A singularly perturbed Kirchhoff problem revisited

In this paper, we revisit the singularly perturbation problem -(epsilon(2)a + epsilon b integral(R3) vertical bar del u vertical bar(2)) Delta u + V(x)u = vertical bar u vertical bar(p-1)u in R-3, (0.1) where a, b, epsilon > 0, 1 < p < 5 are constants and V is a potential function. First we establish the uniqueness and nondegeneracy of positive solutions to the limiting Kirchhoff problem - (a + b integral(R3) vertical bar del u vertical bar(2)) Delta u + u = vertical bar u vertical bar(p-1)u in R-3. Then, combining this nondegeneracy result and Lyapunov-Schmidt reduction method, we derive the existence of solutions to (0.1) for epsilon > 0 sufficiently small. Finally, we establish a local uniqueness result for such derived solutions using this nondegeneracy result and a type of local Pohozaev identity. (C) 2019 Elsevier Inc. All rights reserved.