Minimax Theory and its Applications 07 (2022), No. 2, 207--252
Copyright Heldermann Verlag 2022
Existence and Local Uniqueness of Normalized Multi-Peak Solutions to a Class of Kirchhoff Type Equations
Leilei Cui
Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan, P. R. China
leileicuiccnu@163.com
Gongbao Li
Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan, P. R. China
ligb@ccnu.edu.cn
Peng Luo
Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan, P. R. China
luopeng@whu.edu.cn
Chunhua Wang
Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan, P. R. China
chunhuawang@ccnu.edu.cn
[Abstract-pdf]
We study the existence and local uniqueness of multi-peak solutions to the following Kirchhoff type equations \begin{equation*} -\Big(a+b_{\lambda} \int_{\mathbb R^3}|\nabla u_{\lambda}|^2\Big)\Delta u_{\lambda} +\big(\lambda+V(x)\big)u_{\lambda} = \beta_{\lambda} u_{\lambda}^{p}, \end{equation*} where $u_{\lambda}\in H^{1}(\mathbb{R}^3)$, $u_{\lambda}>0$ in $\mathbb R^3$, with normalized $L^{2}$-constraint, that is, \begin{equation*} \int_{\mathbb{R}^3}u_{\lambda}^2=1, \end{equation*} where $a > 0$, $p\in(1,5)$ are constants, $\lambda, b_{\lambda}, \beta_{\lambda} > 0$ are parameters, and $V(x)\colon \mathbb{R}^3\to\mathbb{R}^1$ is a bounded continuous function. Physicists are very interested in normalized solutions. Compared to finding multi-pick solutions to the equation without normalized $L^{2}$-constraint one is facing here some new difficulties in getting normalized solutions to the equation. We first prove that for the case of $3 < p < 5$, there exist sequences $\{b_{\lambda}\}_{\lambda}$ and $\{\beta_{\lambda}\}_{\lambda}$ such that for any sufficiently large $\lambda > 0$, one can construct multi-peak solutions $u_{\lambda}$ of some given form to the above equation by using the Lyapunov-Schmidt reduction method under some mild assumptions on the function $V(x)$. In the proof of the above existence result, we consider the three cases of $p=11/3$, $3 < p < 11/3$ and $11/3 < p < 5$ separately, which correspond to the cases of mass critical, subcritical and supercritical in physics respectively. Then, applying the blow-up technique and the local Pohozaev identities we obtain a uniqueness result of multi-peak solutions for the case of $3 < p < 5$. The difficulties caused by the nonlocal term and normalized $L^{2}$-constraint are overcome.
Keywords: Kirchhoff type equations, multi-peak normalized solutions, Lyapunov-Schmidt reduction, local Pohozaev identity, existence and local uniqueness.
MSC: 35J20, 35J60, 35J92.
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