Research article Special Issues

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

  • Received: 09 November 2023 Revised: 22 January 2024 Accepted: 05 August 2024 Published: 19 August 2024
  • 35J60, 35B33

  • 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 $.

    Citation: Qi Li, Yuzhu Han, Bin Guo. A critical Kirchhoff problem with a logarithmic type perturbation in high dimension[J]. Communications in Analysis and Mechanics, 2024, 16(3): 578-598. doi: 10.3934/cam.2024027

    Related Papers:

  • 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 $.



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