In this article, we study a class of convective diffusive elliptic problem with Dirichlet
boundary condition and measure data in variable exponent spaces. We begin by introducing an
approximate problem via a truncation approach and Yosida’s regularization. Then, we apply the
technique of maximal monotone operators in Banach spaces to obtain a sequence of approximate
solutions. Finally, we pass to the limit and prove that this sequence of solutions converges to at
least one weak or entropy solution of the original problem. Furthermore, under some additional
assumptionsontheconvectivediffusiveterm,weprovetheuniquenessoftheentropysolution.

Sobolev spaces, variable exponent, entropy solution, maximal monotone graph, Radon measure.