, Subharmonics near an equilibrium for some secondorder Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 819–834. 4. S. J. Li and J. Q. Liu, Some existence theorems on multiple critical points and their applications, Kexue Tongbao, 17 (1984), 1025–1027. 5. , Morse theory and asymptotically linear Hamiltonian systems, J. Diﬀerential Equations, 78 (1989), 53–73. 6. S. J. Li and M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl. 189 (1995), 6–32. 7. J. Liu, The Morse index of a saddle point, Systems Sci. Math. Sci. 2 (1989), 32–39. HOMOLOGICAL LOCAL LINKING 189 8. J. Mawhin and M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, #74, SpringerVerlag, New York, 1989. 9. K. Perera, Critical groups of critical points produced by local linking with applications, preprint. 10. E.A. de B. e Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Anal. 16 (1991), 455–477. 11. Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 43–58. 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