Existence of critical points for noncoercive functionals with critical Sobolev exponent

Abstract

We consider the following noncoercive functional with the critical Sobolev exponent $I(u)=\frac{1}{2}{\int }_{\mathrm{\Omega }}\frac{1}{(1+|u|{)}^{2\alpha }}|\mathrm{\nabla }u{|}^2\mathrm{d}x-\frac{\lambda }{q}{\int }_{\mathrm{\Omega }}|u{|}^q\mathrm{d}x-\frac{1}{2^{\ast }(1-\alpha )}{\int }_{\mathrm{\Omega }}|u{|}^{2^{\ast }(1-\alpha )}\mathrm{d}x,$where $\lambda >0$, $0<\alpha <(N+2)/2N$, $2<q<2^{\ast }(1-\alpha )$, $2^{\ast }=2N/(N-2)$. We prove the existence of a nontrivial critical point via the mountain pass theorem and the multiplicity via Clark's theorem.

COMMUNICATED BY:

• Xingfu Zou

Keywords:

• Critical points
• noncoercive functional
• critical Sobolev exponent
• elliptic equations

2010 Mathematics Subject Classifications:

• 35J60
• 35J65

Acknowledgments

The authors thank the referees for their careful reading and valuable comments.

Disclosure statement

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

Additional information

Funding

Supported by the National Natural Science Foundation of China (11871478), Hunan Provincial Natural Science Foundation (2019JJ40355) and Degree and Graduate Education Reform Research Project of Hunan Province (2020JGYB031).