Multi-bump solutions to Carrier's problem

Abstract

We study the existence of well-known singularly perturbed BVP problem ε 2 y″=1− y 2−2 b(1− x 2) y, y(−1)= y(1)=0 introduced by G.F. Carrier. In particular, we show that there exist multi-spike solutions, and the locations of interior spikes are clustered near x=0 and are separated by an amount of O( ε|ln ε|), while only single spikes are allowed near the boundaries x=±1.