Three-term asymptotics for the boundary layers of semilinear elliptic eigenvalue problems

We consider the nonlinear eigenvalue problem −Δu=λ f(u) in Ω u=0 on ∂Ω, where Ω is a ball or an annulus in R^N (N ≥ 2) and λ > 0 is a parameter. It is known that if λ >> 1, then the corresponding positive solution u_λ develops boundary layers under some conditions on f. We establish the asymptotic formulas for the slope of the boundary layers of u_λ with the exact second term and the ‘optimal’ estimate of the third term.