# Separable solutions of quasilinear Lane–Emden equations

### Laurent Véron

Université François Rabelais, Tours, France### Alessio Porretta

Università di Roma, Italy

## 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.

## Cite this article

Laurent Véron, Alessio Porretta, Separable solutions of quasilinear Lane–Emden equations. J. Eur. Math. Soc. 15 (2013), no. 3, pp. 755–774

DOI 10.4171/JEMS/375