the HÃÂ©non equation

the HÃÂ©non equation The 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}|\nabla 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