Electronic Journal of Qualitative Theory of Differential Equations

In this paper, we study the existence of innitely many homoclinic solutions for a class of second order Hamiltonian systems with general potentials near the origin. Recent results from the literature are generalized and signicantly improved.


Introduction and main results
Consider the following second order Hamiltonian system ü − L(t)u + W u (t, u) = 0, ∀ t ∈ R, (HS) where u = (u 1 , . . ., u N ) ∈ R N , L ∈ C R, R N 2 is a symmetric matrix-valued function, and W u (t, u) denotes the gradient of W (t, u) with respect to u.Here, as usual, we say that a solution u of (HS) is homoclinic (to 0) if u ∈ C 2 R, R N , u(t) ≡ 0, and u(t) → 0 as |t| → ∞.
With the aid of variational methods, the existence and multiplicity of homoclinic solutions for (HS) have been extensively investigated in the literature over the past several decades (see, e.g.,  and the references therein).Many early papers (see, e.g., [1-3, are either independent of t or periodic in t.Compared to the periodic case, the problem is quite different in nature for the nonperiodic case due to the lack of compactness of the Sobolev embedding.After the work of Rabinowitz and Tanaka [17], there are many papers (see, e.g., [4,5,7,9,[11][12][13][14][18][19][20][21][22][23][24][25][26][27]) concerning the nonperiodic case.For this case, the function L plays an important role.Actually, most of these mentioned papers assumed that L is either coercive or uniformly positively definite.Besides, we also note that all these papers required W (t, u) to satisfy some kind of growth conditions at infinity with respect to u, such as superquadratic, asymptotically quadratic or subquadratic growth.
In the recent paper [28], the authors obtained infinitely many homoclinic solutions for (HS) without any conditions assumed on W (t, u) for |u| large.To be precise, W (t, u) in that paper is only locally defined near the origin with respect to u, but L is required to satisfy a very strong coercivity condition.Motivated by [28], in the present paper, we will study the existence of infinitely many homoclinic solutions for (HS) in the case where L is unnecessarily coercive, and W (t, u) is still only locally defined near the origin with respect to u.More precisely, we make the following assumptions: ) is even in u and W (t, 0) ≡ 0, where B δ (0) is the open ball in R N centered at 0 with radius δ.
(W 3 ) There exist a constant > 0, a closed interval I 0 ⊂ R and two sequences of positive Our main result reads as follows.
Remark 1.2.Compared to Theorem 1.1 in [28], the matrix-valued function L in our Theorem 1.1 is not required to satisfy the coercivity condition (L 1 ) or the technical condition (L 2 ) of Theorem 1.1 in [28].In addition, our Theorem 1.1 also essentially improves some related results in the existing literature.It is easy to see that the conditions of our Theorem 1.1 are weaker than those of Theorem 1.2 in [12,18,19].Indeed, there is a function W which satisfies conditions (W 1 )-(W 3 ) but does not satisfy the corresponding conditions of Theorem 1.2 in [12,18,19].For example, let where > 0 small enough and α ∈ (1 + , 2).Then it is easy to check that W satisfies 2 Variational setting and proof of the main result Then E is a Hilbert space and we denote by • the associated norm.Moreover, we write E * for the topological dual of E with norm • E * , and Consequently, there exists τ p > 0 such that where In order to prove our main result via the critical point theory, we need to modify W (t, u) for u outside a neighborhood of the origin to get W (t, u) as follows.
Now we introduce the following modified Hamiltonian system and define the variational functional Φ associated with ( HS) by (2.5) Proposition 2.1.Let (L 0 ), (W 1 ) and (W 2 ) be satisfied.Then Ψ ∈ C 1 (E, R) and Ψ : for all u, v ∈ E, and nontrivial critical points of Φ on E are homoclinic solutions of ( HS).
Proof.First, we show that Φ and Ψ are both well defined.For notational simplicity, we set and always use these notations in the sequel.Since For any u ∈ E, by (2.1), (2.3) and the Hölder inequality, we have where | • | µ i denotes the usual norm of in L µ i (R, R) and τ µ * i is the constant given in (2.1) for i = 1, 2. This together with (2.5) implies that Φ and Ψ are both well defined.
Next, we prove Ψ ∈ C 1 (E, R) and Ψ : E → E * is compact.For any given u ∈ E, define an associated linear operator J (u) : E → R by By (2.1), (2.4) and the Hölder inequality, there holds where c 1 is the constant given in (2.4).This implies that J (u) is well defined and bounded.
By (2.4), for any η ∈ [0, 1], there holds Therefore, for any u, v ∈ E, by the Mean Value Theorem and Lebesgue's Dominated Convergence Theorem, we have EJQTDE, 2013 No. 54, p. 5 where θ(t) ∈ [0, 1] depends on u, v, s.This implies that Ψ is Gâteaux differentiable on E and the Gâteaux derivative of (2.10) Consequently, there exists a constant D 0 > 0 such that For any > 0, by (W 2 ), there exists T > 0 such that where τ ∞ is the constant given in (2.1).Now for any > 0, combining (2.14) and (2.15), we have This means that J is completely continuous.Thus Ψ ∈ C 1 (E, R) and (2.6) holds with Ψ = J .Consequently, Ψ is completely continuous.This together with the reflexivity of Hilbert space E implies that Ψ is compact.In addition, due to the form of Φ in (2.5), we know that Φ ∈ C 1 (E, R) and (2.7) also holds.
Finally, a standard argument shows that nontrivial critical points of Φ on E are homoclinic solutions of ( HS).The proof is completed.2 We will use the following variant symmetric mountain pass lemma due to Kajikiya [29] to prove that ( HS) possesses a sequence of homoclinic solutions.Before stating this theorem, we first recall the notion of genus.
Let E be a Banach space and A a subset of E. A is said to be symmetric if u ∈ A (Φ 2 ) For each k ∈ N, there exists an Then either (i) or (ii) below holds.
(i) There exists a critical point sequence {u k } such that Φ(u k ) < 0 and lim k→∞ u k = 0.
(ii) There exist two critical point sequences {u k } and , and {v k } converges to a non-zero limit.
In order to apply Theorem 2.2, we will show in the following lemmas that the functional Φ defined in (2.5) satisfies conditions (Φ 1 ) and (Φ 2 ) in Theorem 2.2.
Proof.We first prove that Φ is bounded from below.By (2.5) and (2.8), there holds Since ν < 2, it follows that Φ is bounded from below.
Next, we show that Φ satisfies (PS) condition.Let {u n } n∈N ⊂ E be a (PS)-sequence, i.e., for some D 1 > 0. By (2.16) and (2.17), we have for some u 0 ∈ E. By virtue of the Riesz Representation Theorem, Φ : E → E * and Ψ : E → E * can be viewed as Φ : E → E and Ψ : E → E respectively.This together with (2.7) yields Thus Φ satisfies (PS) condition.The proof is completed.2 Lemma 2.4.Let (L 0 ), (W 1 ) and (W 3 ) be satisfied.Then for each k ∈ N, there exists an Proof.We follow the idea of dealing with elliptic problems in Kajikiya [29].Let d 0 be the length of the closed interval I 0 in (W 3 ).For any fixed k ∈ N, we divide I 0 equally into k closed sub-intervals and denote them by I i with 1 ≤ i ≤ k.Then the length of each I i is a ≡ d 0 /k.For each 1 ≤ i ≤ k, let t i be the center of I i and J i be the closed interval centered at t i with length a/2 .Choose a function ϕ ∈ C ∞ 0 (R, R N ) such that |ϕ(t)| ≡ 1 for t ∈ [−a/4, a/4], ϕ(t) ≡ 0 for t ∈ R \ [−a/2, a/2], and |ϕ(t)| ≤ 1 for all t ∈ R. Now for Then it is easy to see that and EJQTDE, 2013 No. 54, p. 9 Evidently, V k is homeomorphic to the unit sphere in R k by an odd mapping.Thus For any s ∈ (0, b) and u = k i=1 r i ϕ i ∈ W k , by (2.5), (2.20) and (2.21), we have where the last equality holds by the definition of W in (2.2) and the fact that |sr i ϕ i (t)| < b for all 1 ≤ i ≤ k.Observing the definition of V k , for every u = k i=1 r i ϕ i ∈ W k , there exists some integer 1 ≤ i u ≤ k such that |r iu | = 1.Then it follows that Here we use the fact that |δ n r iu ϕ iu (t)| ≡ δ n for t ∈ J iu .Note that δ n → 0 and M n → ∞ as n → ∞ in (W 3 ).Then we can choose n 0 ∈ N large enough such that the right-hand side of (2.26) is negative.Define implies −u ∈ A. Denote by Γ the family of all closed symmetric subset of E which does not contain 0. For any A ∈ Γ, define the genus γ(A) of A by the smallest integer k such that there exists an odd continuous mapping from A to R k \ {0}.If there does EJQTDE, 2013 No. 54, p. 7 not exist such a k, define γ(A) = ∞.Moreover, set γ(φ) = 0.For each k ∈ N, letΓ k = {A ∈ Γ | γ(A) ≥ k}.Theorem 2.2 ( [29, Theorem 1]).Let E be an infinite dimensional Banach space and Φ ∈ C 1 (E, R) an even functional with Φ(0) = 0. Suppose that Φ satisfies (Φ 1 ) Φ is bounded from below and satisfies (PS) condition.

. 27 ) 2
Then we haveγ(A k ) = γ(W k ) = k and sup u∈A k Φ(u) < 0.The proof is completed.Now we are in the position to give the proof of our main result.Proof of Theorem 1.1.Lemmas 2.3 and 2.4 show that the functional Φ defined in (2.5) satisfies conditions (Φ 1 ) and (Φ 2 ) in Theorem 2.2.Therefore, by Theorem 2.2, we get a sequence nontrivial critical points {u k } for Φ satisfying Φ(u k ) ≤ 0 for all k ∈ N and u k → 0 in E as k → ∞.By virtue of Proposition 2.1, {u k } is a sequence of homoclinic solutions of ( HS).Since E is continuously embedded into L ∞ , then it follows that max t∈R |u k (t)| → 0 as k → ∞.Hence, there exists k 0 ∈ N such that u k is a homoclinic solution of (HS) for each k ≥ k 0 .This ends the proof.2 Theorem 1.1.Suppose that (L 0 ) and (W 1 )-(W 3 ) are satisfied.Then (HS) possesses a sequence of homoclinic solutions {u k } such that max t∈R |u k EJQTDE, 2013 No. 54, p. 2 .19) By Proposition 2.1, Ψ : E → E is also compact.Combining this with (2.17) and (2.18), the right-hand side of (2.19) converges strongly in E and hence u n k 24))By (W 3 ),(2.21)andthethedefinition of V k , there holdsI iu \J iu W (t, sr iu ϕ iu )dt + i =iu I i W (t, sr i ϕ i )dt ≥ − d 0 s 2 , (2.25) where d 0 is given at the beginning of the proof.For each δ n ∈ (0, b), combining (W 3 ), EJQTDE, 2013 No. 54, p. 10 (2.2) and (2.21)-(2.25),we have Φ(δ n u) ≤ C k δ 2 (t, δ n r iu ϕ iu )dt W