# the HÃÂ©non equation

the HÃÂ©non equation
The Henon equation and its associated energy functional^{} can be written as

$$\{\begin{array}{ccc}-\mathrm{\Delta}u={|x|}^{r}{|u|}^{q-1}u,\hfill & x\in \mathrm{\Omega}\hfill & \\ u=0,\hfill & x\in \partial \mathrm{\Omega},\hfill & \end{array}J(u)={\int}_{\mathrm{\Omega}}\left[\frac{1}{2}{|\nabla u|}^{2}-\frac{1}{q+1}{|x|}^{r}{|u|}^{q+1}\right]\mathit{d}x$$ |

where $u\in {H}_{0}^{1}(\mathrm{\Omega})$, $\mathrm{\Omega}\subset {\mathbb{R}}^{N}(N\ge 2)$ is an open bounded domain, $r\ge 0$ and $$.

Title | the HÃÂ©non equation |
---|---|

Canonical name | TheHnonEquation |

Date of creation | 2013-03-11 19:28:19 |

Last modified on | 2013-03-11 19:28:19 |

Owner | linor (11198) |

Last modified by | (0) |

Numerical id | 1 |

Author | linor (0) |

Entry type | Definition |