Radial Symmetry and Decay Rate of Variational Ground States in the Zero Mass Case

Abstract

P.-L. Lions raised the question whether variational ground state solutions of the semilinear Dirichlet problem \begin{eqnarray*} - \Delta w &=& f(w) \mbox{in} {\Bbb R}^n, w(x) &\to& 0 \mbox{as} |x| \to \infty \end{eqnarray*} 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 Nirenberg and its refinements do not apply. Nevertheless we give an affirmative answer to the question of Lions. We prove that every variational 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. For general nonlinearities we show that the same is true outside a compact set. This is a consequence of our main result, the second-order decay estimate w(r) = c,r2-n (1+O(r-2 )) \mbox{in the $C^1$-sense.} In addition we obtain an integral representation for t...