A singularly perturbed Kirchhoff problem revisited

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In this paper, we revisit the singularly perturbation problem(0.1)−(ϵ2a+ϵb∫R3|∇u|2)Δu+V(x)u=|u|p−1uin R3, where a,b,ϵ>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∫R3|∇u|2)Δu+u=|u|p−1uin R3. Then, combining this nondegeneracy result and Lyapunov-Schmidt reduction method, we derive the existence of solutions to (0.1) for ϵ>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.