We consider solutions to the elliptic linear equation
\[
Lu:=\underset{i,j=1}{\overset{n}{\sum}}\frac{\partial}{\partial x_{i}}\left(a_{ij}\left(x\right)\frac{\partial u}{\partial x_{j}}\right)=0\qquad\qquad\left(1\right)
\]
of second order in an unbounded domain
\[
\left\{ x=\left(x',x_{n}\right)\::\:\mid x'\mid<Ax_{n}^{\sigma}+B,0<x_{n}<\infty\right\} ,0\leq\sigma\leq1,
\]
in $\mathbf{R}^{n}$. We study the asymptotic behiaviour as $x_{n}\rightarrow\infty$
of the solutions of $\left(1\right)$ satisfying the nonlinear boundary
condition
\[
\frac{\partial u}{\partial N}-b\left(x\right)\mid u\left(x\right)\mid^{p-1}u\left(x\right)=0\qquad\qquad\left(2\right)
\]
on the lateral surface
\[
S=\left\{ x\epsilon\partial Q,\;0<x_{n}<\infty\right\} ,
\]
where p>0, b(x)$\geq b_{0}$ >0. We show that a global solution of
the problem can exist not for all values of parameters p, $\sigma$
and indicate these values. The boundary problem in the cylinder was
studied by us in $\left[1\right]$,$\left[2\right]$. The obtained
results generalize some results of B. Hu in $\left[4\right]$.
