Entropy solutions for a doubly nonlinear elliptic problem with variable exponent

We study the boundary value problem b ( u ) − div a ( x , ∇ u ) = f in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R N ( N ⩾ 3 ) and div a ( x , ∇ u ) is a p ( x ) -Laplace type operator with p ( . ) : Ω → [ 1 , + ∞ ) a measurable function and b a continuous and nondecreasing function from R → R . We prove the existence and uniqueness of an entropy solution for L 1 -data f.