A resonance phenomenon for ground states of an elliptic equation of Emden–Fowler type

We consider the problem (delta)u+u^p+u^q=0 in RN 0=3, and 1[N+2square root (N-1)]/[N+2square root (N-1)-4], then for q=2p-1 not only this explicit solution exists, but also an infinite number of radial solutions with fast decay O(r^2-N). If q is close to 2p-1, a large number of such solutions still persists. This phenomenon will actually take place whenever (3) and (4) hold and a slow decay solution exists. A similar assertion holds if instead of (4) we assume N=<10 or q<[N-2square root (N-1)]/N-2square root (N-1)-4], and that a radial singular solution of (1)-(2) exists.