Essentially fully anisotropic Orlicz functions and uniqueness to measure data problem

Studying elliptic measure data problem with strongly nonlinear operator whose growth is described by the means of fully anisotropic $N$-function, we prove the uniqueness for a broad class of measures. In order to provide it, the framework of capacities in fully anisotropic Orlicz-Sobolev spaces is developed and the~capacitary characterization of a~bounded measure is given. Moreover, we give an example of an anisotropic Young function $\mathrm{\Phi }$, such that $\mathrm{\mid }\xi {\mathrm{\mid }}^p\lesssim \mathrm{\Phi }(\xi )\lesssim \mathrm{\mid }\xi {\mathrm{\mid }}^p{\log }^{\alpha }(1+\mathrm{\mid }\xi \mathrm{\mid })$, with arbitrary $p\ge 1$, $\alpha >0$, but so irregularly growing that % we call it essentially fully anisotropic. In fact, the Orlicz–Sobolev–type space generated by $\mathrm{\Phi }$ indispensably requires fully anisotropic tools to be handled.