The Existence and Local Uniqueness of Multi-Peak Positive Solutions to a Class of Kirchhoff Equation

Abstract

In the present paper, we consider the nonlocal Kirchhoff problem $ - \left( {{\varepsilon ^2}a + \varepsilon b\int_{{\mathbb{R}^3}} {|\nabla u{|^2}} } \right)\Delta u + u = Q(x){u^p},u > 0\;\text{in}\;{\mathbb{R}^3}$, where a, b p > 5} and ǫ < 0 is a parameter. Under some assumptions on Q(x), we show the existence and local uniqueness of positive multi-peak solutions by Lyapunov-Schmidt reduction method and the local Pohozaev identity method, respectly.