Spectral asymptotics and bifurcation for nonlinear multiparameter elliptic eigenvalue problems

This paper is concerned with the nonlinear multiparameter elliptic eigenvalue problem u′′(r) + N − 1 r u′(r) + μu(r)− k ∑ i=1 λifi(u(r)) = 0, 0 < r < 1, u(r) > 0, 0 ≤ r < 1, u′(0) = 0, u(1) = 0, where N ≥ 1, k ∈ N and μ, λi ≥ 0 (1 ≤ i ≤ k) are parameters. The aim of this paper is to study the asymptotic properties of eigencurve μ(λ, α) = μ(λ1, λ2, · · · , λk, α) with emphasis on the phenomenon of bifurcation from the first eigenvalue μ1 of −4|D and on gaining a clearer picture of the bifurcation diagram. Here, α > 0 is a normalizing parameter of eigenfunction associated with μ(λ, α). To this end, we shall establish asymptotic formulas of μ(λ, α) as |λ| → ∞, 0. Received by the editors November 1995. Communicated by J. Laubin. 1991 Mathematics Subject Classification : 35P30.