In this paper, we study a class of Kirchhoff equation with steep potential well and concave–convex nonlinearities as follows: $\begin{array}{cc}-\left(a+b{\int }_{{\mathbb{R}}^N}{|\nabla u|}^2\mathrm{d}x\right)△u+\lambda V\left(x\right)u=f\left(x\right){\left|u\right|}^{p-2}u+g\left(x\right){\left|u\right|}^{q-2}u\text{in}{\mathbb{R}}^N,\hfill & \hfill \end{array}$where $a,b>0$, $N\ge 3$, $1<q<2<p<min\{4,2^{\ast }\}$, $2^{\ast }=2N∕(N-2)$ and $V\in C({\mathbb{R}}^N,\mathbb{R})$ is a steep potential well. Such problem has been rarely studied for the case $2<p<min\{4,2^{\ast }\}$ since the appearance of the term $\left({\int }_{{\mathbb{R}}^N}{|\nabla u|}^2\mathrm{d}x\right)△u$. By combining the Ekeland variational principle and the filtration of Nehari manifold, we prove the multiplicity of positive solutions for the above problem when $b$ is sufficiently small and $\lambda$ is large enough. Recent results from the literature are extensively improved and extended.

在本文中，我们研究了一类具有陡峭势阱和凹凸非线性的Kirchhoff方程，如下所示： $\begin{array}{cc}-\left(\mathrm{一种}+b{\int }_{{\mathbb{[R}}^{ñ}}{|\nabla ü|}^{\mathrm{2个}}\mathrm{d}X\right)△ü+\lambda \mathrm{伏特}（X）ü=F（X）{\left|ü\right|}^{p-\mathrm{2个}}ü+G（X）{\left|ü\right|}^{q-\mathrm{2个}}ü\text{在}{\mathbb{[R}}^{ñ}，\hfill & \hfill \end{array}$在哪里 $\mathrm{一种}，b>0$， $ñ\ge 3$， $\mathrm{1个}<q<\mathrm{2个}<p<分\{4，{\mathrm{2个}}^{\ast }\}$， ${\mathrm{2个}}^{\ast }=\mathrm{2个}ñ∕（ñ-\mathrm{2个}）$ 和 $\mathrm{伏特}\in C（{\mathbb{[R}}^{ñ}，\mathbb{[R}）$潜力巨大。这种情况很少针对这种情况进行研究$\mathrm{2个}<p<分\{4，{\mathrm{2个}}^{\ast }\}$ 自该词出现以来 $\left({\int }_{{\mathbb{[R}}^{ñ}}{|\nabla ü|}^{\mathrm{2个}}\mathrm{d}X\right)△ü$。通过结合Ekeland变分原理和Nehari流形的过滤，我们证明了上述问题时正解的多重性。$b$ 足够小 $\lambda$足够大。文献的最新结果得到了广泛的改进和扩展。