Morse indices and bifurcations of positive solutions of Δu + f(u) = 0 on ℝ^N

We consider the semilinear elliptic equation Δu + f(u) = 0, on ℝ^N, where f is of class C¹ and satisfies the conditions f(0) = 0, f′(0) < 0. By a ground state of this equation we mean a positive solution that decays to zero at infinity. Any such solution is necessarily radially symmetric about some point. If N = 1, the ground state, if it exists, is always unique (up to a shift in x), nondegenerate and has Morse index equal to one. We show that none of these statements is valid in general if N ≥ 2.