D s , 2 ( R N ) versus C ( R N ) local minimizers

Abstract

Let s ∈ ( 0 , 1 ), N > 2 s and D s , 2 ( R N ) : = { u ∈ L 2 s ∗ ( R N ) : ‖ u ‖ D s , 2 ( R N ) : = ( C N , s 2 ∬ R 2 N | u ( x ) − u ( y ) | 2 | x − y | N + 2 s d x d y ) 1 2 < ∞ } , where 2 s ∗ : = 2 N N − 2 s is the fractional critical exponent and C N , s is a positive constant. We consider functionals J : D s , 2 ( R N ) → R of the type J ( u ) : = 1 2 ‖ u ‖ D s , 2 ( R N ) 2 − ∫ R N b ( x ) G ( u ) d x , where G ( t ) : = ∫ 0 t g ( τ ) d τ, g : R → R is a continuous function with subcritical growth at infinity, and b : R N → R is a suitable weight function. We prove that a local minimizer of J in the topology of the subspace V s : = { u ∈ D s , 2 ( R N ) : u ∈ C ( R N )  with  sup x ∈ R N ( 1 + | x | N − 2 s ) | u ( x ) | < ∞ } must be a local minimizer of J in the D s , 2 ( R N )-topology.