2015/03/27 by Nassif Ghoussoub, Ghoussoub, Nassif, Shaya Shakerian +1 · 1 citation
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems
paper · pdf · doi:10.48550/arxiv.1503.08193
We consider the problem of attainability of the best constant in the\nfollowing critical fractional Hardy-Sobolev inequality: n
mu
gamma,s(
Rn):=
inf
limits_u
in H^
frac
alpha2 (
Rn)
setminus\n
0
frac
int
Rn |(-
Delta)^
frac
alpha4u|2 dx -
gamma\n
int
Rn
frac|u|2|x|
alphadx (
int
Rn\n
frac|u|^2
alpha^*(s)|x|sdx)^
frac22
alpha^*(s),\n where 0\≤ s<\α<2, n>\α,\n2\α^*(s):= frac2(n-s)n-\α, and \γ \∈ \ℝ.\nThis allows us to establish the existence of nontrivial weak solutions for the\nfollowing doubly critical problem on Rn,\n \
left
\(-\n
Delta)^
frac
alpha2u-
gamma
fracu|x|
alpha · amp;=\n|u|^2
alpha^*-2 u +
frac|u|^2
alpha^*(s)-2u|x|s · amp;
textin \n
Rn
hfill u · amp; · gt;0 · amp;
textin
Rn,
. nwhere 2\α^*:= frac2 nn-\α is the critical\n\α-fractional Sobolev exponent, and \γ < \γH:=2^\α\n\(\Γ2(\(n+\α)/(4)))/(\Γ2(\(n-\α)/(4))), the latter\nbeing the best fractional Hardy constant on Rn.\n