An estimate on the eigenvalues in bifurcation for gradient mappings

This paper provides supplementary information on the Krasnoselʹskiĭ bifurcation theorem for gradient mappings ∇φ(u)=λu in a real Hilbert space. Strengthening the differentiability assumption, we prove a better estimate of the form λ r =λ 0 +O(r p−1 ) for the bifurcating solutions (u r ,λ r ) on the sphere S r . Applications are given to nonlinear elliptic eigenvalue problems of the form −Δu=μ(u+f(x,u)), x∈Ω, u=0, x∈∂Ω .