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

Abstract

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 \(p \geq 3\) 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 \(q\).