WEAK SOLUTION FOR FRACTIONAL p(x)-LAPLACIAN PROBLEM WITH DIRICHLET-TYPE BOUNDARY CONDITION

In the present paper, we prove the existence and uniqueness result of weak solutions to a class of fractional p(x)-Laplacian problem with Dirichlet-type boundary condition, the main tool used here is the varitional method combined with the theory of fractional Sobolev spaces with variable exponent. «ï ®¤­®£® a« áã § ¤ ç ÷§ ¤à®¡®¢ ̈¬ p(x)-«  ̄« á÷ ­®¬ § £à ­ ̈ç­®î ã¬®¢®î â ̈ ̄ã  ̈à ̈å«¥ ¤®¢¥¤¥­® â¥®à¥¬ã  ̄à® ÷á­ã¢ ­­ï â  õ¤ ̈­÷áâì á« ¡a®£® à®§¢'ï§aã.  ̈a®à ̈áâ®¢ãîâìáï ¢ à÷ æ÷©­ ̈© ¬¥â®¤ ÷ â¥®à÷ï ¤à®¡®¢ ̈å  ̄à®áâ®à÷¢ ®¡®«¥¢ §¬÷­­®£®  ̄®àï¤aã.

in porous media are modeled by the following equation, \partialc(p) \partialt = \nabla a[k(c(p))(\nabla p + e)], (1.2) where p is the unknown pressure, c volumetric moisture content, k the hydraulic conductivity of the porous medium, a the heterogeneity matrix and - e is the direction of gravity.
On the other hand, attention has been paid to the study of partial dierential equations involving the p(x)-Laplacian operators (see [13], [14], [15], [16], [18], [21] and the references therein). So the natural question that arises is to see which result can be obtained, if we replace the p(x)-Laplacian operator by its fractional version (the fractional p(x)-Laplacian operator). Currently, as far as we know, the only results for fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacian operator are obtained by [4], [5], [12], [17] and [25]. In particular, the authors generalized the last operator to fractional case. Then, they introduced an appropriate functional space to study problems in which a fractional variable exponent operator is present.
In [6] and [26], the authors used the Browder-Minty Theorem to establish the existence of weak solutions, they proved the boundedness, the coerciveness, the hemi-continuity, and the monotonicity condition of the operator to achieve their work. Motiveted by the ideas in [6] and [26], we will show the existence and uniqueness of weak solutions for problem (1.1) in the fractional Sobolev space with variable exponent, using the variational method under the conditions on \alpha , \Theta and f (see (H 1 ), (H 2 ) and (H 3 ) below) . In the particular case when \Theta = 0, the existence of weak solutions for problem (1.1) was treated by several authors (see for example [5] and [17]). The plan of our paper is divided into three sections, organized as follows: In Section 2, we present some preliminaries on fractional Sobolev spaces with variable exponent and some basic tools to prove Theorem 3.2 . In Section 3, we introduce the assumptions and we give the denition of weak solution of problem (1.1), we nish this section by proving the main result.
(2.2) For 0 < s < 1, we dene the fractional Sobolev space with variable exponent via the Gagliardo approach as follows: It is the variable exponent seminorm. For simplicity, we omit the set \Omega from the notation. The space W s,q(x),p(x,y) (\Omega ) is a Banach space with the norm By W s,q(x),p(x,y) 0 (\Omega ), we denote the subspace of W s,q(x),p(x,y) (\Omega ) which is the closure of compactly supported functions in \Omega with respect to norm \| \cdot \| W s,q(x),p(x,y) (\Omega ) . In particular, if q(x) = p(x) for all x \in \Omega , we denote W s,q(x),p(x,y) (\Omega ) and W s,q(x),p(x,y) 0 (\Omega ) by W s,p(x,y) (\Omega ) and W s,p(x,y) 0 (\Omega ) (see [17]), respectively. Denition 2.1. Let p : \Omega \times \Omega -\rightar]1, +\infty [, be a continuous variable exponent and let s \in It is easy to see that \| \cdot \| \rho p(x,y) is a norm which is equivalent to norm \| \cdot \| W s,q(x),p(x,y) (\Omega ) .
We have the following properties: . We would like to mention that the continuous and compact embedding theorem is proved in [17] under the assumption q(x) > p(x) for all x \in \Omega . Here, we give a slightly dierent version of compact embedding theorem assuming that q(x) = p(x) for all x \in \Omega , which can be obtained by following the same discussions in [17]. Theorem 2.4. Let \Omega \subset \BbbR N be a smooth bounded domain and s \in (0, 1). Let p(x, y) be continuous variable exponent with s \times p(x, y) < N for all (x, y) \in \Omega \times \Omega . Let 2.1 and 2.2 be satised. Assume that r : \Omega -\rightar (1, +\infty ) is a continuous variable exponent such that Then, there exists a positive constant C = C(N, s, p, r, \Omega ) such that, for any u \in Moreover, this embedding is compact.
Let q \prime \in C + (\Omega ) be the conjugate exponent of q, that is, 1 q(x) + 1 q \prime (x) = 1 for all x \in \Omega , then we have the following H older-type inequality : Lemma 2.6 ([16]). (H older-type inequality). If u \in L q(x) (\Omega ) and v \in L q \prime (x) (\Omega ), then Denition 2.7 ([19]). Let Y be a reexive Banach space and let P be an operator from Y to its dual Y \prime . We say that P is monotone if and only if Theorem 2.8 ([19]). Let Y be a reexive real Banach space and P : Y -\rightar Y \prime be a bounded operator, hemi-continuous, coercive and monotone on space Y . Then, the equation P u = h has at least one solution u \in Y for each h \in Y \prime . Lemma 2.9 ([1]). For \xi , \eta \in \BbbR N and 1 < p < \infty , we have Lemma 2.10. For a \geq 0, b \geq 0 and 1 \leq p < +\infty , we have (a + b) p \leq 2 p - 1 (a p + b p ).

Assumptions and main result
In this section, we will introduce the concept of weak solutions for problem (1.1) and we will state the existence and uniqueness result for this type of solutions. Firstly, we cite the following assumptions (H 1 ) : \alpha is a non decreasing continuous real function dened on \BbbR , surjective such that \alpha (0) = 0 and there exists a positive constant \lambda 1 such that | \alpha (z)| \leq \lambda 1 | z| p(x) - 1 for all z \in \BbbR and x \in \Omega . (H 2 ) : \Theta is a continuous function from \BbbR to \BbbR such that for all real numbers x, y, we have | \Theta (x) -\Theta (y)| \leq \lambda 2 | x -y| , where \lambda 2 is a real constant such that 0 < \lambda 2 < 1 2 .
Our main result of this work is the following Theorem The proof of existence part of Theorem 3.2 is divided into several steps. \bullet Step 1. The operator T is bounded.
On the one hand, we use H older-type inequality, hypothesis (H 2 ) and Lemma 2.10, we have for any u, v \in W s,p(x,y) 0 (\Omega ), where C 0 = 2 p + - 1 (\lambda p + - 1 2 + 1) . This implies that A 1 is bounded. On the other hand, using again H older-type inequality, hypothesis (H 1 ) and Theorem 2.4, we get where C 1 , C 2 are two constants of continuous embedding given by Theorem 2.4. Then A 2 is bounded. This allows us to deduce that A is bounded. Finally, by H older-type inequality, we get immediately the boundedness of operator L . Hence, the operator T is bounded. \bullet Step 2. The operator T is hemi-continuous.
Then, by Dominated Convergence Theorem, we deduce that This implies that the operator A 1 is continuous on W s,p(x,y) 0 (\Omega ). Secondly, by application of hypothesis (H 1 ), we get immediately the continuity of operator A 2 . Therefore, T is hemi-continuous on W s,p(x,y) 0 (\Omega ).
\bullet Step 3 . The operator T is coercive.
For that, it suces to prove that A is monotone. Firstly ,we have by application of hypothesis (H 1 ) that where C 0 and C 4 are the two constants getting in the proof of boundedness and coerciveness of operator T and  .3) This implies that A 1 is monotone. Therefore T is monotone. Hence, the existence of weak solution for problem (1.1) follows from Theorem 2.8.
Uniqueness part. Let u and w be two weak solutions of problem (1.1). As a test function for the solution u, we take v = u -w in equality (3.1) and for the solution w we take v = w -u as a test function in (3.1), we have By summing up the two above equalities, we get \int \Omega (u -w) 2 dx + \langle A 1 u -A 1 w, u -w\rangle + \int \Omega \Bigl( \alpha (u) -\alpha (w) \Bigr) (u -w) dx = 0.
This implies that u = w a.e in \Omega .