Precise Spectral Asymptotics for the Dirichlet Problem −u t + g u t = λ sin u t

We consider the nonlinear eigenvalue problem on an interval −u t + g u t = λ sin u t u t > 0 t ∈ I = −T T u ±T = 0 where λ > 0 is a parameter and T > 0 is a constant. It is known that if λ 1, then the corresponding solution has boundary layers. In this paper, we characterize λ by the boundary layers of the solution when λ 1from a variational point of view. To this end, we parameterize a solution pair λ u by a new parameter 0 < < T, which characterizes the boundary layers of the solution, and establish precise asymptotic formulas for λ with exact second term as → 0. It turns out that the second term is a constant which is explicitly determined by the nonlinearity g.