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

Abstract

We show radial symmetry of positive solutions to the H\'{e}non equation $-\Delta u = |x|^{-\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 _{|x|\to \infty } |x|^{\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 $N\geq 4$. We also establish radial symmetry for related and more general problems in $\mathbb{R}^N $ and $\mathbb{R}^N \setminus \{ 0\} $.