Bulletin of the
Korean Mathematical Society BKMS

In this paper, we consider the following Kirchhoff type equation on the whole space $$\left\{\aligned &-(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\triangle u = u^{5} + \lambda k(x)g(u), \ x\in \mathbb{R}^{3},\\ & u\in\mathcal{D}^{1,2}(\mathbb{R}^{3}),\endaligned\right.$$ where $\lambda>0$ is a real number and $k, g$ satisfy some conditions. We mainly investigate the existence of ground state solution via variational method and concentration-compactness principle.

Keywords: Kirchhoff type equation, critical nonlinearity, concentration-compactness principle, ground state solution

MSC numbers: Primary 35J20, 35J60, 58E05

Supported by: This work was financially supported by SSF ZR2021MA096, ZR2020MA005.