UTMS 2013 – 6 May 15 , 2013 Structure of the positive radial solutions for the supercritical

We are interested in the structure of the positive radial solutions of the supercritical Neumann problem in a unit ball  ε ( Urr + N−1 r Ur ) − U + U = 0, 0 < r < 1, Ur(1) = 0, U > 0, 0 < r < 1, where N is the spatial dimension and p > pS := (N + 2)/(N − 2), N ≥ 3. We show that there exists a sequence {εn}n=1 (ε1 > ε2 > · · · → 0) such that this problem has infinitely many singular solutions {(εn, U∗ n)}n=1 ⊂ R × (C(0, 1) ∩ C(0, 1]) and that the nonconstant regular solutions consist of infinitely many smooth curves in the (ε, U(0))-plane. It is shown that each curve blows up at εn and if pS < p < pJL, then each curve has infinitely many turning points around εn. Here, pJL := { 1 + 4 N−4−2 √ N−1 (N ≥ 11), ∞ (2 ≤ N ≤ 10). In particular, the problem has infinitely many solutions if ε ∈ {εn}n=1. We also show that there exists ε̄ > 0 such that the problem has no nonconstant regular solution if ε > ε̄. The main technical tool is the intersection number between the regular and singular solutions.