Abstract
We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical exponent problem
(
-
Δ
)
s
u
s
=
|
u
s
|
2
s
⋆
-
2
u
s
,
u
s
∈
D
0
s
(
Ω
)
,
2
s
⋆
:=
2
N
N
-
2
s
,
(-\Delta)^{s}u_{s}=\lvert u_{s}\rvert^{2_{s}^{\star}-2}u_{s},\quad u_{s}\in D^%
{s}_{0}(\Omega),\,2^{\star}_{s}:=\frac{2N}{N-2s},
where s is any positive number, Ω is either
ℝ
N
{\mathbb{R}^{N}}
or a smooth symmetric bounded domain, and
D
0
s
(
Ω
)
{D^{s}_{0}(\Omega)}
is the homogeneous Sobolev space. Depending on the kind of symmetry considered, solutions can be sign-changing. We show that, up to a subsequence, a l.e.s.s.
u
s
{u_{s}}
converges to a l.e.s.s.
u
t
{u_{t}}
as s goes to any
t
>
0
{t>0}
. In bounded domains, this convergence can be characterized in terms of an homogeneous fractional norm of order
t
-
ε
{t-\varepsilon}
. A similar characterization is no longer possible in unbounded domains due to scaling invariance and an incompatibility with the functional spaces; to circumvent these difficulties, we use a suitable rescaling and characterize the convergence via cut-off functions. If t is an integer, then these results describe in a precise way the nonlocal-to-local transition. Finally, we also include a nonexistence result of nontrivial nonnegative solutions in a ball for any
s
>
1
{s>1}
.
What sticks? Ephemerality, permanence and local transition pathways