Existence and multiplicity of positive solutions of a critical Kirchhoff type elliptic problem in dimension four

In this paper, we study a Kirchhoff type elliptic problem, \[ \begin{cases} \displaystyle - \Big ( 1+\alpha \int_{\Omega}|\nabla u|^2dx \Big ) \Delta u =\lambda u^q+u^3,\ u > 0 \text{ in }\Omega,\\ u=0\text{ on }\partial \Omega, \end{cases} \] where $\Omega\subset \mathbb{R}^4$ is a bounded domain with smooth boundary $\partial \Omega$ and we assume $\alpha,\lambda > 0$ and $1\le q < 3$. For $q=1$, we prove the existence of possibly multiple solutions for $\alpha > 0$ and $\lambda\ge \lambda_1$ in suitable intervals, where $\lambda_1 > 0$ is the first eigenvalue of $-\Delta$ on $\Omega$. On the other hand for $q\in (1,3)$, we show the existence of a solution for all small $\alpha > 0$ and all $\lambda > 0$. We establish the results by the method of the Nehari manifold and the concentration compactness analysis for Palais-Smale sequences.