Renormalized solutions for some boundary value elliptic problem with L1−data in generalized Sobolev spaces

The aim of this paper is to investigate an existence of renormalized solutions forsome boundary value elliptic problem of the form −div(a(x, u,u) + (x, u)) +g(x, u,u) =f in ,in the framework of Musielak-Orlicz spaces, where the term satisfies the naturalgrowth condition, the functionghas a natural growth with respect to its third argument andwithout sign condition, no 2-condition is assumed on the Musielak function, and fL1()