Uniqueness and nonuniqueness of nodal radial solutions of sublinear elliptic equations in a ball

The following Dirichlet problem equation(1.1){Δu+K(|x|)f(u)=0inB,u=0on∂B, is considered, where B={x∈RN:|x|<1}, N≥2N≥2, K∈C2[0,1]K∈C2[0,1] and K(r)>0K(r)>0 for 0≤r≤10≤r≤1, f∈C1(R), sf(s)>0sf(s)>0 for s≠0s≠0. Assume moreover that ff satisfies the following sublinear condition: f(s)/s>f′(s)f(s)/s>f′(s) for s≠0s≠0. A sufficient condition is derived for the uniqueness of radial solutions of (1.1) possessing exactly k−1k−1 nodes, where k∈N. It is also shown that there exists K∈C∞[0,1]K∈C∞[0,1] such that (1.1) has three radial solutions having exactly one node in the case N=3N=3.