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• W2911892502 abstract "In the present paper, we establish the uniqueness and nondegeneracy of positive energy solutions to the Kirchhoff equation begin{eqnarray*} -left(a+bint_{mathbb{R}^{3}}|nabla u|^{2}right)Delta u+u=|u|^{p-1}u & & text{in }mathbb{R}^{3}, end{eqnarray*} where $a,b>0$, $1<p<5$ are constants. Then, as applications, we derive the existence and local uniqueness of solutions to the perturbed Kirchhoff problem begin{eqnarray*} -left(epsilon^2a+epsilon bint_{mathbb{R}^{3}}|nabla u|^{2}right)Delta u+V(x)u=|u|^{p-1}u & & text{in }mathbb{R}^{3} end{eqnarray*} for $epsilon>0$ sufficiently small, under some mild assumptions on the potential function $V:mathbb{R}^3to mathbb{R}$. The existence result is obtained by applying the Lyapunov-Schmidt reduction method. It seems to be the first time to study singularly perturbed Kirchhoff problems by reduction method, as all the previous results were obtained by various variational methods. Another advantage of this approach is that it gives a unified proof to the perturbation problem for all $pin (1,5)$, which is quite different from using variational methods in the literature. The local uniqueness result is totally new. It is obtained by using a type of local Pohozaev identity, which is developed quite recently by Deng, Lin and Yan in their work On the prescribed scalar curvature problem in $mathbb{R}^N$, local uniqueness and periodicity. (see J. Math. Pures Appl. (9) {bf 104}(2015), 1013-1044)." @default.
• W2911892502 created "2019-02-21" @default.
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• W2911892502 date "2017-03-16" @default.
• W2911892502 modified "2023-09-23" @default.
• W2911892502 title "Uniqueness and Nondegeneracy of positive solutions to Kirchhoff equations and its applications in singular perturbation problems" @default.
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