Precise asymptotic formulas for variational eigencurves of semilinear two-parameter elliptic eigenvalue problems

We consider the two-parameter nonlinear eigenvalue problem¶−Δu = μu − λ(u + up + f(u)),  u > 0 in Ω,  u = 0 on ∂Ω,¶where p>1 is a constant and μ,λ>0 are parameters. We establish the asymptotic formulas for the variational eigencurves λ=λ(μ,α) as μ→∞, where α>0 is a normalizing parameter. We emphasize that the critical case from a viewpoint of the two-term asymptotics of the eigencurve is p=3. Moreover, it is shown that p=5/3 is also a critical exponent from a view point of the three-term asymptotics when Ω is a ball or an annulus. This sort of criticality for two-parameter problems seems to be new.