Abstract:

In this paper, we consider the following Kirchhoff type problem (3.36) $ \begin{equation}\left\{ \begin{array}{l}-\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2{\rm d}x\Big)\triangle u+V(x)u=|u|^{p-2}u+\varepsilon|u|^4u, \, \, \, x\in\mathbb{R}^3, \\ u\in H^{1}(\mathbb{R}^3), \end{array} \right. \end{equation}$ where $a>0$, $b>0$, $4< p < 6$ and $V (x)\in L_{\rm loc}^{\frac{3}{2}}(\mathbb{R}^3)$ is a given nonnegative function such that $\lim\limits_{|x|\rightarrow\infty}V(x): =V_{\infty}$. Under suitable conditions on $V(x)$, we prove that the existence of ground state solutions for small $\varepsilon$.

Key words: Kirchhoff type problem, Critical nonlinearity, Ground state