Multiple positive solutions for an elliptic problem involving a critical Sobolev exponent

Abstract

In this work, we study the multiple results of positive solutions for a class of elliptic problem involving a critical Sobolev exponent $-\mathrm{△}u+\lambda V(x)u=\mu f(x)u^{q-1}+g(x)u^{2^{\ast }-1}\mathrm{in}{\mathbb{R}}^N,$we prove that the problem has at least k distinct positive solutions. The methods used here are based on combining minimax methods, Nehari manifold and Ekeland's variational principle.

Keywords:

• Multiple positive solutions
• critical Sobolev exponent
• Nehari manifold
• minimax methods
• Ekeland's variational principle
• elliptic equations

2010 Mathematics Subject Classifications:

• 35A15
• 35B33
• 35J35

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the NSFC [grant numbers 11571176, 12001274] and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China [grant number 19KJB110014].