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.