Abstract

We consider the problem ?u + up + uq = 0 in ?N, (1) 0<u(x) ? 0 as x ? + ?, (2) where N ? 3, and 1 <p<N+2/N-2<q. (3) Lin and Ni observed that if one further has q = 2p-1, then there exists an explicit radial solution of (1)-(2) of the form u(r) = (A/B+r2)1/p-1. However, the question of existence in the general range has remained basically open. We prove that if the range of p is further restricted to p > N + 2N - 1/N + 2N - 1 - 4, (4) then for q = 2p-1 not only this explicit solution exists, but also an infinite number of radial solutions with fast decay O(r2-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-2N-1/N-2N-1-4, and that a radial singular solution of (1)-(2) exists. © 2002 Elsevier Inc. All rights reserved.