Yes there are infinite Morse index solutions for $N\geq 10$. See remark 1(i) in the second paper you reference.
[...] for every N ≥ 10 the equation (1.1) possesses a radial stable solution. The existence of such a solution is a consequence of the analysis performed in [12], as was remarked in [6].
(For your second request you should be a bit more specific exactly what you are interested in.)