Asymptotic formulas for boundary layers and eigencurves for nonlinear elliptic eigenvalue problems

Nonlinear elliptic eigenvalue problem -Deltau + u(p) = lambdau in Omega, u > 0 in Omega, u = 0 on partial derivativeOmega is studied, where Omega subset of R-N (N greater than or equal to 2) is a bounded appropriately smooth domain, p > 1 is a constant and lambda > 0 is an eigenvalue parameter. We establish the precise asymptotic formulas for the boundary layers of u(lambda) as lambda --> infinity. Moreover, we establish the precise asymptotic formulas for eigencurve lambda(alpha) (associated with eigenfunction u(alpha) with parallel tou(alpha)parallel to(2) = alpha) as alpha --> infinity. In particular, the exact second term of "the width of the boundary layers" (partial derivativeu(lambda)/partial derivativev)(z) (z is an element of partial derivativeOmega)(lambda --> infinity) and lambda(alpha) (alpha --> infinity) are obtained when Omega are a ball and an annulus.