The third solution for a Kirchhoff-type problem with a critical exponent

The aim of this paper is to analyze the multiplicity of solutions for a nonhomogeneous Kirchhoff-type problem on R4 with a critical exponent. The first two positive solutions are deduced by using the variational method, and the third solution is found with the help of Brezis-Lieb's lemma and Mazur's lemma. Moreover, we obtain a new compactness condition and improve on the existing results in the literature.