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

Existence of a positive, decaying radial solution to the problem Δu− u+ u + λu = 0 in R , 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 nonlinearities 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.