Existence and multiplicity of homoclinic solutions for a second-order Hamiltonian system

In this paper, we find new conditions to ensure the existence of one nontrivial homoclinic solution and also infinitely many homoclinic solutions for the second order Hamiltonian system ü− a(t)|u|p−2u +∇W(t, u) = 0, t ∈ R, where p > 2, a ∈ C(R, R) with inft∈R a(t) > 0 and ∫ R ( 1 a(t) )2/(p−2)dt < +∞, and W(t, x) is, as |x| → ∞, superquadratic or subquadratic with certain hypotheses different from those used in previous related studies. Our approach is variational and we use the Cerami condition instead of the Palais–Smale one for deformation arguments.


Introduction
Consider the second order Hamiltonian system ü(t) − a(t)|u(t)| p−2 u(t) + ∇W(t, u(t)) = 0, (HS) R) and ∇W(t, x) denotes the gradient of W(t, x) with respect to x.As usual, we say that a solution u of (HS) is homoclinic (to 0) if u(t) → 0 as |t| → ∞.Furthermore, if u ≡ 0, then u is called a nontrivial homoclinic solution.Homoclinic orbits of nonlinear differential equations have long been studied in the dynamical systems literature, generally in a setting involving perturbations and using a Melnikov function.The existence of many homoclinic orbits is a classical problem and the first multiplicity results go back to Poincaré [19] and Melnikov [17].They proved, by means of perturbation techniques, that the system possesses infinitely many homoclinic orbits in the case of N = 1 when the potential depends periodically on time.

Y. Ye
Motivated by the works mentioned above, the main goal of this paper is to find new conditions to guarantee the existence of homoclinic solutions of problem (HS).We are particularly interested in the cases where a(t) satisfies: and W(t, x) satisfies conditions which are more general than (W 1 ).Typical examples, which match our setting but not satisfying Theorems 1.1-1.3,are Examples 1-5.
Remark 1.4.Assumption (V) is weaker than (V 1 ).There are functions a which match our setting but not satisfying (V 1 ).For example, let elsewhere.
We first handle the superquadratic case.Assume furthermore the following hypotheses: (W 4 ) There exist µ > p and L > 1 such that and inf t∈R,|x|=L W(t, x) > 0.
Then problem (HS) possesses at least one nontrivial homoclinic solution.Moreover, if W(t, x) is even in x, then problem (HS) possesses an unbounded sequence of homoclinic solutions Remark 1.11.We mention that the monotonicity condition like (W 7 ) was used in Jeanjean [12] to obtain one positive solution for a semilinear problem in R N , in [14] to get infinitely many solutions for quasilinear elliptic problems setting on a bounded domain, and in [10] to compute the critical points of the energy functional and obtain nontrivial solutions via Morse theory.It turns out that if for fixed (t, then (W 7 ) is satisfied.
Next we consider the subquadratic case.Assume that: (W 9 ) W(t, 0) ≡ 0, and there exist constants The paper is organized as follows.After presenting some preliminaries, we prove the above existence and multiplicity results for the superquadratic and subquadratic cases in turn.

Notation.
Throughout the paper we denote by c, c i the various positive constants which may vary from line to line and are not essential to the problem.

Preliminaries
We shall construct the variational setting under condition (V).For a nonnegative measurable function a and a real number s > 1, define the weighted Lebesgue space and associated with it the norm We define, for any r ∈ [1, +∞], with the usual norms , where a(t) is the function introduced in (V).It is easy to check that E is a reflexive Banach space under the norm we have which implies that E is continuously embedded into H 1 .So E is continuously embedded into L r for 2 ≤ r ≤ +∞, and hence, for each r ∈ [2, +∞], there is τ r > 0 such that Furthermore, we have the following lemma.

Y. Ye
Lemma 2.1.If assumption (V) is satisfied, then the embedding E → L r is compact for 2 ≤ r ≤ +∞.
Proof.We adapt an argument in Ding [8].Let K ⊂ E be a bounded set.Then there is for any ε > 0, we take R 0 > 0 large enough such that for all 1 ≤ r ≤ +∞.Thus, there are u 1 , u 2 , . . ., u m ∈ K such that for any u ∈ K, there is Hence, using Hölder's inequality, (2.5) and (2.3), we obtain The above arguments yield that K has a finite ε-net and so is precompact in L 2 .
For any n ∈ N, t ∈ R and u ∈ E, one has by the Hölder inequality.Particularly, for any R > 0 and u, v ∈ K, we obtain For any ε > 0, first choosing n sufficiently large such that and then R 1 large enough satisfying by (2.4).It follows that sup Again, using the Sobolev compact embedding theorem, there are u 1 , u 2 , . . ., u m ∈ K such that for any u ∈ K, there is which, together with (2.6), shows that we see immediately that K is precompact in L r .
(i) For u ∈ E, there holds (ii) Given α, β > 0, there is c > 0 such that for every u ∈ E, there holds Proof.The conclusion follows easily from the definition of • .

The superquadratic case
By assumptions (V) and (W 2 ), the energy functional associated to problem (HS) on E given by for all u, v ∈ E. It is routine to show that any nontrivial critical point of I is a classical solution of system (HS) with u(±∞) = 0.
To find the critical points of I, we shall show that I satisfies the Cerami condition, i.e., (u n ) ⊂ E has a convergent subsequence whenever {I(u n )} is bounded and (1+ u n ) I (u n ) → 0 as n → ∞.Such a sequence is then called a Cerami sequence.
We show that (u n ) is bounded.Arguing indirectly, assume that u n → ∞ as n → ∞.We consider w n := u n / u n .Then, up to a subsequence, we get and Combining this with (W 3 ), we obtain and then, using (W 4 ), Hence we obtain by the second limit of (3.2).Here, and in what follows, o(1) denotes a quantity which goes to zero as n → ∞.On the other hand, (W 4 ) implies where Therefore, This, jointly with (3.5), contradicts the fact w n = 1.
Passing to a subsequence, u n u weakly in E, u n → u in L 2 and u n (t) → u(t) for a.e.t ∈ R. The boundedness of (u n ) implies that for some M > 1.Thus, using (3.3) and (W 3 ), where c 2 := εM p−2 + δ −1 max δ≤|x|≤M W(x).It is easy to see that there holds and therefore by (3.11) and the fact u n → u in L 2 we obtain Taking for all n.By (W 2 ), for any ε > 0 (< 1/3), there exists a ε > 0 such that which implies that Since σ > 0, using (W 6 ), (3.14) and (2.2), we can take b ε ≥ L 1 so large that and then, using (W 3 ), Therefore, a combination of (3.15)-(3.17)shows that and consequently (3.13) holds.Now, noting which, jointly with (3.13), shows that On the other hand, by (W 5 ), we have This, jointly with (3.18), produces a contradiction since w n = 1.Thus (u n ) is bounded in E, and hence contains a subsequence, relabeled (u n ) which converges to some u ∈ E weakly in E and strongly in L 2 .Arguing as in the latter part of the proof of Lemma 3.1, we conclude that the (C) condition is satisfied.
Proof.As in the proof of Lemma 3.1, it suffices to consider the case w = 0 and w = 0.If w = 0, inspired by [12], we choose a sequence (s n ) ⊂ R such that For any m ≥ 1 and wn := √ mw n , we have wn 0 in E and wn → 0 in L ∞ .Combining this with (V) and (3.4), we have, for sufficiently large n, and then, using Lemma 2.2 (ii), by the arbitrariness of m.Observing I(0) = 0 and {I(u n )} is bounded, one sees that for n large enough, s n ∈ (0, 1) and Combining this with (W 7 ), we obtain a contradiction with (3.19).
If w = 0, the proof follows the same lines as that of Lemma 3.1, and therefore is omitted.
We shall apply the mountain pass theorem (see [21,Theorem 2.2]) and the symmetric mountain pass theorem (see [21,Theorem 9.12]) to prove our results.In the linking theorem, it is usually supposed that the functional Φ satisfies the stronger Palais-Smale condition.Nevertheless, the Cerami condition is sufficient for the deformation lemma (see [4]), and therefore for the linking theorem to hold.Proposition 3.4.Let E be a real Banach space and Φ ∈ C 1 (E, R) satisfying the Cerami condition (C).Suppose that Φ(0) = 0 and: (i) there exist ρ, α > 0 such that Φ| ∂B ρ (0) ≥ α; (ii) there is an e ∈ E\B ρ (0) such that Φ(e) ≤ 0.
Then Φ possesses an unbounded sequence of critical values.

Y. Ye
Lemma 3.7.Let (V) and (W 2 )-(W 4 ) be satisfied.Then, for any finite dimensional subspace E ⊂ E, there holds Proof.The equivalence of the norms on the finite dimensional space E implies there exists Combining (3.6) with (W 3 ) and (3.4), we obtain where . Consequently, using (3.21), (3.20) and (2.2), we obtain, Lemma 3.8.Let (V) and (W 1 ) be satisfied and Then, for any finite dimensional subspace E ⊂ E, there holds so that

The subquadratic case
Inspired by [28], we shall extend W to an appropriate W ∈ C 1 (R × R N , R) and introduce the following Hamiltonian systems ü − a(t)|u| p−2 u + ∇ W(t, u) = 0, ∀t ∈ R.
Then, applying variational methods, we show that system (4.1)possesses a sequence of homoclinic solutions, which converges to zero in L ∞ norm, and consequently, obtain infinitely many solutions for the original problem (HS).
(C 2 ) W(t, 0) ≡ 0, and there exist constants i.e., (C 2 ) holds.Now we define the variational functional Φ associated to system (4.1)by: and then, using the Hölder inequality, Thus Φ is well defined.In addition, we have the following lemma.
Proof.For Φ ∈ C 1 (E, R), it suffices to show it for the functional ψ(u) = R W(t, u)dt.It follows from (C 2 ) and the Young inequality that, for u, v ∈ E and s ∈ This completes the proof.
We shall make use of the new version of symmetric mountain pass lemma of Kajikiya (see [13]) to prove Theorem 1.13.Let E be a Banach space and Γ := {A ⊂ E\ {0} : A is closed and symmetric with respect to the origin} .
For A ∈ Γ, the genus γ(A) of A is defined as being the least positive integer k such that there is an odd mapping h ∈ C(A, R k )\ {0}.If k does not exist, we set γ(A) = +∞.Furthermore, by definition, γ(∅) = 0.
In the sequel, we only recall the properties of the genus that will be need throughout the paper.See [21] for more information on this subject.(i) There is an odd continuous mapping from A to B, then γ(A) ≤ γ(B).
(iii) The n-dimensional sphere S n has a genus of n + 1 by the Borsuk-Ulam theorem.Proposition 4.4.Let E be an infinite dimensional Banach space and Φ ∈ C 1 (E, R) be even, Φ(0) = 0 and satisfies the following conditions: (i) Φ is bounded from below and satisfies the Palais-Smale condition (PS), i.e., (u n ) ⊂ E has a convergent subsequence whenever {Φ(u n )} is bounded and Φ (u n ) → 0 as n → ∞.
(ii) For each k ∈ N, there exists an A k ∈ Γ such that γ(A k ) = k and sup u∈A k Φ(u) < 0.
(1) There exists a sequence {u k } such that Φ (u k ) = 0, Φ(u k ) < 0 and {u k } converges to zero.As δ n → 0 + and M n → +∞ (n → ∞), we choose n 0 > 0 large enough such that the right side of the last inequality is negative.Now, letting We have Consequently, by Proposition 4.4, Φ has infinitely many nontrivial solutions (u k ) such that u k → 0 in E as k → ∞.By (2.2), u k → 0 in L ∞ .Hence, for k large, they are homoclinic solutions of (HS).This completes the proof.
Set (u n ) be a Cerami sequence.We verify that (u n ) is bounded.Assuming the contrary, u n → ∞, w n := u n / u n w in E and w n (t) → w(t) for a.e.t ∈ R after passing to a subsequence.We claim that |u n − u| p dt + o(1), which yields that u n → u in E. This completes the proof.Lemma 3.2.Let (V), (W 2 )-(W 3 ) and (W 5 )-(W 6 ) be satisfied.ThenI satisfies the (C) condition.Proof.