We consider regular solutions of the elliptic problem (1) $\Delta u-|\nabla u|^q+\lambda u^p=0$, and $u>0$ for $x\in\Omega $, $u=0$ on $x\in\partial \Omega $, where $\Omega \subset \bold R^n$ is a smooth, bounded domain, $u\colon \overline{\Omega }\rightarrow \bold R$ is $C^2$, and $q, p, \lambda$ are parameters satisfying $q, p>1$ and $\lambda >0$. The motivation for studying problem (1) comes from a related parabolic problem, $(2)\ v_t=\Delta v-|\nabla v|^q+v^p\ (t>0,\ x\in\Omega )$, $v(0, x)=\varphi(x)\ge 0\ (x\in\Omega )$, $v(t, x)=0\ (t>0$, $x\in\partial \Omega )$, where now $v=v(t, x)$. Previously, we investigated whether there exist initial values $\varphi$ for which the resulting solution of (2) blows up in finite time. It turns out that a natural candidate for such a $\varphi$ is a solution of (1). More precisely, assume either $(3)\ 1<q< 2p/(p+1)$, $n/2<(p+1)/(p-1)$, $\lambda>0$ is sufficiently small, or $(4)\ q=2p/(p+1)$, $p$ is sufficiently large, $0<\lambda\le 2/(p+1)$. It follows that if $\varphi$ is a solution of (1), then the resulting solution of (2) blows up in finite time.

Thus we are led to investigate under what conditions there exist solutions of (1) and whether these solutions are unique.

Chipot, M; Weissler, F (1988). *On the elliptic problem Δu-|∇u| q +λu p =0.* In: Ni, W M; Peletier, L A; Serrin, J. Nonlinear diffusion equations and their equilibrium states I (Berkeley, CA, 1986). New York: Springer, 237-243.

## Abstract

We consider regular solutions of the elliptic problem (1) $\Delta u-|\nabla u|^q+\lambda u^p=0$, and $u>0$ for $x\in\Omega $, $u=0$ on $x\in\partial \Omega $, where $\Omega \subset \bold R^n$ is a smooth, bounded domain, $u\colon \overline{\Omega }\rightarrow \bold R$ is $C^2$, and $q, p, \lambda$ are parameters satisfying $q, p>1$ and $\lambda >0$. The motivation for studying problem (1) comes from a related parabolic problem, $(2)\ v_t=\Delta v-|\nabla v|^q+v^p\ (t>0,\ x\in\Omega )$, $v(0, x)=\varphi(x)\ge 0\ (x\in\Omega )$, $v(t, x)=0\ (t>0$, $x\in\partial \Omega )$, where now $v=v(t, x)$. Previously, we investigated whether there exist initial values $\varphi$ for which the resulting solution of (2) blows up in finite time. It turns out that a natural candidate for such a $\varphi$ is a solution of (1). More precisely, assume either $(3)\ 1<q< 2p/(p+1)$, $n/2<(p+1)/(p-1)$, $\lambda>0$ is sufficiently small, or $(4)\ q=2p/(p+1)$, $p$ is sufficiently large, $0<\lambda\le 2/(p+1)$. It follows that if $\varphi$ is a solution of (1), then the resulting solution of (2) blows up in finite time.

Thus we are led to investigate under what conditions there exist solutions of (1) and whether these solutions are unique.

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## Additional indexing

Item Type: | Book Section, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | regular solutions; existence; uniqueness; support |

Language: | English |

Date: | 1988 |

Deposited On: | 28 Oct 2009 13:37 |

Last Modified: | 14 Sep 2016 13:39 |

Publisher: | Springer |

Series Name: | Mathematical Sciences Research Institute Publications. |

Number: | 12 |

ISSN: | 0940-4740 |

ISBN: | 0-387-96771-0 |

Official URL: | http://www.springer.com/mathematics/analysis/book/978-0-387-96771-4 |

Related URLs: | http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0699.35102 http://www.recherche-portal.ch/primo_library/libweb/action/search.do?fn=search&mode=Advanced&vid=ZAD&vl%28186672378UI0%29=isbn&vl%281UI0%29=contains&vl%28freeText0%29=0-387-96771-0 |

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