A unified approach to symmetry for semilinear equations associated to the Laplacian in R^N

We show radial symmetry of positive solutions to the Henon equation -Delta u = vertical bar x vertical bar(-l)u(q) in R-N \{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(vertical bar x vertical bar ->infinity) vertical bar x vertical bar(gamma)u(x) = L, for some numbers gamma is an element of (0, N - 2) and L > 0. Moreover, some examples of non-radial solutions are given for q > N+1/N-3 and N >= 4. We also establish radial symmetry for related and more general problems in R-N and R-N \ {0}. (C) 2020 Elsevier Inc. All rights reserved.