Du Bois–Reymond Type Lemma and Its Application to Dirichlet Problem with the p(t)–Laplacian on a Bounded Time Scale

This paper is devoted to study the existence of solutions and their regularity in the p(t)–Laplacian Dirichlet problem on a bounded time scale. First, we prove a lemma of du Bois–Reymond type in time-scale settings. Then, using direct variational methods and the mountain pass methodology, we present several sufficient conditions for the existence of solutions to the Dirichlet problem.


Introduction
Variational methods and critical point theory have been very successful in obtaining existence and multiplicity results for nonlinear ordinary or partial differential equations, as well as for nonlinear difference equations submitted to various boundary conditions. See, for example, [1][2][3][4][5] and their references.
The aim of this paper is to use those methodologies for the study of the Dirichlet problem for a dynamic equation on a bounded time scale T involving the p(t)-Laplacian, −∆ p(t) u(t) := − ∆ ∆t |∆ w u(t)| p(t)−2 ∆ w u(t) = f t, u σ (t) , t ∈ T u(a) = u(b) = 0 . (1) In this equation, ∆ w denotes a weak derivative operator defined in terms of the ∆integral on a time scale (see Section 2 for precise definitions), p : T → (1, ∞) is a measurable and essentially bounded function with an essential lower bound larger than one, and f is a ∆-Carathéodory function.
A partial motivation is the paper of Xian-Ling Fan and Qi-Hu Zhang [6] dealing with a similar problem in the case of a partial differential equation. Such equations are known to describe mathematical models of various phenomena arising in the study of elastic mechanics [7] or image restoration [8]. Early variational approaches on Dirichlet problems with p-Laplacian are quoted in [9], extensions to p(x)-Laplacian are given in [6,10], and some generalizations (anisotropic problems) are described in the paper [4]. Since the research was conducted in discrete and continuous settings separately, it seems interesting to demonstrate that a sort of unification is also possible with the use of a timescale notion considered with some type of measure that has not been vastly exploited but which appears indispensable. For boundary value problems on time scales, one can consult [11]. Since we take the definition of the ∆-measure from [11], it is necessary to provide additional proof regarding the absolutely continuity of functions defined over subsets containing the maximum of the bounded time scale T.
The underlying Lebesgue and Sobolev spaces with variable exponents, where the variational approach takes place, are defined in Section 2, where their required properties are proved. The first paper on the variable exponent Lebesgue and Sobolev spaces L p(x) (Ω) and W m,p(x) (Ω), Ω ⊂ R n is due to Kováčik and Rákosník [12] and was developed in [13]. Some earlier papers on the Lebesgue and Sobolev spaces on time scales are [11,14], and we refer to [15] for further basic information on the variable exponent Lebesgue and Sobolev spaces on time scales.
The variational treatment of problem (1) requires proving a so-called du Bois-Reymond Lemma in this new frame to make the link between the critical points of the action functional and the solutions of the boundary value problem. This is done in Section 3 (Lemma 3). It also requires a careful study of the differentiability and other properties of the action functional. This is the object of Section 4.
We are now ready to apply in Section 5 the direct method of the calculus of variations to prove the existence of a solution to problem (1) when F(t, u) := t 0 f (s, u) ∆s is bounded above by an expression of the form c 1 |u| + c 2 |u| β β + c 3 , where the c i are positive constants, c 2 is sufficiently small and β ∈ (1, ess inf T p] (Theorem 2). This is the essence of Theorem 2.
When F(t, u) grows faster than ess sup T p at infinity, the action functional need not have a minimum, but the simplest of the minimax method, namely the mountain pass lemma, may be used to prove the existence of a nontrivial solution to problem (1) when f (t, 0) = 0 and f (t, u) is sufficiently 'flat' in u near u = 0. This is done in Theorem 4, where the growth of F for large u is governed by a suitable Ambrosetti-Rabinowitz condition and f (t, u) → 0 when u → 0 faster than |u| ess sup T p−1 .

Variable Exponent Lebesgue and Sobolev Spaces on Time Scales
In this section, we recall some basic facts concerning functions defined on time scales (see [11,14,16,17]) and discuss the variable exponent Lebesgue and Sobolev spaces on time scales (see [15]).
Let T be a bounded time scale. We define a = inf{s ∈ T}, b = sup{s ∈ T}. (2) Since T is bounded, a, b ∈ T. Define the forward jump operator σ : T → T by If σ(t) > t, then the point t ∈ T is said to be right-scattered. If σ(t) = t, then t ∈ T is called a right-dense point. The backward jump operator : T → T is as follows: If (t) < t, then we say that the point t ∈ T is left-scattered. If (t) = t, the point t ∈ T is called left-dense.
The function u extends u to [a, b], and it enables us to establish equivalence between Lebesgue ∆-integrable and integrable functions. [11]) and µ L denotes the classical Lebesgue measure. Hence, all subsets of the time scale T containing b are of a finite ∆−measure, and this is the main difference from the approach given in [14]. We denote u σ (t) = u(σ(t)) for t ∈ T, where σ is defined in (3). If u ∈ C(T), then u σ ∈ C rd (T). Moreover, one has Let us denote T κ = T \ ( (sup T), sup T], where : T → T is defined in (4). In this way, we remove from the time scale T its left-scattered maximum, when necessary. Alternatively, it can be written as We recall that u : T → R is ∆−differentiable at t ∈ T κ if there exists a finite number f ∆ (t) with the property that given any ε > 0, there is a neighborhood U ⊂ T of t such that for all s ∈ U. If u is ∆-differentiable at every t ∈ T κ , then u is said to be ∆-differentiable. Moreover, if u is ∆-differentiable at t, then u is continuous at t, and so, if u is ∆-differentiable then u ∈ C(T). Denote by C 1 rd (T) the set of functions u ∈ C(T), which are ∆−differentiable on T κ , and their ∆−derivatives are rd-continuous on T κ with the norm Given a function u : T → R, we consider an auxiliary function which extends u to the real interval [a, b], u : [a, b] → R defined as Lemma 1. The following statements are equivalent (i) u maps every ∆−null subset of T into a null set; (ii) u maps every null subset of [a, b] into a null set.

Proof.
From [17], we know that conditions (i) and (ii) are equivalent in the case when the point b defined in (2) does not contain ∆−null subsets of time scale T. Therefore, since we adopted the approach to the ∆−measure from [11], it is sufficient to show that The total variation of u on T is given by a ∈ R, we say that u is a function of bounded variation on T. A direct consequence of the definition of u is the following result. Proposition 1. Ref. [17]. Let u : T → R and u : [a, b] → R be the extension of u to [a, b] defined in (7). Then, u is of bounded variation on T if and only if u is of bounded variation on [a, b].
We denote by AC(T) the set of all absolutely continous functions over T.
The following results establish a criterion for absolute continuity on the time scale T.
Ref. [17]. A function u : T → R is absolutely continuous on T if the following conditions hold true (i) u is continuous and of bounded variation on T; (ii) u maps every ∆-null subset of T into a null set.
Proposition 3. Ref. [17]. Function u : T → R is absolutely continuous on T if and only if the extension function u defined in (7) Consider a measurable function p : T → (1, ∞) and assume that it is bounded, i.e., and we write p ∈ L ∞ + (T). By M(T), we denote the set of all equivalence classes of real ∆-measurable functions defined on T being equal ∆-a.e. on T. The variable exponent Lebesgue space L p(t) (T) consists of all measurable functions u ∈ M(T) for which the ρ p(·) -modular is finite, i.e., The Luxemburg-type norm on this space is defined as Equipped with this norm, L p(t) (T) is separable and reflexive if p ∈ L ∞ + (T). For estimates, one can use the following inequalities.

Proposition 7.
There exist functions f 1 , Note that these inequalities imply the equivalence of convergence in norm and in modular.
Let p, q ∈ L ∞ + (T) and p, q be conjugative on the time scale T, e.g., Proposition 10. Ref. [15]. For every u ∈ L p(t) (T) and v ∈ L q(t) (T), the following Hölder inequality holds: We define the variable exponent Sobolev space on time scales by Then, W 1,p(t) (T), · W 1,p(t) (T) is separable and reflexive if p ∈ L ∞ + (T).
We denote by C ∞ rd (T) (respectively C ∞ (T)) the set of continuous functions over T which are of n times rd-continuously (respectively continuously) ∆-differentiable on T κ for any n ∈ N. We define W 1,p(t) 0
In the classical one-dimensional situation of W 1,p(t) (I) with I = (a, b), each element u has a continuous representative u (see (7)) in its equivalence class for equality ∆−a.a., and W 1,p(t) 0 (I) can be characterized as the set of u ∈ W 1,p(t) (I) such that u(a) = 0 = u(b).

Recall that there exists
Consequently, one can consider the space W 1,p(t) 0 (T) with the following equivalent norm It is known that the following continuous embeddings hold and Moreover, we recall that the following embeddings are compact. Since any element of W 1,p − (T) is absolutely continuous (see [14]), we know that the same holds for any u ∈ W 1,p(t) (T), which implies that any element of W 1,p(t) (T) is ∆-differentiable ∆-a.e. on T. If u ∈ W 1,p(t) 0 (T), then u is continuous, which implies that u σ ∈ C rd (T) and (6) holds. Consequently, by (12) and (13), there are A, C, C 1 > 0 such that

Du Bois−Reymond Type Lemma
In this section, we will prove a du Bois-Reymond type lemma for nondifferentiable functions.
By (9), we estimate for any u ∈ L p(t) (T). Consequently, for u ∈ L p(t) (T), where q is the function given in (9). By (10) and (15), for any u, v ∈ L p(t) (T), is well defined.

Lemma 3. If h ∈ L q(t) (T) and
for ∆-a.a. t ∈ T and v ∆ ∈ L p(t) (T). By (17), we have Since ρ q(·) is a modular, we have h(t) = c for ∆−a.a. t ∈ T and the lemma follows.
The following lemma plays a key role in the next section.

The p(t)-Laplacian Dirichlet Problem
Let X := W 1,p(t) 0 (T). The following assumptions upon f and p are made: Let us consider the following problem: where u ∈ X, a and b are defined in (2) and σ is a forward jump operator given in (3).
We say that u ∈ X is a weak solution to (20) if (21) for every v ∈ X.
We define the functional ϕ : X → R by where for ∆−a.a. t ∈ T and x ∈ R. Moreover, let us denote and for u ∈ X.
Observe that if f satisfies Assumption (F), then also F is an L 1 -Carathéodory function over T × R and thus, t → F(t, u σ (t)) belongs to L 1 (T). Consequently, ϕ 2 is well defined, which implies that ϕ is well defined. (24) is continuously differentiable on X at any u ∈ X and
Let u n → u in L p(t) (T) and (v n ) be a subsequence of (u n ). Let v n l and g be given as in Lemma 2. Then, from Lemma 2 and Proposition 5, one has Since v n l (t) → u(t) for ∆−a.a. t ∈ T, it follows from Lebesgue Dominated Convergence Theorem that ∆t → 0, as l → ∞, but then, since any subsequence (ξ p(t) (v n )) has a subsequence (ξ p(t) (v n l )) convergent to the same limit, ∆t → 0, as n → ∞.
If u n → u in X, then, by (14) , u n → u in C(T) and there exists d > 0 such that |u n (t)| ≤ d for n ∈ N, which implies that |u σ n (t)| ≤ d for n ∈ N and ∆−a.a. t ∈ T. Since f is L 1 −Carathéodory function, there is f d ∈ L 1 (T) such that, for n ∈ N and for ∆−a.a. t ∈ T, we have | f (t, u σ n (t))| ≤ f d (t). Let u n → u in X. Then, from (14), u σ n → u σ in L p(t) (T). Now, as in the second part of the proof of Lemma 5, using Lemma 2, one can show that f (t, u σ n (t)) → f (t, u σ (t)) for ∆-a.a. t ∈ T, as n → ∞. Applying the Lebesgue Dominated Convergence Theorem, ϕ 2 is continuously differentiable. (22) is also a weak solution to (20). Now, taking h 1 (t) := − f (t, u σ (t)) and h 2 (t) := ∆ w u(t) for t ∈ T in Lemma 4, we obtain that a possible solution to (21) is a solution to problem (20).

Remark 2. From Lemmas 5 and 6, a critical point of functional ϕ defined in
Moreover, from Lemma 4, the function is absolutely continuous on T. Consequently, a weak solution to problem (20) is a classical solution.
We now provide some properties of the operator ϕ 1 that will be needed in next Sections.
It is easy to verify that the following holds : if p 0 ∈ (1, ∞), then for all x, y ∈ R n . Lemma 7. The mapping ϕ 1 : X → X * is coercive and strictly monotone.
Proof. Since ϕ 1 is continuous, one can easily see that ϕ 1 is hemicontinuous, i.e., for all u, v, w ∈ X, the mapping h → ϕ 1 (u + hv)(w) is continuous on [0, 1]. Now, the statement follows from the fact that for a monotone operator, demicontinuity and hemicontinuity are equivalent.
Lemma 10. If u n u in X and then u n → u in X.
Proof. It suffices to show that (ϕ 1 ) −1 is continuous. Let z n , z ∈ X * , z n → z. Then, there are u n , u ∈ X, such that ϕ 1 (u n ) = z n and ϕ 1 (u) = z. Since ϕ 1 is continuous, (u n ) is bounded in X. Without loss of generality, let u n v. Then, we have From Lemma 10, u n → v in X. Consequently, u n → u in X.
Proof. The proof is analogous to the proof of Theorem 2 with one exception. If β = p − , then to show that ϕ is weakly coercive, no inequality of the type (36) is needed.

Remark 3.
Assume that the following condition holds (F)' f : T × R → R is a ∆-Carathéodory function over T × R. Then, (F)' together with (37) and Proposition 9 guarantee that Lemmas 5 and 6 hold.

Example 1. Notice that condition (37) is satisfied if
where G(t, x) = x 0 g(t, s)ds and g(t, x) ≤ c 1 for ∆−a.a. t ∈ T, x ∈ R. In particular, problem (20) with f given by has a weak solution for every bounded function g ∈ L 1 (T).

Existence of a Nontrivial Solution Using the Mountain Pass Theorem
The existence conditions of Theorem 2 are satisfied when F(t, x) does not grow too fast when x → ∞. We now use another tool of the variational calculus, namely a minimax instead of a minimum characterization of a critical point of the functional, to prove the existence of a nontrivial solution of problem (20) when f (t, x) tends fast enough to 0 when x → 0 (insuring the existence of the trivial solution) and fast enough to infinity when x → ∞.
We say that C 1 -functional ϕ : X → R satisfies the Palais-Smale condition, denoted (PS), if any sequence (u n ) n∈N in X, such that (ϕ(u n )) n∈N is bounded and ϕ (u n ) → 0 as n → ∞, admits a convergent subsequence.
Proof. Assume that (u n ) n∈N is a sequence such that u n ∈ X for n ∈ N, (ϕ(u n )) n∈N is bounded and ϕ (u n ) X * → 0 as n → ∞. First, we shall show that (u n ) n∈N is bounded. Let ε > 0. Since ϕ (u n ) X * → 0, we obtain that there exists n 0 ∈ N, such that ϕ (u n ) X * < ε for n ≥ n 0 . Thus, we have ϕ (u n )(u n ) ≥ −ε u n X for n ≥ n 0 . Moreover, ϕ (u n )(u n ) = T |∆ w u n (t)| p(t) ∆t − T f (t, u σ n (t))u σ n (t)∆ t = ρ p(·) (∆ w u n ) − T f (t, u σ n (t))u σ n (t)∆t for n ∈ N, where ρ p(·) is the modular defined in (8). Since f is the L 1 -Carathéodory function over T × R, integrals The existence of nontrivial solutions to problem (20) will be shown using the Mountain Pass Theorem of Ambrosetti and Rabinowitz [23], which we recall here in the following form.