On the Gibbons conjecture for equations involving the $p$-Laplacian

In this paper we prove the validity of Gibbons conjecture for the quasilinear elliptic equation $ -Delta_p u = f(u) $ on $mathbb{R}^N.$ The result holds true for $(2N+2)/(N+2) < p < 2$ and for a very general class of nonlinearity $f$.