Multi-bump solutions to Carrier's problem (Q1868774)

The author deals with the following singularly perturbed BVP \[ \varepsilon ^2y''=1-y^2-2b(1-x^2)y, \quad y(-1)=y(1)=0, \] where \(b\) is a constant and \(\varepsilon\) is a small positive parameter. The problem was posed by Carrier. The purpose of the paper is to use shooting arguments and techniques to study the existence of solutions to (1) and the locations where the solutions have sharp changes as \(\varepsilon\to 0\) when \(b\neq 0\). In particular, the existence of multi-spike solutions is proved. It is also shown that the locations of interior spikes are allowed to be near the boundaries \(x=\pm 1.\)