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

Abstract

AbstractIn the present paper, we consider the following Kirchhoff type problem$$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$whereb> 0,p∈ (4, 6), the potential$V\in C(\mathbb R^3,\mathbb R)$andɛis 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.