Separable solutions of quasilinear Lane–Emden equations

  • Laurent Véron

    Université François Rabelais, Tours, France
  • Alessio Porretta

    Università di Roma, Italy


For 0<p1<q0<p-1<q and either ϵ=1\epsilon=1 or ϵ=1\epsilon=-1, we prove the existence of solutions of Δpu=ϵuq-\Delta_pu=\epsilon u^q in a cone CSC_S, with vertex 00 and opening SS, vanishing on CS\partial C_S, under the form u(x)=xβω(xx)u(x)=|x|^{-\beta}\omega(\frac{x}{|x|}). The problem reduces to a quasilinear elliptic equation on SS 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