Abstract:

In this paper, we will discuss the following Kirchhoff equation
$\left\{\begin{array}{ll}-\left(a+b\int_{\mathbb{R}^{3}}{|\nabla u{{|}^{2}}}\right)\triangle u+u=\left(1+\varepsilon g(x)\right) u^{p}, x\in\mathbb{R}^{3}, \\u\in H^{1}\left(\mathbb{R}^{3}\right), \end{array}\right.$
where $\varepsilon$, $a$, $b$ are positive constants, $1＜ p＜5,g(x)\in L^{\infty}\left(\mathbb{R}^{3}\right)$. When $g(x)$ satisfy some conditions, we use perturbation method prove that there exists a $\varepsilon_{0}$, if $0＜\varepsilon ＜\varepsilon_{0}$ there are many solutions for above problem.

Key words: Kirchhoff equations, Multiple solutions, Perturbation method, Nonlinear