Geometry of Phase Space and Solutions of Semilinear Elliptic Equations in a Ball

We consider the problem (1) ⎧⎨ ⎩ −∆u = up + λu in B , u > 0 in B , u = 0 on ∂B, where B denotes the unit ball in RN , N ≥ 3, λ > 0 and p > 1. Merle and Peletier showed that for p > N+2 N−2 there is a unique value λ = λ∗ > 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 − 2 √ N − 1− 4 or N ≤ 10 , then for λ close to λ∗, a large number of classical solutions of (1) exist. In particular infinitely many solutions are present if λ = λ∗. We establish a similar assertion for the problem ⎧⎨ ⎩ −∆u = λ f(u+ 1) in B , u > 0 in B , u = 0 on ∂B , where f(s) = sp + sq , 1 < q < p, and p satisfies the same condition as above.