  Existence and global regularity results for boundary-value problems of Robin
type for harmonic and polyharmonic functions in $n$-dimensional half-spaces are
offered. The Robin condition on the normal derivative on the boundary of the
half-space is prescribed by a nonlinear function $\mathcal N$ of the relevant
harmonic or polyharmonic functions. General Orlicz type growths for the
function $\mathcal N$ are considered. For instance, functions $\mathcal N$ of
classical power type, their perturbations by logarithmic factors, and
exponential functions are allowed. New sharp boundedness properties in Orlicz
spaces of some classical operators from harmonic analysis, of independent
interest, are critical for our approach.
