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.
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.