Abstract
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.
Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent
