Semi-classical solutions for Kirchhoff type problem with a critical frequency

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.