Some Existence Results for a Kirchhoff Type Logarithmic Problem in Bounded Domains

The main feature of this paper lies in the fact that it is the first attempt to study the existence about the Kirchhoff type problem involving both logarithmic and critical nonlinearities in bounded domains. Since the logarithmic nonlinearity may change sign in the domain and does not satisfy the Ambrosetti-Rabinowitz condition, we cannot obtain the boundedness of \((PS)\) sequence in a standard way. Moreover, the lack of compactness makes our problem become much complex due to the critical nonlinearity. By introducing some truncation techniques and the Nehari manifold method, we overcome these obstacles and obtain the existence of positive solutions for \(N=3\) and \(N\geq 4\) respectively.