Existence and instability of spike layer solutions to singular perturbation problems (Q1865328)

The authors propose an abstract framework to establish existence and to compute the Morse index of spike layer solutions of singularly perturbed semilinear elliptic equations. A nonlinear Lyapunov-Schmidt scheme is used to reduce the problem to one on a normally hyperbolic manifold and the related linearized problem is also analyzed by this reduction. As an application, it is shown the existence of a multi-peak spike layer solution with peaks on the boundary of the domain. Precise estimates of the small eigenvalues of an operator obtained by linearizing a spike layer solution are also established.