Multiplicity of positive solutions for Kirchhoff type problems in $\mathbb R^3$

Abstract

We are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem: \begin{equation*} \begin{cases} \displaystyle - \bigg(\epsilon ^2a + \epsilon b\int_{{\mathbb{R}^3}}{|\nabla u{|^2}} dx \bigg)\Delta u + u = Q(x)|u|^{p-2}u,& x\in \mathbb{R}^3, \hfill \\ u \in H^1(\mathbb{R}^3), \quad u > 0, & x\in \mathbb{R}^3 , \end{cases} \end{equation*} where $\epsilon> 0 $ is a parameter, $a, b> 0$ are constants, $p\in (2, 6)$, and $Q\in C(\mathbb{R}^3)$ is a nonnegative function. We show how the profile of $Q$ affects the number of positive solutions when $\epsilon $ is sufficiently small.