Behaviour at infinity for solutions of a mixed nonlinear elliptic boundary value problem via inversion

Abstract

We study a mixed boundary value problem for the quasilinear elliptic equation $divA(x,\nabla u\left(x\right))=0$ in an open infinite circular half-cylinder with prescribed continuous Dirichlet data on a part of the boundary and zero conormal derivative on the rest. The equation is assumed to satisfy the standard ellipticity conditions with a parameter $p>1$. We prove the existence and uniqueness of bounded weak solutions to the mixed problem and characterize the regularity of the point at infinity in terms of $p$-capacities. For solutions with only Neumann data near the point at infinity we show that they behave in exactly one of three possible ways, similar to the alternatives in the Phragmén–Lindelöf principle.