Abstract

We consider the problem (1) {...-Δu = u p + a;u in B, u > 0 in B, u = 0 on ∂B, where B denotes the unit ball in ℝ N, N ≥ 3, a; > 0 and p > 1. Merle and Peletier showed that for p > N+2/N-2 there is a unique value a; = a; * > 0 such that a radial singular solution exists. This value is the only one at which an unbounded sequence of classical solutions of (1) may accumulate. Here we prove that if additionally p < N-2 N-1/N-2N-1-4 or N ≤ 10, then for a; close to a; *, a large number of classical solutions of (1) exist. In particular infinitely many solutions are present if a; = a; *. We establish a similar assertion for the problem ...-Δu = a;f(u+ 1) in B, u > 0 in B, u = 0 on ∂B, where f(s) = s p + s q, 1 < q < p, and p satisfies the same condition as above. © 2007 American Mathematical Society.