Separable solutions of quasilinear Lane–Emden equations

Abstract

For $0<p-1<q$ and either $\epsilon=1$ or $\epsilon=-1$, we prove the existence of solutions of $-\Delta_pu=\epsilon u^q$ 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.