Periodic Solutions of Second-order Systems with Subquadratic Convex Potential

In this paper, we investigate the existence of periodic solutions for the second order systems at resonance: ¨ u(t) + m 2 ω 2 u(t) + ∇F(t, u(t)) = 0 a.e. t ∈ [0, T], u(0) − u(T) = ˙ u(0) − ˙ u(T) = 0, where m > 0, the potential F(t, x) is convex in x and satisfies some general subquadratic conditions. The main results generalize and improve Theorem 3.7 in J. Mawhin and

Using the variational principle of Clarke and Ekeland together with an approximate argument of H. Brézis [2], Mawhin and Willem [6] proved an existence theorem for semilinear equations of the form Lu = ∇F(x, u), where L is a noninvertible linear selfadjoint operator and F is convex with respect to u and satisfies a suitable asymptotic quadratic growth condition.This result was applied to periodic solutions of first order Hamiltonian systems with convex potential.In [5], the authors considered the second order systems (1.1) with m = 0.They proved that when the potential F satisfies the following assumptions: (A ) F(t, x) is measurable in t for every x ∈ R N , and continuously differentiable and convex in x for a.e.t ∈ [0, T]; (A 1 ) There exists l ∈ L 4 (0, T; R N ) such that then problem (1.1) has at least one solution, see [5,Theorem 3.5].This result was slightly improved in Tang [8] by relaxing the integrability of l and γ.In [12], Tang and Wu dealt with the (β, γ)-subconvex case, i.e., for some γ > 0. Under assumptions (A), (A 3 ) and (1.2) and the subquadratic condition: there exist 0 < µ < 2 and M > 0 such that they obtained the existence result by taking advantage of Rabinowitz's saddle point theorem.
Recently, Tang and Wu [13] extended a theorem established by A. C. Lazer, E. M. Landesman and D. R. Meyers [4] on the existence of critical points without compactness assumptions, using the reduction method, the perturbation argument and the least action principle.As a main application, they successively studied the existence of periodic solutions of problem (1.1) (m = 0) with subquadratic convex potential, with subquadratic µ(t)-convex potential and with subquadratic k(t)-concave potential, which unifies and significantly generalizes some earlier results in [5,8,15,22,23] obtained by other methods.If m = 0, it is a resonance case.Using the dual least action principle and the perturbation technique, Mawhin and Willem [5] also obtained the following theorem.
Theorem A ([5,Theorem 3.7]).Suppose that F(t, x) satisfies conditions (A ), (A 1 ) and the following: (A 2 ) There exist α ∈ (0, (2m + 1)ω 2 ) and γ ∈ L 2 (0, T; R + ) such that Then problem (1.1) has at least one solution in H 1 T , where is a Hilbert space with the norm defined by Motivated by the works mentioned above, in this paper, we are interested in problem (1.1), where the potential is convex and satisfies conditions which are more general than (A 2 ).Applying the abstract critical point theory established in [13], we prove some existence results, which generalize Theorem A and complement the results in [13].The main results are the following theorems.
Theorem 1.1.Suppose that assumption (A) holds and F(t, x) is convex in x for a.e.t ∈ [0, T].Assume that (A 3 ) holds and: for all x ∈ R N and a.e.t ∈ [0, T], and Then problem (1.1) has at least one solution in H 1 T .Remark 1.2.Theorem 1.1 extends Theorem A, since (A 4 ) is weaker than (A 2 ) and assumption (A) holds for functions F in Theorem A (see [13,Remark 1.3] for a proof).There are functions F which match our setting but not satisfying Theorem A. For example, let where l ∈ L 3 (0, T; R N )\L ∞ (0, T; R N ).Then by Young's inequality, one has Evidently, (A 3 ) and (1.4) are satisfied, and is convex by the fact that s > 0 is convex and increasing, and Hence F satisfies all the conditions of Theorem 1.1.But it does not satisfy Theorem A, for (A 2 ) does not hold.
Theorem 1.1 yields immediately the following corollary.
Corollary 1.3.The conclusion of Theorem 1.1 remains valid if we replace (A 4 ) by Remark 1.4.It is easy to see that (A 5 ) is weaker than (A 2 ).So Corollary 1.3 also generalizes Theorem A.
Remark 1.9.Theorem 1.7 generalizes Theorem A. There are functions F satisfying our Theorem 1.7 and not satisfying Theorems A and 1.1.For example, let and l ∈ L ∞ (0, T; R N ).Then F satisfies all the conditions of Theorem 1.7.But obviously F does not satisfy Theorems A and 1.1.
Theorem 1.10.Suppose that assumption (A) holds and F(t, x) is convex in x for a.e.t ∈ [0, T].Assume that (A 3 ) holds and: Then problem (1.1) has at least one solution in H 1 T .

Y. Ye
Remark 1.11.There are functions F satisfying our Theorem 1.10 and not satisfying the results mentioned above.For example, let where β ∈ L 1 (0, T; R + ) with 0 < T 0 β(t) dt < 12(2m+1) T(m+1) 2 and l ∈ L 2 (0, T; R N ).Then one has which is just (1.6) with Thus F satisfies all the conditions of Theorem 1.10.But in the case that F does not satisfy the conditions of Theorems A, 1.1 and 1.7.

Proofs of the theorems
Under assumption (A), the energy functional associated to problem (1.1) given by is continuously differentiable and weakly upper semi-continuous on H 1 T .Furthermore, for all u, v ∈ H 1 T , and ϕ is weakly continuous.It is well known that the weak solutions of problem (1.1) correspond to the critical points of ϕ (see [5]).
We recall an abstract critical point theorem which will be used in the sequel.

Proposition 2.2 ([13, Lemma 5.1]).
Assume that H is a real Hilbert space, f : H × H → R is a bilinear functional.Then g : H → R given by For m > 0, set This completes the proof.
Then for every M > 0, ϕ(v + w) → +∞ as v → ∞, v ∈ H m , uniformly for w ∈ H ⊥ m with w ≤ M. Proof.We prove this assertion by contradiction.Suppose that the statement of the theorem does not hold, then there exist M > 0, c 1 > 0 and two sequences Define the function F : It follows from the continuous differentiability and the convexity of F(t, •) that F is continuously differentiable and convex on R 2N , which yields that F is weakly lower semi-continuous on R 2N .Using (A 3 ), one has Hence, by the least action principle [5, Theorem 1.1], F has a minimum at some (a 0 , b 0 ) ∈ R 2N for which By the convexity of F(t, •), we obtain and then, using assumption (A), (2.2) and (2.1), for all n, which implies that (u n ) is bounded by the equivalence of the norms on the finitedimensional space H m−1 .Combining this with assumption (A), the convexity of F(t, •) and (2.1), we obtain which yields that the sequences (a n ) and (b n ) are also bounded.This contradicts the fact that v n → ∞ as n → ∞.Therefore the conclusion holds.Now we are in the position to prove our theorems.
Proof of Theorem 1.1.According to Proposition 2.1, it remains to show that We follow an argument in [13].Arguing indirectly, assume that there exists a sequence for some c 4 ∈ R. Write u n = a n u n cos(m + 1)ωt + b n u n sin(m + 1)ωt + w n , where a n , b n ∈ R N and w n ∈ H ⊥ m+1 .Then we have, using (1.3), which implies that (w n ) is bounded.Taking v n = u n / u n , then v n = 1, and hence the sequences {a n }, {b n } are bounded.Up to a subsequence, we can assume that a n → a and b n → b as n → ∞ for some a, b ∈ R N .By the boundedness of (w n ), one has w n / u n → 0 as n → ∞.Hence, It follows from Fatou's lemma (see [20]) that lim sup T and the proof is completed.
Proof of Theorem 1.7.First, we claim that there exists a constant a 0 < 2m+1 (m+1) 2 such that The proof is similar to the first part of [13, Proof of Theorem 3.2], for the convenience of the readers we sketch it here briefly.Arguing indirectly, we assume that there exists a sequence which implies that u n = 0 for all n.By the homogeneity of the above inequality, we may assume that