Behavior of solutions of quasilinear elliptic inequalities in an unbounded domain

We consider the solutions of the inequality $Lu\le\varphi(|{\operatorname{grad}u}|)$, where $L$ is a uniformly elliptic homogeneous operator and $\varphi$ is a function increasing faster than any linear function but not faster than $\xi\ln\xi$, in the unbounded domain
$$ \biggl\{x\in\mathbb R^n\biggm| \sum_{i=2}^nx_i^2<\bigl(\psi(x_1)\bigr)^2, -\infty<x_1<\infty\biggr\}, $$
where $\psi$ is a bounded function with bounded derivative. We estimate the growth of the solutions in terms of $\int_0^{x_1}\frac{dr}{\psi(r)}$. For the special case in which $\varphi(\xi)=a\xi\ln\xi+C$, the solutions $u(x_1,x_2,\dots,x_n)$ grow as $\bigl(\int_0^{x_1}\frac{dr}{\varphi(r)}\bigr)^N$, where $N$ is any given number and $a=a(N)$.