Separable solutions of quasilinear Lane-Emden equations

For $0<p-1<q$ and $\ge =\pm 1$, we prove the existence of solutions of $-{\text{Gd}}_{p}u=\ge u^q$ in a cone $C_S$, with vertex 0 and opening $S$, vanishing on $\text{prt}{C}_S$, under the form $u(x)=\mathrm{\mid }x{\mathrm{\mid }}^{\text{gb}}\text{gw}(\frac{x}{\mathrm{\mid }x\mathrm{\mid }})$. 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.