Boundary regularity and Wiener-type criteria at infinity for nonlinear elliptic equations of $p$-Laplace type

We study boundary regularity at the infinity point $\mathbf{\infty }$ for nonlinear elliptic equations of $p$-Laplace type in unbounded open sets $\mathrm{\Omega }\subset {\mathbf{R}}^n$. We consider the case $p\ge n\ge 2$ and characterize the regularity at $\mathbf{\infty }$ by means of Wiener-type integrals. Our approach uses circular inversion, which maps $\mathbf{\infty }$ to the origin and the original nonlinear equation to a similar weighted equation. The Wiener criterion at the origin for such equations is then transformed back to provide Wiener-type criteria at $\mathbf{\infty }$. When $p>n$, the criteria simplify so that $\mathbf{\infty }$ is regular if and only if the boundary $\mathrm{\partial }\mathrm{\Omega }$ is unbounded. For $p=n$ this is not true, as shown by an example. This simplified criterion is also proved for $p$-harmonic functions in unbounded open subsets of Ahlfors $Q$-regular metric measure spaces with $Q<p$, supporting a Poincaré inequality.