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

Abstract

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 $f$.