On the \(A\)-Laplacian (Q1773463)

Summary: We prove, for Orlicz spaces \({\mathbf L}_A(\mathbb{R}^N)\) such that \(A\) satisfies the \(\Delta_2\) condition, the nonsolvability of the \(A\)-Laplacian equation \(\Delta_Au+h=0\) 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.