Separable solutions of quasilinear Lane–Emden equations

Abstract

For $0<p-1<q$ and either $ϵ=1$ or $ϵ=-1$, we prove the existence of solutions of $-{\Delta }_{p}u=ϵu^q$ in a cone $C_S$, with vertex $0$ and opening $S$, vanishing on $\partial C_S$, under the form $u(x)=|x{|}^{-\beta }\omega (\frac{x}{|x|})$. 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.