On vs. local minimizers for a critical functional related to fractional p-Laplacian

Abstract

Let , is a bounded domain with boundary such that and where is the fractional critical Sobolev exponent. Let be a Carathéodory function with and there exists satisfying if , otherwise, such that for all and for some Consider the associated functional defined as where Theorem 1.1 proves that if is a local minimum of I in the -topology, then it is also a local minimum in -topology. This result is useful for proving multiple solutions to the associated Euler–Lagrange equation (P) defined below. Theorem 1.1 given in the present paper can be also extended to more general quasilinear elliptic equations.

Keywords:

• Fractional p-Laplace operator
• elliptic equations
• variational methods
• local minimizers

AMS Subject Classifications:

• 35J65
• 35J20
• 35J70

Acknowledgements

I want to thank the anonymous referees for their carefully reading this paper and their useful comments.

Notes

No potential conflict of interest was reported by the author.