On monotonicity of solutions of Dirichlet problem for some quasilinear elliptic equations in half-spaces

We prove the monotonicity of nonnegative bounded solutions to the Dirichlet problem for a quasilinear elliptic equation of the form $-\Delta_p u=f(u)g(x_n)$ with $p\geqslant3$ in a half-space, where $x_n$ is the normal coordinate of the argument. This assertion implies new nonexistence results for the case $f(u)=u^q$ with the appropriate values of $q$.