Periodic solutions of Euler-Lagrange equations with sublinear pontentials in an Orlicz-Sobolev space setting

In this paper, we obtain existence results of periodic solutions of hamiltonian systems in the Orlicz-Sobolev space $W^1L^{\Phi}([0,T])$. We employ the direct method of calculus of variations and we consider a potential function $F$ satisfying the inequality $|\nabla F(t,x)|\leq b_1(t) \Phi_0'(|x|)+b_2(t)$, with $b_1, b_2\in L^1$ and certain $N$-functions $\Phi_0$.


Introduction
This paper deals with system of equations of the type: d dt D y L(t, u(t), u ′ (t)) = D x L(t, u(t), u ′ (t)) a.e. t ∈ (0, T ), where L ∶ [0, T ] × R d × R d → R, d ⩾ 1, is called the Lagrange function or lagrangian and the unknown function u ∶ [0, T ] → R d is absolutely continuous.
In other words, we are interested in finding periodic weak solutions of Euler-Lagrange system of ordinary differential equations.
The problem (1) comes from a variational one, that is, the equation in (1) is the Euler-Lagrange equation associated to the action integral This topic was deeply addressed for the Lagrange function L p,F (t, x, y) = y p p + F (t, x), for 1 < p < ∞. For example, the classic book [1] deals mainly with problem (1) for the lagrangian L 2,F and through various methods: direct, dual action, minimax, etc. The results in [1] were extended and improved in several articles, see [2,3,4,5,6] to cite some examples. Lagrange functions (3) for arbitrary 1 < p < ∞ are considered in [7,8] and in this case (1) is reduced to the p-laplacian system d dt (u ′ (t) u ′ p−2 ) = ∇F (t, u(t)) a.e. t ∈ (0, T ), In this context, it is customary to call F a potential function, and it is assumed that F (t, x) is differentiable with respect to x for a.e. t ∈ [0, T ] and the following conditions hold: (C) F and its gradient ∇F , with respect to x ∈ R d , are Carathéodory functions, i.e. they are measurable functions with respect to t ∈ [0, T ] for every x ∈ R d , and they are continuous functions with respect to x ∈ R d for a.e. t ∈ [0, T ].
In [9] it was treated the case of a lagrangian L which is lower bounded by a Lagrange function where Φ is an N-function (see section 2 for the definition of this concept). In the paper [9] it was also assumed a condition of bounded oscillation on F . In this current paper we will study a condition of sublinearity (see [3,4,6,8,10]) on ∇F for the lagrangian L Φ,F , or more generally for lagrangians which are lower bounded by L Φ,F . The paper is organized as follows. In section 2, we give preliminaries facts on N-functions and Orlicz-Sobolev spaces of functions. Section 3 is devoted to the main result of this work and it also includes an auxiliary lemma of vital importance. Section 4 contains the proofs and section 5 provides an application of our result to a concrete case.

Preliminaries
For reader convenience, we give a short introduction to Orlicz and Orlicz-Sobolev spaces of vector valued functions. Classic references for these topics are [11,12,13,14].
Hereafter we denote by R + the set of all non negative real numbers. A function Φ ∶ R + → R + is called an N-function if Φ is convex and it also satisfies that In addition, in this paper for the sake of simplicity we assume that Φ is differentiable and we call ϕ the derivative of Φ. On these assumptions, ϕ ∶ R + → R + is a homeomorphism whose inverse will be denoted by ψ. We write Ψ for the primitive of ψ that satisfies Ψ(0) = 0. Then, Ψ is an N-function which is known as the complementary function of Φ.
We recall that an N-function Φ(u) has principal part f (u) if Φ(u) = f (u) for large values of the argument (see [12, p. 16] and [12,Sec. 7] for properties of principal part).
There exist several orders and equivalence relations between N-functions (see [13,Sec. 2.2]). Following [13,Def. 1, we say that the N-function Φ 2 is stronger than the N-function The if and only if for every a > 0 there exists x 0 = x 0 (a) ⩾ 0 such that (4) holds. Finally, we say that Φ 2 is completely stronger than Φ 1 (Φ 1 Φ 2 ) if and only if for every a > 0 there exist We also say that a non decreasing function for every x ⩾ x 0 . We note that η ∈ ∆ ∞ 2 if and only if η η. If x 0 = 0, the function η ∶ R + → R + is said to satisfy the ∆ 2 -condition (η ∈ ∆ 2 ). If there exists x 0 > 0 such that inequality (5) holds for x ⩽ x 0 , we will say that Φ satisfies the ∆ 0 2 -condition (Φ ∈ ∆ 0 2 ). We denote by α η and β η the so called Matuszewska-Orlicz indices of the function η, which are defined next. Given an increasing, unbounded, continuous function η ∶ [0, +∞) → [0, +∞) such that η(0) = 0, we define It is known that the previous limits exist and 0 ⩽ α η ⩽ β η ⩽ +∞ (see [14, p. 84] (see [14,Cor. 11.6]). Therefore If η is an increasing function that satisfies the ∆ 2 -condition, then η is controlled by above and below by power functions ([15, Sec and [14,Thm. 11.13]). More concretely, for every ǫ > 0 there exists a constant K = K(η, ǫ) such that, for every t, u ⩾ 0, Given an N-function Φ we define the modular function Here ⋅ is the euclidean norm of R d . Now, we introduce the Orlicz class The Orlicz space L Φ equipped with the Orlicz norm is a Banach space. By u ⋅ v we denote the usual dot product in R d between u and v.
The following inequality holds for any u ∈ L Φ In fact, u L Φ is the infimum for k > 0 of the right hand side in above expression (see [12,Thm. 10.5] and [17]).
shown that E Φ is the only one maximal subspace contained in the Orlicz class A generalized version of Hölder's inequality holds in Orlicz spaces (see [12,Thm. 9 We define the Sobolev-Orlicz space W 1 L Φ (see [11]) by And, we introduce the following subset of We will use repeatedly the decomposition u = u+ũ for a function and, it is easy to see that for every N-function Φ we have that L ∞ ↪ L Φ ↪ L 1 .
The following simple embedding lemma, whose proof can be found in [9], will be used several times.

Main result
We begin with a lemma which establishes the coercivity of the modular function ρ Φ (u) with respect to certain functions of the Orlicz norm Φ 0 ( u L Φ ). This lemma generalizes [9, Lemma 5.2] in two directions. Namely, certain power function is replaced by a more general N-function Φ 0 and the ∆ 2 -condition on Ψ is relaxed to ∆ ∞ 2 . It is worth noting that the second improvement is more important than the first one. And, we present the result here since the lemma introduces a function Φ * that will play a significant role in the statement of our main theorem.
We point out that this lemma can be applied to more cases than [9, Lemma 5.2]. For example, if Φ(u) = u 2 , Φ 1 and Φ 0 are N-functions with principal parts equal to u 2 log u and u 2 (log u) 2 respectively, then (10) holds for Φ 0 . On the other hand, Φ 0 ( u ) is not dominated for any power function u α with α < 2.
As in [9] we will consider general Lagrange In [9] it was shown that if L ∈ A(a, b, c, λ, f, Φ) then there exists the Gateâux derivative of the integral functional I defined by (2), on the set We observe that the condition (A) on the potential F is equivalent to say that Unlike what is usual in the literature, we do not assume the lagrangian L split into two terms, one of them function of y and the other one function of (t, x). We only suppose that L is lower bounded by a function of this type. More precisely, we assume that for every (t, In addition, as usual we suppose that the time integral of F satisfies certain coercivity condition, see (A 6 ) below. However, all these hypotheses are not enough. It is also necessary to assume extra conditions on the potential F . Several hypotheses were tested in the past years. The so called subconvexity of F was tried in [4,2,6] for semilinear equations and in [18,8] for p-laplacian systems. Potentials F satisfying the following inequality were studied in [9]. Regarding (11), it is interesting to notice that such inequality is equivalent to say the condition F (t, ⋅) BO ∈ L 1 ([0, T ]), where ⋅ BO denotes the seminorm defined in [19, p. 125] on the space of functions of bounded variations.
In [3,8] the authors dealt with the p-laplacian case with potentials F such that where b 1 , b 2 ∈ L 1 ([0, T ]) and α < p. Such potentials F were called sublinear nonlinearities. In this paper, we are interested in studying this type of potentials, but with more general bounds on ∇F which include N-functions instead of power functions; namely, we will consider inequalities like Next, we give our main result. Here, we will amend an erroneous assumption made in the end of the proof of [9, Thm. 6.2]. There, it was assumed without discussion that a minimum of I was on the domain of differentiability of I.
Then the action integral I has a minimum u ∈ x, y) is strictly convex with respect to y ∈ R d and D y L(0, x, y) = D y L(T, x, y) then u solves the problem (1).

Proofs
The following result is analogous to some lemmata in W 1,p , see [18,Lem. 1].
By the decomposition u = u +ũ and some elementary operations, we get It is known that L ∞ ↪ L Φ , i.e. there exists and, applying Sobolev's inequality, we obtain Wirtinger's inequality, that is there exists C 2 = C 2 (T ) > 0 such that Therefore, from (12), (13) and (9), we get For the converse, we observe that , and the property under consideration is proved.  Proof. We can assume that Ψ ∉ ∆ 0 2 . Consequently, from Lemma 4.2 we have that the right continuous derivative ψ of Ψ does not satisfy the ∆ 0 2 -condition. Therefore, we obtain a sequence of positive numbers x n , n = 1, 2, . . ., such that x n → 0, 2x n+1 < x n < 2x n and ψ(2x n ) > 2ψ(x n ).
Next, we will use induction to prove that We suppose n = 1. Then items 2 and 3 are obvious. From (14) we have and this inequality implies 1.
It remains to show the inequality Ψ * (x) ⩽ aΨ(x), for every a > 1 and sufficiently large x. We take x 0 sufficiently large to have The last statement of the lemma is consequence of Ψ(ax) > aΨ(x) when a > 1.
The following lemma is essentially known, because it is basically a consequence of the fact that Ψ ∈ ∆ ∞ 2 if and only if Ψ Ψ, [13,Prop. 4,p. 20] and [13,Cor. 10,p. 30]. However, we prefer to include an alternative proof, as we do not see clearly that the results of [13] contemplate the case of N-functions satisfying the ∆ 2 -condition.
2. There exists an N-function Φ * such that Remark 2. We want to emphasize that the difference between the conclusions in item 2 of the previous lemma is that (15) holds for s ⩾ 0 or s ⩾ 1 depending on Ψ ∈ ∆ 2 or Ψ ∈ ∆ ∞ 2 , respectively.
Definition 4.5. We define the functionals and

respectively.
Proof. Theorem 3.2 . By the decomposition u = u +ũ, Cauchy-Schwarz's inequality and (A 5 ), we have First, by Hölder's and Sobolev-Wirtinger's inequalities we estimate I 2 as follows where Note that, since Φ ′ 0 is an increasing function and Φ ′ 0 (x) ⩾ 0 for x ⩾ 0, then In this way, we have for every s ∈ [0, 1]. Now, inequality (18), Hölder's and Sobolev-Wirtinger's inequalities imply that where C 2 = C 2 (T, b 1 L 1 ) and C 3 = C 3 (T ). Next, by Young's inequality with complementary functions Φ 1 and Ψ 1 , we obtain We have that any N-function Φ 0 satisfies the inequality xΦ ′ 0 (x) ⩽ Φ 0 (2x) (see [13, p. 17] The previous observations imply that From (19), (20), (21) and (17), we have with C 5 depending on Φ 0 , Φ 1 , T, b 1 L 1 and b 2 L 1 . Finally, using (A 4 ), (16) and (22), we get Let u n be a sequence in W 1 L Φ with u n W 1 L Φ → ∞ and we have to prove that I(u n ) → ∞. On the contrary, suppose that for a subsequence, still denoted by u n , I(u n ) is upper bounded, i.e. there exists M > 0 such that I(u n ) ⩽ M. As u n W 1 L Φ → ∞, from Lemma 4.1, we have u n + u ′ n L Φ → ∞. Passing to a subsequence is necessary, still denoted u n , we can assume that u n → ∞ or u ′ n L Φ → ∞. Now, Lemma 3.1 implies that the functional J C 5 ,Φ 1 (u ′ ) is coercive; and, by (A 6 ), the functional From the condition (A) on F , we have that on a bounded set the functional H C 5 ,Ψ 1 ○Φ ′ 0 (u n ) is lower bounded and also J C 5 ,Φ ′ 0 (u ′ n ) ⩾ 0. Therefore, I(u n ) → ∞ as u n W 1 L Φ → ∞ which contradicts the initial assumption on the behavior of I(u n ).
Let {u n } ⊂ W 1 L Φ be a minimizing sequence for the problem inf{I(u) u ∈ W 1 L Φ }. Since I(u n ), n = 1, 2, . . ., is upper bounded, the previous part of the proof shows that {u n } is norm bounded. Hence, by virtue of [9, Cor. 2.2], we can assume, taking a subsequence if necessary, that u n converges uniformly to a T -periodic continuous function u. As {u ′ n } is a norm bounded sequence in L Φ , there exists a subsequence, again denoted by u ′ n , that converges to a function v ∈ L Φ in the weak* topology of L Φ . From this fact and the uniform convergence of u n to u, we obtain that [1, p. 6]) and u ∈ W 1 L Φ ([0, T ], R d ). Now, taking into account the relations [L 1 ] * = L ∞ ⊂ E Ψ and L Φ ⊂ L 1 , we have that u ′ n converges to u ′ in the weak topology of L 1 . Consequently, from the semicontinuity of I (see [9, Lemma 6.1]) we get For the second part of the theorem, assume that u is a minimum of I with d(u ′ , E Φ ) < λ. Since I is Gâteaux differentiable at u (see [9,Thm. 3.2]), therefore for every v ∈ W 1 L Φ T . From [9, Eq. (26)] we have D y L(t, u(t), u ′ (t)) ∈ L Ψ ([0, T ], R n ) ↪ L 1 ([0, T ], R n ); and, from [9, Eq. (24)], it follows that D x L(t, u(t), u ′ (t)) ∈ L 1 ([0, T ], R n ). Consequently, from [1, p. 6] (note that W 1 L Φ T includes the periodic test functions) we obtain the absolutely continuity of D y L(t, u(t), u ′ (t)) and that the differential equations in (1) are satisfied. The strict convexity of L(t, x, y) with respect to y and the T -periodicity with respect to t imply the boundary conditions in (1) (see [9,Thm. 4.1]).

An example
In this section we develop an application of our main result so that the reader can appreciate the innovations that brings.
The main novelty of our work is that we obtain existence of minima of I associated with lagrangian functions L(t, x, y) that do not satisfy a power-like grow condition on y.
In fact, it is possible to apply Theorem 3.2 to lagrangians L = L(t, x, y) with exponential grow on the variable y. For example, suppose that L(t, x, y) = f (y) + F (t, x), with f ∶ R n → R differentiable, strictly convex and f (y) ⩾ e y . We define for n ⩾ 1 Φ(y) = e y − n−1 i=0 y i i! .