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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 growthof~the~monotone vector field $A$ with respect to the $(z,\xi)$ variables isexpressed through some $N$-functions $B$ and $P$. We do not require anyparticular type of growth condition of such functions, so we deal with problemsin nonreflexive spaces. When the problem involves measure data and weaklymonotone operator, we prove existence. For $L^1$-data problems with stronglymonotone operator we infer also uniqueness and regularity of~solutions andtheir gradients in the scale of Orlicz-Marcinkiewicz spaces.

Elliptic problems with growth in nonreflexive Orlicz spaces and with measure or $L^1$ data