A critical Kirchhoff problem with a logarithmic type perturbation in high dimension

Abstract

In this paper, the following critical Kirchhoff-type elliptic equation involving a logarithmic-type perturbation$ -\Big(a+b\int_{\Omega}|\nabla u|^2\mathrm{d}x\Big)\Delta u = \lambda|u|^{q-2}u\ln |u|^2+\mu|u|^2u $is considered in a bounded domain in $ \mathbb{R}^{4} $. One of the main obstructions one encounters when looking for weak solutions to Kirchhoff problems in high dimensions is that the boundedness of the $ (PS) $ sequence is hard to obtain. By combining a result by Jeanjean [27] with the mountain pass lemma and Brézis–Lieb's lemma, it is proved that either the norm of the sequence of approximation solutions goes to infinity or the problem admits a nontrivial weak solution, under some appropriate assumptions on $ a $, $ b $, $ \lambda $, and $ \mu $.

Keywords

• critical
• kirchhoff equation
• logarithmic perturbation
• high dimension
• mountain pass lemma