The Brezis-Nirenberg result for the Kirchhoff-type equation in dimension four

Abstract

In this article, we study the Brezis-Nirenberg result for Kirchhoff-type equation in dimension four. Using the variational method, the existence of positive solutions and the multiplicity of solutions are obtained. It is worth to point out that a global Palais-Smale compactness condition for the critical problem is obtained. Thanks to this global Palais-Smale compactness condition, k pairs of distinct solutions are obtained, where .

Keywords:

• Kirchhoff-type equation
• critical exponent
• positive solutions
• multiplicity
• variational method

AMS Subject Classifications:

• 35A15
• 35B09
• 35B33

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by National Science Foundation of China [grant number 11471267]; Science and Technology Foundation of Guizhou Province [grant number LH[2016]7033]; the Fundamental Research Funds for the Central Universities [grant number XDJK2014C161].