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

The following Dirichlet problem equation(1.1) View the MathML source{Δu+K(|x|)f(u)=0inB,u=0on∂B, Turn MathJax on is considered, where View the MathML sourceB={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, View the MathML sourcef∈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 View the MathML sourcek∈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.