RADIAL SYMMETRY AND DECAY-RATE OF VARIATIONAL GROUND-STATES IN THE ZERO MASS CASE

P.-L. Lions raised the question whether variational ground state solut ions of the semilinear Dirichlet problem - Delta w = f(w) in R-n, w(x) --> 0 as \x\ --> infinity are radial with constant sign. We consider the zero mass case f(0) = f'(0) = 0 without regularity assumptions for the nonlinearity. The celebrated symmetry result of Gidas, Ni, and Ni renberg and its refinements do not apply. Nevertheless we give an affi rmative answer to the question of Lions. We prove that every variation al ground state is either strictly positive or strictly negative. For positive nonlinearities positive solutions are radially symmetric with respect to some point and strictly decreasing in radial direction. Fo r general nonlinearities we show that the same is true outside a compa ct set. This is a consequence of our main result, the second-order dec ay estimate w(r) = cr(2-n) (1 + O(r(-2))) in the C-1-sense. In additio n we obtain an integral representation for the constant c.