The Existence and Local Uniqueness of Multi-Peak Solutions to a Class of Kirchhoff Type Equations

In this paper, we study the existence and local uniqueness of multi-peak solutions to the Kirchhoff type equations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ - \left({{\varepsilon ^2}a + \varepsilon b\int_{{\mathbb{R}^3}} {|\nabla u{|^2}}} \right)\,\,\Delta u + V(x)u = {u^p},\,\,\,\,\,\,u > 0\,\,\,\,\,{\rm{in}}\,\,\,{\mathbb{R}^3},$$\end{document} which concentrate at non-degenerate critical points of the potential function V(x), where a, b > 0, 1 < p < 5 are constants, and ε > 0 is a parameter. Applying the Lyapunov-Schmidt reduction method and a local Pohozaev type identity, we establish the existence and local uniqueness results of multi-peak solutions, which concentrate at {ai}1≤i≤k, where {ai}1≤i≤k are non-degenerate critical points of V(x) as ε → 0.