Non-Uniqueness of Positive Ground States of Non-Linear Schrödinger Equations

Existence of a positive, decaying radial solution to the problem ?u-u+up+?uq=0 in ?N, when ?>0 and 1<q<p<(N+2)/(N-2) has been known for a long time. For ?=0, it is well known that this solution is unique. While uniqueness conditions for rather general non-linearities have been found, the issue has remained elusive for this problem. We prove that uniqueness is in general not true. We find that if N=3, 1<q<3, ? is fixed sufficiently large, and p<5 is taken sufficiently close to 5, then there are at least three positive decaying radial solutions. © 2012 London Mathematical Society.