On the $A$-Laplacian

Abstract

We prove, for Orlicz spaces ${\mathbf{L}}_A\left({ℝ}^N\right)$ such that $A$ satisfies the ${\Delta }_2$ condition, the nonresolvability of the $A$-Laplacian equation ${\Delta }_{A}u+h=0$ on ${ℝ}^N$, where $\int h\ne 0$, if ${ℝ}^N$ 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 ${\mathbf{L}}_A\left({ℝ}^N\right)$ if ${ℝ}^N$ is $A$-hyperbolic.