On properties of solutions of a class of nonlinear second-order equations

The boundary value problem
$$ Lu=f(|u|) \quad \text {in}\quad \Omega , \qquad u\big|_{\partial \Omega }=w, $$
is studied, where $\Omega$ is an arbitrary, possibly unbounded, open subset of $R^n$, $L=\sum\limits_{i,j=1}^n\dfrac \partial {\partial x_i} \biggl(a_{ij}(x)\dfrac \partial {\partial x_j}\biggr)$ is a differential operator of elliptic type with measurable coefficients, and $w$, $f$ are some functions.