In this paper, a Kirchhoff problem with critical nonlinearity in a smooth bounded domain of \(\mathbb {R}^4\) , subject to Dirichlet boundary conditions is investigated. In dimension four, the Sobolev critical exponent is \(2^* = 4\) , which brings a significant challenge for studying the problem from a variational perspective. By using of the method of truncation, the Mountain Pass Lemma and the second concentration compactness principle, the multiplicity of solutions for the critical Kirchhoff problem in \(\mathbb {R}^4\) is derived. The result obtained in this paper not only complement but also improve upon earlier findings in the literature.
