In this paper, we discuss the existence of multiple solutions of the generalized quasilinear Schrödinger equation of Kirchhoff-type $\begin{array}{rl}& (1+b{\int }_{{\mathbb{R}}^3}{g}^2(u)|\mathrm{\nabla }u{|}^2\mathrm{d}x)[-\mathrm{div}(g^2(u)\mathrm{\nabla }u)+g(u)g^{\prime }(u)|\mathrm{\nabla }u{|}^2]\\ & +V(x)u=f(x,u),x\in {\mathbb{R}}^3\end{array}$ where b>0 is a parameter, $g\in C^1(\mathbb{R},{\mathbb{R}}^{+})$. In preceding approaches, the order of the nonlinear term $f(x,t)$ about t at infinity is usually supposed to be higher than $|t{|}^3$. But in this paper, this condition is not necessary. Under weaker condition ${\int }_0^{t}f(x,s)\mathrm{d}s\ge \mu |G(t){|}^r-\theta |G(t){|}^{r-1}$, where $G(t)$ is the primitive function of $g(t)$, $2<r<6$, we get the existence of infinitely many solutions of the equation. Our work can be regarded as the extension of those such as [J. Zhang et al. Existence of multiple solutions of Kirchhoff type equation with sign-changing potential. Appl Math Comput. 2014;242:491–499] and related works.

在本文中，我们讨论了基尔霍夫型广义拟线性薛定谔方程的多重解的存在性$\begin{array}{rl}& (1+b{\int }_{{\mathbb{R}}^3}{G}^2(你)|\mathrm{\nabla }你{|}^2\mathrm{d}X)[-\mathrm{div}(G^2(你)\mathrm{\nabla }你)+G(你)G^{\text{'}}(你)|\mathrm{\nabla }你{|}^2]\\ & +五(X)你=F(X,你),X\in {\mathbb{R}}^3\end{array}$其中b > 0 是一个参数，$G\in C^1(\mathbb{R},{\mathbb{R}}^{+})$. 在前面的方法中，非线性项的阶$F(X,吨)$大约在无穷远处的t通常应该高于$|吨{|}^3$. 但在本文中，这个条件不是必须的。在较弱的条件下${\int }_0^{吨}F(X,s)\mathrm{d}s\ge \mu |G(吨){|}^r-\theta |G(吨){|}^{r-1}$， 在哪里$G(吨)$是的原始函数$G(吨)$,$2<r<6$，我们得到方程的无限多解的存在性。我们的工作可以看作是诸如[J. 张等人。具有变符号势的基尔霍夫型方程的多重解的存在性。应用数学计算。2014;242:491–499] 和相关著作。