On exponential asymptotics at infinity of lower Q-homeomorphisms on the complex plane

The definition of a $Q$-homeomorphism with respect to $p$-modulus is a natural generalization of the geometric definition of a quasiconformal mapping: if $Q(z)\leqslant K<\infty$ a.e. (almost everywhere), then $f$ is quasiconformal as $p=2$ on the complex plane $\mathbb{C}$, see Definition A, p. 21--22 in \cite{A}, $f$ has local Lipschitz property as $1<p<2$, and $f^{-1}$ is local Lipschitz continuous as $p>2$, moreover the bounds for $p$ are sharp, see \cite{Ge}. The class of $Q$-homeomorphisms with respect to $p$-modulus as $p=2$ was first considered in the work \cite{MRSY3}. The definition of a ring $Q$-homeomorphism with respect to $p$-modulus generalizes and localizes the definition of a $Q$-homeomorphism with respect to $p$-modulus and is motivated by the ring definition of quasiconformal mappings in the sense of Gehring, see \cite{Ger2}, introduced originally by V.~Ryazanov, U.~Srebro, and E.~Yakubov on the complex plane as $p=2$, see \cite{DegBe}. The concept of lower $Q$-homeomorphisms with respect to $p$-modulus has also proven to be a very useful tool for studying planar and spatial mappings. This definition has also geometric character and is motivated by Gehring's ring definition of quasiconformity. The theory of lower and ring $Q$-homeomorphisms is applied in the study of boundary value problems for Beltrami equations on the complex plane, as well as spatial homeomorphisms of Sobolev classes and more general Orlicz-Sobolev classes, see \cite{GSSCAOT,RSSY,SalAASF,ARSUMB2018,ARSDOP2019,ASZbPrIM,RSSUMZ2013,RSSUMZ2016,SalZbPrI2015,SalMStud2015,SalUMBOS2016}. The theory of lower and ring $Q$-homeomorphisms with respect to $p$-modulus can also be applied to mappings that are quasiconformal in the mean, see \cite{Golberg2,Kru}. Recently, lower and ring $Q$-homeomorphisms with respect to nonconformal modulus have been applied to studying the properties of regular solutions of nonlinear Beltrami equations, see \cite{GolSal,SalStef23,PSSTM}. Furthermore, the theory of ring $Q$-homeomorphisms can be applied to the study of mappings with finite distortion that belong to Orlicz-Sobolev classes $W^{1,\varphi}_{\rm loc}$ under a Calderon-type condition, and, in particular, to Sobolev classes $W^{1,p}_{\rm loc}$ as $p>n-1$, див. \cite{G1}. In the paper, asymptotic behavior at infinity of lower $Q$-homeomorphisms with respect to the $p$-module as $1<p<2$ in the complex plane has been investigated. A sufficient condition on the norm of the function $Q$ under which the mapping exhibits exponential growth at infinity has been found. An example to demonstrate the sharpness of the obtained results has been constructed.