Asymptotic formulas and critical exponents for two-parameter nonlinear eigenvalue problems

Abstract

We study the nonlinear two-parameter problem −u″(x)+λu(x)q=μu(x)p, u(x)>0, x∈(0,1), u(0)=u(1)=0. Here, 1<q<p are constants and λ,μ>0 are parameters. We establish precise asymptotic formulas with exact second term for variational eigencurve μ(λ) as λ→∞. We emphasize that the critical case concerning the decaying rate of the second term is p=(3q−1)/2 and this kind of criticality is new for two-parameter problems.