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One-dimensional nonlinear boundary value problems with variable exponent

In this paper, a class of nonlinear differential boundary value problems with variable exponent is investigated. The existence of at least one non-zero solution is established, without assuming on the nonlinear term any condition either at zero or at infinity. The approach is developed within the framework of the Orlicz-Sobolev spaces with variable exponent and it is based on a local minimum theorem for differentiable functions.


1.
Introduction. Differential equations with non-standard growth have been a very active field of investigation in recent years, since they can be used even in the study of various physical phenomena as for instance the modelling of electrorheological fluids (see for instance [17]) or in the analysis of the image restoration (see for instance [1]). In particular, the existence and multiplicity of solutions for boundary value problems with nonlinear equations driven by the p(x)−laplacian operator have been investigated in several papers (see for instance [2], [5], [6], [8], [13], [14] and references therein). A natural setting for the study of differential equations is given by the Orlicz-Sobolev spaces. Indeed, the theory of variable exponent Lebesgue and Sobolev spaces can be inserted in the more general theory of Orlicz spaces (see [16]) and the solutions of differential problems can be seen as elements of such spaces. We recall that detailed investigations on basic properties of these generalized spaces have been made first by Kovácik and Rákosník in [15] and later, with different methods, by Fan and Zhao in [11]. For a complete overview on this topics, we also refer to the monographes [7] and [9].
The aim of this paper is to investigate the following nonlinear boundary value problem with variable exponent 1] a ≥ 0, p ∈ C([0, 1]), with min [0,1] p > 1, and λ is a positive real parameter.
Precisely, the existence of one non-zero solution to (D p(x) λ suitable algebraic inequality involving the primitive of the nonlinear term f (see condition (8) in Theorem 3.1). It is worth noticing that no condition at zero or at infinity on the nonlinear term is assumed, but only a proper behavior of f in an appropriate range, possibly even far from zero, and not involving infinity. As an example, we present here a very special case of our result. Precisely, denoting, as in Section 2, p − = min [0,1] p and p + = max [0,1] p, we have the following. Theorem 1.1. Let g : R → R be a nonnegative continuous function such that admits at least one non-zero weak solutionū such that 0 ≤ū(x) < 4 for all x ∈ [0, 1] .
We wish to underline again that the key assumption of Theorem 1.1 can be satisfied also from nonlinear terms g which are not (p − − 1)−sublinear at zero and which have a completely arbitrary behavior at infinity (see Example 3.2). On the other hand, the key assumption of our main result, Theorem 3.1, is surely satisfied when g is (p − − 1)−sublinear at zero (see Theorem 3.4 and Example 3.1).
The paper is arranged as follows. In Section 2, our main tool, that is the local minimum theorem (see Theorem 2.2) is recalled, as well as the basic definitions and properties of variable exponent Lebesgue and Sobolev spaces are collected. In particular, a proof of the Poincaré inequality in our context is pointed out (Proposition 2), so that it is underlined that it is not necessary to assume the log-Hölder condition on p (see Remark 1). In Section 3, our main result, that is Theorem 3.1, is established and some special cases (Theorems 3. 2. Basic notations and preliminary results. In this section, we recall some notations, definitions and basic properties on the variable Sobolev and Lebesgue spaces which will be used later and we refer to [7], [9], [11], [15], [16] for more details and a complete overview on this topic. On L p(x) ([0, 1]) we consider the norm The generalized Sobolev space W 1,p(x) ([0, 1]) is defined by putting and it is endowed with the following norm Taking also (1) Now, by the Hölder inequality we prove that in our context the Poincaré inequality holds true without further assumptions on the variable exponent p (see Remark 1). To be precise, we have the following result.
Then, one has , from Proposition 1 one has u ∈ L 1 ([0, 1]). Therefore, by standard computations one has for all x ∈ [0, 1]. Now, again from Proposition 1, one has for all x ∈ [0, 1] and so the first inequality is proved. Now, from the previous inequality one has for which one has and the second inequality is verified. Finally, let be a family of functions u ∈ W 1,p(x) 0 p − for all u ∈ , for which the functions in are equi-continuous. Hence, the classical Ascoli-Arzelà Theorem ensures the conclusion.
Remark 1. Note that in Proposition 2, it is not necessary that the exponent p satisfies the log-Hölder condition (see [9,  in L q(x) , with q continuous and sub-critical, is compact (see also [10] and [15]). We recall that, in different contexts, in order to prove the Poincaré inequality, the log-Hölder condition on p must be assumed (see [ (2)). Actually, as it can be easily seen, the same proof gives a better estimate of such a constant, that is, . Clearly, we also have that for all u ∈ W 1,p(x) 0 ([0, 1]).
The following proposition shows that this latest norm is equivalent to the usual one. (1) holds. Then, one has On the other hand, we have dx.
From the definition of norm, taking also (3) into account, we have dx ≤ 1.

GABRIELE BONANNO, GIUSEPPINA D'AGUÌ AND ANGELA SCIAMMETTA
Therefore, it follows Hence, one has and the proof is complete.
Remark 4. Clearly, from Proposition 2 and Proposition 3 we obtain, in particular, the following inequality Throughout the sequel, f : [0, 1] × R → R is an L 1 −Carathéodory function, that is: 1 for all u ∈ X. It is well known that Φ is sequentially weakly lower semicontinuous, it is in C 1 , Φ : X → X * is an homeomorphism and one has for all u, v ∈ X (see for instance [14,Proposition 2.5]). Moreover, put for all u ∈ X. Arguing as in the classical case, it is easy to verify that Ψ is sequentially weakly continuous, it is in C 1 and one one has for all u, v ∈ X. Further, owing to Proposition 2, it follows that Ψ is compact. We also recall that u : Finally, we recall the following definition given in [3]. Φ(u n ) < r for each n ∈ N; has a convergent subsequence.
In order to obtain the existence of one non-zero solution to (D p(x) λ ), our main tool is a recent result obtained in [4, Theorem 2.3], recalled below, which is a consequence of the local minimum theorem established in [3].
The following is a weak maximum principle which guaranties the nonnegativity of the weak solution under appropriate hypothesis on the nonlinear term. Proof. Consider the problem Arguing as [8, Remark 3.1] we obtain our thesis.
Remark 5. Arguing as in [11,Theorem 1.3], it is easy to verify that (4). Moreover, taking Remark 4 into account, one has 3. Main result. In this section, we present our results. The main result is Theorem 3.1, where the existence of one non-zero solution is established, without assuming condition at zero or at infinity. Moreover, some special cases are pointed out, that is, Theorems 3.2, 3.3 and 3.4. In particular, Theorem 3.4 ensures the conclusion assuming the (p − − 1)−sublinearity at zero of the nonlinear term. Finally, two concrete examples of application are given. In the first example, the (p − − 1)−sublinearity at zero of the nonlinear term is satisfied, while in the second one, such a condition is not verified.
Proof. Let X = W 1,p(x) 0 ([0, 1]) be the generalized Sobolev space endowed with the norm defined in (3) and let Φ, Ψ be the functionals as defined in (4) and (5) respectively. As seen in Section 2, Φ and Ψ satisfy all regularity assumptions requested in Theorem 2.2 and, owing to [3, Proposition 2.1], the functional I λ = Φ − λΨ verifies (P S) [r] condition for each r > 0. So, in order to apply Theorem 2.2 it is enough to verify condition (6). To this end, definẽ Therefore, one has Moreover, from d < c and (8) one has max{d p − ; d p + } < 2p − p + (4 p + + 2 a 1 ) min{c p − ; and this is an absurd for which our claim is proved. Hence, it follows that 0 < Φ(ũ) < r.
Further, we observe that from (7) one has u ∞ ≤ max (p + Φ(u))  F (x, d) dx that is, sup and condition (6) the functional I λ = Φ − λΨ admits at least one non-zero critical pointū ∈ X such that Φ(ū) < r. Hence, arguing as seen before, one has |ū(x)| < c for all x ∈ [0, 1], and, taking into account that the critical points of I λ are precisely the weak solutions to problem (D p(x) λ ) and Lemma 2.3, our conclusion is achieved. Now, we point out the following particular case of Theorem 3.1.