Uniqueness of single-peaked solutions to singularity perturbed semilinear Dirichlet problems

We are concerned with the uniqueness of single-peaked solutions to the following problem integral-epsilon 2ou + u = Q(x)up-1, in ohm, u > 0, in ohm, u = 0, on partial differential ohm, where epsilon > 0 is a small parameter, and N >= 3, 2 < p < 2N/(N - 2). By local Pohozaev identity and blow-up analysis, we show the uniqueness of single-peaked solutions under certain assumptions on asymptotic behavior of Q(x) and its first derivatives near the critical point. Here the degeneracy of the critical point is also allowed, which gives a partial answer to the question proposed in Cao et al. (1998). (c) 2021 Elsevier Ltd. All rights reserved.