Separable solutions of quasilinear Lane–Emden equations

Alessio Porretta; Laurent Véron

Journal of the European Mathematical Society (2013)

  • Volume: 015, Issue: 3, page 755-774
  • ISSN: 1435-9855

Abstract

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For 0 < p - 1 < q and either ϵ = 1 or ϵ = - 1 , we prove the existence of solutions of - Δ p u = ϵ u q in a cone C S , with vertex 0 and opening S , vanishing on ∂ C S , of the form u ( x ) = x - β ω ( x / x ) . The problem reduces to a quasilinear elliptic equation on S and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.

How to cite

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Porretta, Alessio, and Véron, Laurent. "Separable solutions of quasilinear Lane–Emden equations." Journal of the European Mathematical Society 015.3 (2013): 755-774. <http://eudml.org/doc/277354>.

@article{Porretta2013,
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 $\partial C_S$, of the form $u(x)=\left|x\right|^\{-\beta \}\omega (x/\left|x\right|)$. The problem reduces to a quasilinear elliptic equation on $S$ and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.},
author = {Porretta, Alessio, Véron, Laurent},
journal = {Journal of the European Mathematical Society},
keywords = {quasilinear elliptic equations; $p$-Laplacian; cones; Leray-Schauder degree; quasilinear elliptic equations; -Laplacian; cones; Leray-Schauder degree},
language = {eng},
number = {3},
pages = {755-774},
publisher = {European Mathematical Society Publishing House},
title = {Separable solutions of quasilinear Lane–Emden equations},
url = {http://eudml.org/doc/277354},
volume = {015},
year = {2013},
}

TY - JOUR
AU - Porretta, Alessio
AU - Véron, Laurent
TI - Separable solutions of quasilinear Lane–Emden equations
JO - Journal of the European Mathematical Society
PY - 2013
PB - European Mathematical Society Publishing House
VL - 015
IS - 3
SP - 755
EP - 774
AB - 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 $\partial C_S$, of the form $u(x)=\left|x\right|^{-\beta }\omega (x/\left|x\right|)$. The problem reduces to a quasilinear elliptic equation on $S$ and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.
LA - eng
KW - quasilinear elliptic equations; $p$-Laplacian; cones; Leray-Schauder degree; quasilinear elliptic equations; -Laplacian; cones; Leray-Schauder degree
UR - http://eudml.org/doc/277354
ER -

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