Abstract

We show radial symmetry of positive solutions to the Hénon equation −Δu=|x|−ℓuq in RN∖{0}, where ℓ≥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|→∞⁡|x|γu(x)=L, for some numbers γ∈(0,N−2) and L>0. Moreover, some examples of non-radial solutions are given for [Formula presented] and N≥4. We also establish radial symmetry for related and more general problems in RN and RN∖{0}.