eu LIOUVILLE-TYPE RESULT FOR QUASILINEAR ELLIPTIC PROBLEMS WITH VARIABLE EXPONENT

In this paper we discuss several aspects of relations between the Keller-Osserman property and the validity of the Liouville-type theorem for quasilinear elliptic equation with variable exponent that include non-homogeneous p(.)-Laplace equations for p measurable function. AMS Subject Classification: 35J92, 31C15, 35B53, 31A05


Introduction
The study of singular quasilinear elliptic equation, and specifically boundary blow-up problems, has attracted considerable attention starting with the original work of Bieberbach (1916) [6].This approach was introduced in the last decades of the need to provide answers to important questions of the nonlinear problems with variable exponent for example see [18], [5].
Set in order to obtain and even formulate Liouville's theorem, for example, for p(.)-superharmonic functions on R d , d ≥ 3, one needs to compare an arbitrary p(.)-superharmonic function with a constant which is a trivial p(.)subharmonic function.In this paper, we study the quasilinear elliptic equation where A : R d × R d → R d and B : R d × R → R, are Carathéodory functions satisfying the structure conditions given in assumptions (H0), (H1), (H2), (H3), and (H4) below.
This problem appears in the study of quantum mechanics, classical statistical and Hamiltonian mechanics we refer to [13].There are also related mathematical results in ergodic theory.In [2] Baalal and Boukricha study the same problem when the function p(x) = p is constant.Here we provide a more elementary proof, which in particular we are interested in the nonlinear potential theory associated with variable exponent of quasilinear elliptic equations (1).
For the existence and uniqueness of p(.)-solutions u ∈ W 1,p(x) (Ω) where 1 < p(x) < d for all x ∈ Ω, of the variational Dirichlet problem associated with the quasilinear elliptic equation (1) see [3], these p(.)-solutions are obtained by the p(.)-obstacle problem.The Harnack inequality is valid as it is a fundamental tool for the crucial question for the regularity of weak p(.)-solutions for the equation (1) with nonstandard p(x)-growth conditions, see [4].
In [15], [19] Keller and Osserman prove that a necessary and sufficient condition for the considered problem to have an entire solution is that B. In the case where p(x) = p is constant the validity of the Keller-Osserman property with variable exponent see [5].
The attention is focused on those problems whose p(.)-solution of (1) depends on some Liouville-type theorem.Let us mentioned, in this respect, some examples of Liouville's theorem: Cauchy and Liouville, Hadamard and Liouville, Poisson and Liouville, De Giorgi and Liouville, Harnack and Liouville then Moser and Liouville.The potential theory focuses more on the Poison-Liouville theorem for more details see [12].
The contribution of this paper is to extend some of the results in [2] to the equation (1) under general assumptions below.We discuss several aspects of relations between the Keller-Osserman property and the validity of the Liouville-type theorem for quasilinear elliptic equation with variable exponent that include p(x)-Laplace equations for p measurable function.
This paper deal with a crucial question concerning the Non-homogeneity of p(x)-Laplacian in the presence of the nonlinear term given as B. Because of the Non-homogeneity of p(x)-Laplacian, p(x)-Laplacian problems are more complicated than those of p-Laplacian ones; and another difficulty of this paper is that B cannot be represented as b(x)f (u).
In the first section, we introduce some generalization and position of the problem.In second section we give some basic facts about variable exponent spaces and a rough overview of properties of p(.)-solutions of the prototype equality.In Section 3, we generalize, with detailed proofs, we prove that the Keller-Osserman property in (R d , H) is valid; i.e., every open ball admits a regular Evans function, we discuss several aspects of relations between the Keller-Osserman property and the validity of the Liouville-type theorem for quasilinear elliptic equation with variable exponent.

Preliminaries
We define the Lebesgue space with variable exponent L p(.) (Ω) as the set of all measurable functions p : Ω →]1, +∞[ called a variable exponent and we denote p − := ess inf x∈Ω p(x) and p + := ess sup x∈Ω p(x).
For each open bounded subset Ω of R d (d ≥ 2), we denote We introduce also the convex modular If the exponent is bounded, i.e., if p + < ∞, then the expression defines a norm in L p(.) (Ω), called the Luxemburg norm.
Let p * (x) be the critical Sobolev exponent of p(x) defined by for p(x) < d
The following comparison principle is useful for the potential theory associated with equation (1): Lemma 3. Suppose that u is a p(.)-supersolution and v is a p(.)-subsolution on Ω such that lim sup for all y ∈ ∂Ω and if both sides of the inequality are not simultaneously +∞ or −∞, then v ≤ u in Ω.
Throughout the paper we suppose that the functions A : R d × R d → R d is a Carathéodory function satisfying the following assumptions: for a.e.x ∈ Ω, all ξ ∈ R d , where k is a positive bounded function lying in L p ′ (x) (Ω) and β, ν > 0.
In this paper we suppose that the function B : R d × R → R is given Carathéodory function and the following condition is satisfied: where g is a positive bounded function lying in L p ′ (x) (Ω) and p(x) ≤ δ(x) + 1 < p * (x)

Keller-Osserman Property
For every open set U we shall denote by U (U ) the set of all relatively compact open regular subset V in U with V ⊂ U .By previous section and in order to obtain an axiomatic nonlinear potential theory, we shall investigate the harmonic sheaf associated with (1) and defined as follows: For every open subset U of R d (d ≥ 1), we set Element in the set H(U ) are called harmonic on U .
In this section, we prove that functions for which the comparison principle holds are monotone limits of p(.)-supersolutions see [17], [1].

Definition 4.
We say that a function u : Ω → (−∞, +∞] is p(.)superharmonic in Ω if: 1. u is lower semicontinuous; 2. u is finite almost everywhere and; We also introduce the Keller-Osserman property, the p(.)-regularEvans functions apparently for the first time to proof Liouville theorem for the equation (1) with variable exponent.
For an investigation of regular Evans functions see [7], [10], [20], [21]. then for x ∈ B, we obtain By the comparison principle we have H B n ≤ ψ for every n ∈ N and therefore, the increasing sequence (H B n) n of harmonic functions is locally uniformly bounded on B.
The Bauer convergence property implies that u = sup n H B n ∈ H(B) see [2].
Therefore we have lim inf

Liouville Theorem with Variable Exponent
In this section we prove the Liouville theorem, since that the original proof of Baalal and Boukricha is based on the explicit knowledge of positive supersolutions of the equation ( 1) with p > 1 positive constant on arbitrary open balls, with the further property that Evans functions explose at the boundary of the considered balls.This fact is crucially related to the shape of the non-linear function B(t) = |t| p−1 t and it does not easily extend to more general functions B see [8].
For this reason, we give here a different proof based on Theorem (9).This approach has the advantage to apply to a larger class of problems.We may now state our main results.First we have to consider a lemma: Lemma 10.Under the assumptions in Theorem 9, for every ball B=B(x 0 , R) with center x 0 and radius R and for every u ∈ H(U ), Proof.From the proof of the previous theorem, if for every n ≥ 2 and Then we obtain the inequality Since −u is a p(.)-solution of similarly equation, we get with the same constant c as before.Then we have the desired inequality.
The proof of the Liouville theorem uses an idea of A. Baalal end A. Boukricha in [2], who proved the corresponding Liouville theorem for quasilinear elliptic equation with Keller-Osserman condition.Here we make use of various have a Liouville like theorem associated with variable exponent.

3 .Theorem 5 .
If h is a p(.)-solution in D ⊂⊂ Ω, continuous in D and u ≥ h on ∂Ω, then u ≥ h in D. Let u be a p(.)-supersolution in with the property u(x) = ess lim inf x→y u(y) for all x ∈ Ω Then u is p(.)-superharmonic.Let H be the sheaf of continuous p(.)-solutions related to the equation (1) and we consider ball B=B(x 0 , R) with center x 0 and radius R and for every u ∈ H(B), Definition 6.A function u ∈ H + (B) is called p(.)-regularEvans function for H and B if lim B∋x→z u(x) = +∞ for every regular point z in the boundary of B.
x→z u(x) ≥ n for every z in ∂B, thus lim x→z u(x) = +∞ for every z in ∂B and u is a p(.)-regularEvans function.Since B is an arbitrary ball.