Abstract: Let s∈(0,1), N>2s and
Ds,2(RN):={u∈L2s∗(RN):‖u‖Ds,2(RN):=(CN,s2∬R2N|u(x)−u(y)|2|x−y|N+2sdxdy)12<∞},
where 2s∗:=2NN−2s is the fractional critical exponent and CN,s is a positive constant. We consider functionals J:Ds,2(RN)→R of the type
J(u):=12‖u‖Ds,2(RN)2−∫RNb(x)G(u)dx,
where G(t):=∫0tg(τ)dτ, g:R→R is a continuous function with subcritical growth at infinity, and b:RN→R is a suitable weight function. We prove that a local minimizer of J in the topology of the subspace
Vs:={u∈Ds,2(RN):u∈C(RN) with supx∈RN(1+|x|N−2s)|u(x)|<∞}
must be a local minimizer of J in the Ds,2(RN)-topology.
