  We investigate solutions to nonlinear elliptic Dirichlet problems of the type
\[ \left\{\begin{array}{cl} - {\rm div} A(x,u,\nabla u)= \mu &\qquad \mathrm{
in}\qquad \Omega, u=0 &\qquad \mathrm{ on}\qquad \partial\Omega,
\end{array}\right. \] where $\Omega$ is a bounded Lipschitz domain in
$\mathbb{R}^n$ and $A(x,z,\xi)$ is a Carath\'eodory's function. The growth
of~the~monotone vector field $A$ with respect to the $(z,\xi)$ variables is
expressed through some $N$-functions $B$ and $P$. We do not require any
particular type of growth condition of such functions, so we deal with problems
in nonreflexive spaces. When the problem involves measure data and weakly
monotone operator, we prove existence. For $L^1$-data problems with strongly
monotone operator we infer also uniqueness and regularity of~solutions and
their gradients in the scale of Orlicz-Marcinkiewicz spaces.
