In the present paper, we consider the following Kirchhoff type problem $$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$where b > 0, p ∈ (4, 6), the potential $V\in C(\mathbb R^3,\mathbb R)$ and ɛ is a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e., $\inf _{\mathbb R^N} V=0$. As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.

中文翻译：

---

具有临界频率的基尔霍夫型问题的半经典解

在本文中，我们考虑以下基尔霍夫类型问题$$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p- 2}v \quad {\rm in}\ \mathbb{R}^3, $$在哪里b> 0,p∈ (4, 6)，势$V\in C(\mathbb R^3,\mathbb R)$和ε是一个正参数。对于具有几类临界频率势的问题，通过变分方法获得了半经典状态解的存在性和多重性，即$\inf _{\mathbb R^N} V=0$. 对于基尔霍夫型问题，文献中对临界频率情况的研究很少，尤其是在无穷大时可能会消失。