Asymptotic Formulas for Boundary Layers and Eigencurves for Nonlinear Elliptic Eigenvalue Problems
Tetsutaro Shibata · Communications in Partial Differential Equations · 2003
Nonlinear elliptic eigenvalue problem −Δu + up = λuin Ω u > 0in Ω, u = 0on ∂Ωis studied, where (N ≥ 2) is a bounded appropriately smooth domain, p > 1is a constant and λ > 0is an eigenvalue parameter. We establish the precise asymptotic formulas for the boundary layers of uλ as λ→∞. Moreover, we establish the precise asymptotic formulas for eigencurve λ(α)(associated with eigenfunction uα with ) as α→ ∞. In particular, the exact second term of “the width of the boundary layers” () (λ →∞) and λ(α)(α →∞) are obtained when Ωare a ball and an annulus.