Isotropic problem with non homogeneous Dirichlet boundary condition and L^1-data

The purpose of this paper is to study an isotropic problem with non homogeneous Dirichlet boundary condition and L_1 data in a variable-exponent Sobolev space. We start by showing the existence and uniqueness of the weak solution when the source term is bounded. Finally, we use a problem-based approach to prove the existence and uniqueness of entropy solution when the source term is integrable.