Existence and Asymptotic-Behavior of Nodal Solutions for Semilinear Elliptic-Equations

The elliptic equation Δu + ƒ(u) = 0 in Rn is discussed in the case where ƒ(u) = up−1u (u ≥ 1), = uq−1u (u < 1), 1 < p < (n + 2)/(n − 2) < q, and n ≥ 3. It is shown that any radially symmetric solution behaves, as x → ∞, like one of three types: (i) c x−(n−2) for some c ≠ 0, (ii) c*x−2/(q − 1), (iii) −c* x−2/(q−1) where c* = [2(q − 1)−1 (n − 2 − 2/(q − 1))]1/(q−1) > 0. It is further proved that for any k ≥ 0 there exist at least three radially symmetric solutions which have exactly k zeros in Ihe interval 0 ≤ x < ∞ and which behave like (i), (ii), and (iii), respectively.