A critical Kirchhoff equation with a logarithmic type perturbation in dimension 4

Abstract

In this paper, a critical Kirchhoff type elliptic equation involving a logarithmic type perturbation is considered in a bounded domain in . Three main features of the problem bring essential difficulties when proving the existence of weak solutions. The first one is the nonlocal term which makes the structure of the corresponding energy functional more complicated, the second one is that the problem is critical in the sense that the energy functional lacks compactness, and the third one is the appearance of the logarithmic term which satisfies neither the standard monotonicity condition nor the Ambrosetti–Rabinowitz condition. Moreover, the boundedness of the sequence is hard to obtain for Kirchhoff problems in dimension 4. By combining a result by Jeanjean (Proc. Roy. Soc. Edinburgh Sect. A, 1999, 129: 787–809) and a recent estimate by Deng et al. (Adv. Nonlinear Stud., 2023, 23: No. 20220049) with the mountain pass lemma and Brézis‐Lieb's lemma, it is proved that either the norm of the sequence of approximation solution goes to infinity or the problem admits a nontrivial weak solution, under appropriate assumptions on the parameters.