Fully anisotropic elliptic problems with minimally integrable data

We investigate nonlinear elliptic Dirichlet problems whose growth is driven by a general anisotropic N-function, which is not necessarily of power--type and need not satisfy the Delta_2 nor the nabla _2-condition. Fully anisotropic, non-reflexive Orlicz-Sobolev spaces provide a natural functional framework associated with these problems. Minimal integrability assumptions are detected on the datum on the right-hand side of the equation ensuring existence and uniqueness of weak solutions. When merely integrable, or even measure, data are allowed, existence of suitably further generalized solutions -- in the approximable sense -- is established. Their maximal regularity in Marcinkiewicz--type spaces is exhibited as well. Uniqueness of approximable solutions is also proved in case of L^1--data.