Existence result for a nonlinear elliptic problem by topological degree in Sobolev spaces with variable exponent

Abstract
The aim of this paper is to establish the existence of solutions for a nonlinear elliptic problem of the form







{





A

(
u
)

=
f




i
n



Ω





u
=
0




o
n




∂
Ω








\left\{ {\matrix{{A\left( u \right) = f} \hfill & {in} \hfill & \Omega \hfill \cr {u = 0} \hfill & {on} \hfill & {\partial \Omega } \hfill \cr } } \right.




where
A
(
u
) = −div
a
(
x
,
u
, ∇
u
) is a Leray-Lions operator and
f
∈

W
−1,p
′

(.)
(Ω) with
p
(
x
) ∈ (1, ∞). Our technical approach is based on topological degree method and the theory of variable exponent Sobolev spaces.