the HÃÂ©non equation

the HÃÂ©non equationThe Henon equation and its associated energy functional can be written as

\\;\\left\\{\\begin{array}[]{lll}-\\Delta u=|x|^{r}|u|^{q-1}u,&\\!x\\in\\Omega\\\\u=0,&\\!x\\in\\partial\\Omega,\\end{array}\\right.\\;J(u)=\\int_{\\Omega}\\Big{[}\\frac{1%}{2}|\abla u|^{2}-\\frac{1}{q+1}|x|^{r}|u|^{q+1}\\Big{]}dx

where $u\\in H_{0}^{1}(\\Omega)$ , $\\Omega\\subset{\\mathbb{R}}^{N}(N\\geq 2)$ is an open bounded domain, $r\\geq 0$ and $1<q\\leq\\frac{N+2}{N-2}$ .

the HÃÂ©non equation

the HÃÂ©non equation