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

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