For $0 < p-1 < q$ and $\ge=\pm 1$, we prove the existence of solutions of $-\Gd_pu=\ge u^q$ in a cone $C_S$, with vertex $0$ and opening $S$, vanishing on $\prt C_S$, under the form $u(x)=|x|^\gb\gw(\frac{x}{|x|})$. The problem reduces to a quasilinear elliptic equation on $S$ and existence is based upon degree theory and homotopy methods. We also obtain a non-existence result in some critical case by an integral type identity.