A unified approach to symmetry for semilinear equations associated to the Laplacian in ${\mathbb{R}}^N$

We show radial symmetry of positive solutions to the H'{e}non equation $-\mathrm{\Delta }u=\mathrm{\mid }x{\mathrm{\mid }}^{-\mathrm{\ell }}{u}^q$ in ${\mathbb{R}}^N\setminus \{0\}$, where $\ell \geq 0$,$q>0$ and satisfy further technical conditions. A new ingredient is a maximum principle for open subsets of a half space. It allows to apply the Moving Plane Method once a slow decay of the solution at infinity has been established, that is ${\lim }_{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}\mathrm{\mid }x{\mathrm{\mid }}^{\gamma }u(x)=L$, for some numbers $\gamma \in (0,N-2)$ and $L>0$. Moreover, some examples of non-radial solutions are given for $q>\frac{N+1}{N-3}$ and $N\ge 4$. We also establish radial symmetry for related and more general problems in ${\mathbb{R}}^N$ and ${\mathbb{R}}^N\setminus \{0\}$.