We prove, for Orlicz spaces $\mathbf{L}_{A}(\mathbb{R}^N)$ such
that $A$ satisfies the $\Delta _2$ condition, the nonresolvability of the $A$ -Laplacian equation $\Delta _{A}u+h=0$ on $\mathbb{R}^N$ , where $\int h\neq 0$ , if $\mathbb{R}^N$ is $A$ -parabolic. For a large class of Orlicz spaces including Lebesgue spaces $\mathbf{L}^p$ ( $p>1$ ), we also prove that the same equation, with any bounded measurable function $h$ with compact support, has a solution with gradient in
$\mathbf{L}_{A}(\mathbb{R}^N)$ if $\mathbb{R}^N$ is $A$ -hyperbolic.