Non-uniqueness of positive ground states of non-linear Schrödinger equations

Existence of a positive, decaying radial solution to the problem Δ u − u + u p + λ u q = 0 in R N , when λ>0 and 1<q<p<(N+2)/(N−2) has been known for a long time. For λ=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, λ is fixed sufficiently large, and p<5 is taken sufficiently close to 5, then there are at least three positive decaying radial solutions.