ON THE FRACTIONAL LANE-EMDEN EQUATION
JUAN DÁVILA, LOUIS DUPAIGNE, AND JUNCHENG WEI
Abstract. We classify solutions of finite Morse index of the fractional LaneEmden equation
(−∆)s u = |u|p−1 u in Rn .
1. Introduction
Fix an integer n ≥ 1 and let pS (n) denote the classical Sobolev exponent:

 +∞ if n ≤ 2
pS (n) = n + 2

if n ≥ 3
n−2
A celebrated result of Gidas and Spruck [2020, 20] asserts that there is no positive
solution to the Lane-Emden equation
(1.1)
− ∆u = |u|p−1 u
in Rn ,
whenever p ∈ (1, pS (n)). For p = pS (n), the same equation is known to have
(up to translation and rescaling) a unique positive solution, which is radial and
explicit (see Caffarelli-Gidas-Spruck [44, 4]). Let now pc (n) > pS (n) denote the
Joseph-Lundgren exponent:

+∞ if n ≤ 10

√
pc (n) = (n − 2)2 − 4n + 8 n − 1

if n ≥ 11
(n − 2)(n − 10)
This exponent can be characterized as follows: for p ≥ pS (n), the explicit singular
2
solution us (x) = A|x|− p−1 is unstable if and only if p < pc (n). It was proved by
Farina [1818, 18] that (1.1) has no nontrivial finite Morse index solution whenever
1 < p < pc (n), p 6= pS (n).
Through blow-up analysis, such Liouville-type theorems imply interior regularity
for solutions of a large class of semilinear elliptic equations: they are known to be
equivalent to universal estimates for solutions of
(1.2)
− Lu = f (x, u, ∇u)
in Ω,
where L is a uniformly elliptic operator with smooth coefficients, the nonlinearity
f scales like |u|p−1 u for large values of u, and Ω is an open set of Rn . For precise
statements, see the work of Polacik, Quittner and Souplet [2626, 26] in the subcritical setting, as well as its adaptation to the supercritical case by Farina and two of
the authors [1111, 11].
In the present work, we are interested in understanding whether similar results
hold for equations involving a nonlocal diffusion operator, the simplest of which
1
2
J. DÁVILA, L. DUPAIGNE, AND J. WEI
is perhaps the fractional laplacian. Given s ∈ (0, 1), the fractional version of the
Lane-Emden equation1 reads
(−∆)s u = |u|p−1 u in Rn .
(1.3)
Here we have assumed that u ∈ C 2σ (Rn ), σ > s and
ˆ
|u(y)|
(1.4)
dy < +∞,
(1
+
|y|)n+2s
n
R
so that the fractional laplacian of u
ˆ
s
(−∆) u(x) := An,s
Rn
u(x) − u(y)
dy
|x − y|n+2s
is well-defined (in the principal-value sense) at every point x ∈ Rn . The normalizing
2s−1 Γ( n+2s )
2
constant An,s = 2πn/2 |Γ(−s)|
is of the order of s(1 − s) as s converges to either 0
or 1.
The aforementioned classification results of Gidas-Spruck and Caffarelli-GidasSpruck have been generalized to the fractional setting (see Y. Li [2424, 24] and
Chen-Li-Ou [88, 8]). The corresponding fractional Sobolev exponent is given by

 +∞ if n ≤ 2s
pS (n) = n + 2s

if n > 2s
n − 2s
Our main result is the following Liouville-type theorem for the fractional LaneEmden equation.
Theorem 1.1. Assume that n ≥ 1 and 0 < s < σ < 1. Let u ∈ C 2σ (Rn ) ∩
L1 (Rn , (1 + |y|)n+2s dy) be a solution to (1.3) which is stable outside a compact set
i.e. there exists R0 ≥ 0 such that for every ϕ ∈ Cc1 (Rn \ BR0 ),
ˆ
(1.5)
p
|u|p−1 ϕ2 dx ≤ kϕk2Ḣ s (Rn ) .
Rn
• If 1 < p < pS (n) or if pS (n) < p and
(1.6)
p
Γ( n2 −
s
s
p−1 )Γ(s + p−1 )
s
s
Γ( p−1
)Γ( n−2s
− p−1
)
2
>
2
Γ( n+2s
4 )
n−2s 2 ,
Γ( 4 )
then u ≡ 0;
• If p = pS (n), then u has finite energy i.e.
ˆ
kuk2Ḣ s (Rn ) =
|u|p+1 < +∞.
Rn
If in addition u is stable, then in fact u ≡ 0.
Remark 1. For p > pS (n), the function
(1.7)
2s
us (x) = A|x|− p−1
1Unlike local diffusion operators, local elliptic regularity for equations of the form (1.2) where
this time L is the generator of a general Markov diffusion, cannot be captured from the sole
understanding of the fractional Lane-Emden ´equation. For example, further investigations will be
needed for operators of Lévy symbol ψ(ξ) = S n−1 |ω · ξ|2s µ(dω), where µ is any finite symmetric
measure on the sphere S n−1 .
ON THE FRACTIONAL LANE-EMDEN EQUATION
where
µ
Ap−1 = λ
2s
n − 2s
−
2
p−1
3
¶
and where
(1.8)
λ(α) = 22s
Γ( n+2s+2α
)Γ( n+2s−2α
)
4
4
n−2s−2α
n−2s+2α
Γ(
)Γ(
)
4
4
is a singular solution to (1.3) (see the work by Montenegro and two of the authors
[1212, 12] for the case s = 1/2, and the work by Fall [1616; 16, Lemma 3.1] for the
general case). In virtue of the following Hardy inequality (due to Herbst [2222, 22])
ˆ
φ2
Λn,s
dx ≤ kφk2Ḣ s (Rn )
2s
Rn |x|
with optimal constant given by
Λn,s = 22s
2
Γ( n+2s
4 )
n−2s 2 ,
Γ( 4 )
us is unstable if only if (1.6) holds. Let us also mention that regular radial solutions
in the case s = 1/2 were constructed by Chipot, Chlebik ad Shafrir [99,9]. Recently,
Harada [2121, 21] proved that for s = 1/2, condition (1.6) is the dividing line for
the asymptotic behavior of radial solutions to (1.3), extending thereby the classical
results of Joseph and Lundgren [2323, 23] to the fractional Lane-Emden equation
in the case s = 1/2. A similar technique as in [99, 9] allows us to show that the
condition (1.6) is optimal. Indeed we have:
Theorem 1.2. Assume p > pS (n) and that (1.6) fails. Then there are radial
smooth solutions u > 0 with u(r) → 0 as r → ∞ of (1.3) that are stable.
It is by now standard knowledge that the fractional laplacian can be seen as a
Dirichlet-to-Neumann operator for a degenerate but local diffusion operator in the
higher-dimensional half-space Rn+1
+ :
Theorem 1.3 ([2828, 282525, 2555, 5]). Take s ∈ (0, 1), σ > s and u ∈ C 2σ (Rn ) ∩
L1 (Rn , (1 + |y|)n+2s dy). For X = (x, t) ∈ Rn+1
+ , let
ˆ
ū(X) =
P (X, y)u(y) dy,
Rn
where
and pn,s
P (X, y) = pn,s t2s |X − y|−(n+2s)
´
n+1
is chosen so that Rn P (X, y) dy = 1. Then, ū ∈ C 2 (Rn+1
+ ) ∩ C(R+ ),
t1−2s ∂t ū ∈ C(Rn+1
+ ) and

∇ · (t1−2s ∇ū) = 0



ū = u


1−2s
 − lim t
∂t ū = κs (−∆)s u
t→0
where
(1.9)
κs =
Γ(1 − s)
.
22s−1 Γ(s)
in Rn+1
+ ,
on ∂Rn+1
+ ,
on ∂Rn+1
+ ,
4
J. DÁVILA, L. DUPAIGNE, AND J. WEI
Applying Theorem 1.3 to a solution of the fractional Lane-Emden equation, we
end up with the equation
(
−∇ · (t1−2s ∇ū) = 0
in Rn+1
+
(1.10)
1−2s
p−1
− lim t
∂t ū = κs |ū| ū
on ∂Rn+1
+
t→0
An essential ingredient in the proof of Theorem 1.1 is the following monotonicity
formula
n+1
Theorem 1.4. Take a solution to (1.10) ū ∈ C 2 (Rn+1
+ ) ∩ C(R+ ) such that
n+1
t1−2s ∂t ū ∈ C(Rn+1
+ ). For x0 ∈ ∂R+ , λ > 0, let
E(ū, x0 ; λ) =
à ˆ
!
ˆ
p+1
1
κs
2s p−1
−n
1−2s
2
p+1
λ
t
|∇ū| dx dt −
|ū|
dx
2 Rn+1
p + 1 ∂Rn+1
∩B n+1 (x0 ,λ)
∩B n+1 (x0 ,λ)
+
+
ˆ
p+1
s
+ λ2s p−1 −n−1
t1−2s ū2 dσ.
p + 1 ∂B n+1 (x0 ,λ)∩Rn+1
+
Then, E is a nondecreasing function of λ. Furthermore,
µ
¶2
ˆ
p+1
dE
∂ ū
2s ū
2s p−1
−n+1
1−2s
=λ
t
+
dσ
dλ
∂r
p−1r
∂B n+1 (x0 ,λ)∩Rn+1
+
Remark 2. In the above, B n+1 (x0 , λ) denotes the euclidean ball in Rn+1 centered at
x0 of radius λ, σ the n-dimensional Hausdorff measure restricted to ∂B n+1 (x0 , λ),
X
r = |X| the euclidean norm of a point X = (x, t) ∈ Rn+1
+ , and ∂r = ∇ · r the
corresponding radial derivative.
An analogous monotonicity formula has been derived by Fall and Felli [1717, 17]
to obtain unique continuation results for fractional equations. Previously, Caffarelli
and Silvestre derived an Almgren quotient formula for the fractional laplacian in
[55, 5] and Caffarelli, Roquejoffre and Savin [66, 6] obtained a related monotonicity
formula to study regularity of nonlocal minimal surfaces. Another monotonicity
formula for fractional problems was obtained by Cabré and Sire [33, 3] and used by
Frank, Lenzmann and Silvestre [1919, 19].
The proof of Theorem 1.1 follows an approach used in our earlier work with Kelei
Wang [1313, 13] (see also [2929, 29]). First we derive suitable energy estimate (Section 2) and handle the critical and subcritical cases (Section 3). In Section 4 we give
a proof of the monotonicity formula Theorem 1.4. Then we use the monotonicity
formula and a blown-down analysis (Section 6) to reduce to homogeneous singular
solutions. We exclude the existence of stable homogeneous singular solutions in the
optimal range of p (Section 5). Finally we prove Theorem 1.2 in Section 7.
2. Energy estimates
Lemma 2.1. Let u be a solution to (1.3). Assume that u is stable outside some
n
n ) and for x ∈ Rn , define
ball BR
⊂ Rn . Let η ∈ Cc∞ (Rn \ BR
0
0
ˆ
(η(x) − η(y))2
ρ(x) =
dy
(2.1)
|x − y|n+2s
Rn
ON THE FRACTIONAL LANE-EMDEN EQUATION
Then,
ˆ
1
An,s
|u|p+1 η 2 dx + kuηk2Ḣ s (Rn ) ≤
p
p−1
Rn
5
ˆ
u2 ρ dx.
Rn
Proof. Multiply (1.3) by uη 2 . Then,
ˆ
ˆ
p+1 2
|u| η dx =
(−∆)s u uη 2 dx
n
n
R
R
ˆ ˆ
(u(x) − u(y))(u(x)η(x)2 − u(y)η(y)2 )
An,s
dx dy
=
2
|x − y|n+2s
Rn Rn
ˆ ˆ
An,s
u2 (x)η 2 (x) − u(x)u(y)(η 2 (x) + η 2 (y)) + u2 (y)η 2 (y)
=
dx dy
2
|x − y|n+2s
Rn Rn
ˆ ˆ
(u(x)η(x) − u(y)η(y))2 − (η(x) − η(y))2 u(x)u(y)
An,s
=
dx dy
2
|x − y|n+2s
Rn Rn
ˆ ˆ
(η(x) − η(y))2 u(x)u(y)
An,s
= kuηk2Ḣ s (Rn ) −
dx dy
2
|x − y|n+2s
Rn Rn
Using the inequality 2ab ≤ a2 + b2 , we deduce that
ˆ
ˆ
An,s
|u|p+1 η 2 dx ≤
(2.2)
kuηk2Ḣ s (Rn ) −
u2 ρ dx
2
n
n
R
R
Since u is stable, we deduce that
ˆ
ˆ
An,s
(p − 1)
|u|p+1 η 2 dx ≤
u2 ρ dx
2
Rn
Rn
Going back to (2.2), it follows that
ˆ
ˆ
An,s
1
p+1 2
2
|u| η dx ≤
kuηkḢ s (Rn ) +
u2 ρ dx
p
p − 1 Rn
Rn
¤
Lemma 2.2. For m > n/2 and x ∈ Rn , let
(2.3)
η(x) = (1 + |x|2 )−m/2
and
ˆ
ρ(x) =
Rn
(η(x) − η(y))2
dy
|x − y|n+2s
Then, there exists a constant C = C(n, s, m) > 0 such that
¡
¢− n −s
¡
¢− n −s
(2.4)
C −1 1 + |x|2 2
≤ ρ(x) ≤ C 1 + |x|2 2 .
Proof. Let us prove the upper bound first. Since ρ is a continuous function, we
may always assume that |x| ≥ 1. Split the integral
ˆ
(η(x) − η(y))2
dy
|x − y|n+2s
Rn
in the regions |x − y| < |x|/2, |x|/2 ≤ |x − y| ≤ 2|x|, and |x − y| > 2|x|. When
|x − y| ≤ |x|/2,
m+1
|η(x) − η(y)| ≤ C(1 + |x|2 )− 2 |x − y|.
So,
ˆ
ˆ
(η(x) − η(y))2
2 −m−1
|x − y|2−n−2s dy
dy ≤ C(1 + |x| )
n+2s
|x
−
y|
|x−y|≤|x|/2
|x−y|≤|x|/2
¡
¢− n −s
2 −m−s
≤ C(1 + |x| )
≤ C 1 + |x|2 2 .
6
J. DÁVILA, L. DUPAIGNE, AND J. WEI
When |x|/2 ≤ |x − y| ≤ 2|x|,
ˆ
ˆ
(η(x) − η(y))2
−n−2s
dy
≤
C|x|
(η(x)2 + η(y)2 ) dy
n+2s
|x|/2≤|x−y|≤2|x| |x − y|
|y|≤2|x|
n
≤ C|x|−n−2s (|x|−2m+n + 1) ≤ C(1 + |x|2 )− 2 −s ,
where we used the assumption m > n2 . When |x − y| > 2|x|, then |y| ≥ |x| and
η(y) ≤ C(1 + |x|2 )−m/2 . Then,
ˆ
ˆ
(η(x) − η(y))2
1
2 −m
dy
≤
C(1
+
|x|
dy
)
n+2s
n+2s
|x−y|>2|x| |x − y|
|x−y|>2|x| |x − y|
n
≤ C(1 + |x|2 )−m−s ≤ C(1 + |x|2 )− 2 −s .
Let us turn to the lower bound. Again, we may always assume that |x| ≥ 1. Then,
µ ¶−(n+2s) ˆ
ˆ
(η(y) − η(x))2
|x|
ρ(x) ≥
(η(y) − 2−m/2 )2 dy
dy
≥
n+2s
|x
−
y|
2
|y|≤1/2
|y|≤1/2
and the estimate follows.
¤
Corollary 2.3. Let m > n/2, η given by (2.3), R ≥ R0 ≥ 1, ψ ∈ C ∞ (Rn ) be such
that 0 ≤ ψ ≤ 1, ψ ≡ 0 on B1n and ψ ≡ 1 on Rn \ B2n . Let
ˆ
³x´ µ x ¶
(ηR (x) − ηR (y))2
(2.5)
ψ
and ρR (x) =
dy
ηR (x) = η
R
R0
|x − y|n+2s
Rn
There exists a constant C = C(n, s, m, R0 ) > 0 such that for all |x| ≥ 3R0
³ x ´2
³x´
ρR (x) ≤ Cη
|x|−(n+2s) + R−2s ρ
R
R
Proof. Fix x such that |x| ≥ 3R0 . Using the definition of ηR and Young’s inequality,
we have
³ ³ ´
³ ´´2
¡y¢ 2
y
x
µ ¶2 ¡ x ¢
ˆ
ˆ
³
´
ψ
−
ψ
(η R − η R
)
R0
R0
1
x 2
y
ρR (x) ≤ η
dy +
ψ
dy
n+2s
n+2s
2
R
|x − y|
R0
|x − y|
Rn
Rn
¢
¡
¢
¡
ˆ
y
x
³ x ´2 ˆ
−η R
)2
(η R
1
≤η
dy
+
dy
n
R
|x − y|n+2s
|x − y|n+2s
B2R
Rn
0
³ x ´2
³x´
≤ Cη
|x|−(n+2s) + R−2s ρ
R
R
and the result follows.
¤
n
Lemma 2.4. Let u be a solution of (1.3) which is stable outside a ball BR
. Take
0
s(p+1)
n n
ρR as in Corollary 2.3 with m ∈ ( 2 , 2 + 2 ). Then, there exists a constant
C = C(n, s, m, p, R0 ) > 0 such that for all R ≥ 3R0 ,
!
È
ˆ
p+1
n−2s p−1
2
2
.
u ρR dx ≤ C
u ρR dx + R
Rn
n
B3R
0
Proof. By Corollary 2.3, if R ≥ |x| ≥ 3R0 , then
ρR (x) ≤ C(|x|−n−2s + R−2s )
ON THE FRACTIONAL LANE-EMDEN EQUATION
and so
ˆ
p+1
7
p+1
4
ρR (x) p−1 ηR (x)− p−1 dx ≤ CRn−2s p−1 .
n \B n
BR
3R0
Similarly, if |x| ≥ R ≥ 3R0 , then
ρR (x) ≤ CR
−2s
µ
¶− n2 −s
|x|2
1+ 2
R
and so
ρR (x)
p+1
p−1
4
− p−1
ηR (x)
≤ CR
p+1
−2s p−1
p+1
2m
µ
¶− n+2s
2
p−1 + p−1
|x|2
1+ 2
R
2m
n+2s p+1
n
Since m ∈ ( n2 , n2 + s p+1
2 ), we have p−1 −
2 p−1 < − 2 and so
ˆ
p+1
p+1
4
ρR (x) p−1 ηR (x)− p−1 dx ≤ CRn−2s p−1 .
n
Rn \B3R
Now,
ˆ
0
ˆ
ˆ
2
Rn
u ρR dx =
n
B3R
ˆ
u ρR dx +
0
µˆ
2
≤
n
B3R
u ρR dx +
0
n
Rn \B3R
Rn
2
|u|p+1 ηR
4
4
−
2
u2 ρR ηR p+1 ηRp+1 dx
0
2
¶ p+1
µˆ
dx
p+1
p−1
Rn
− 4
ηR p−1
ρR
µˆ
ˆ
p+1
n
B3R
p−1
u2 ρR dx + CR(n−2s p−1 ) p+1
≤
0
Rn
¶ p−1
p+1
dx
2
¶ p+1
2
|u|p+1 ηR
dx
By a standard approximation argument, Lemma 2.1 remains valid with η = ηR and
ρ = ρR and so the result follows.
¤
n+2s
Lemma 2.5. Assume that p 6= n−2s
. Let u be a solution to (1.3) which is stable
n
outside a ball BR
and
ū
its
extension,
solving (1.10). Then, there exists a constant
0
C = C(n, p, s, R0 , u) > 0 such that
ˆ
4s
t1−2s ū2 dxdt ≤ CRn+2(1−s)− p−1
n+1
BR
for any R > 3R0 .
Proof. According to Theorem 1.3,
ˆ
ū(x, t) = pn,s
u(z)
Rn
so that
t2s
(|x − z|2 + t2 )
ˆ
ū(x, t)2 ≤ pn,s
So,
ˆ
u(z)2
Rn
t2s
(|x − z|2 + t2 )
È
ˆ
1−2s 2
n+1
BR
t
ˆ
|x|≤R,z∈Rn
2
R
n+2s
2
dz
dz.
t
!
dzdx
n+2s dt
(|x − z|2 + t2 ) 2
n¡
¢− n +1−s o
¢− n +1−s ¡
u2 (z) |x − z|2 2
− |x − z|2 + R2 2
dzdx
ū dxdt ≤ pn,s
≤C
n+2s
2
u(z)
|x|≤R,z∈Rn
0
8
J. DÁVILA, L. DUPAIGNE, AND J. WEI
Split this last integral according to |x − z| < 2R or |x − z| ≥ 2R. Then,
ˆ
n¡
¢− n +1−s ¡
¢− n +1−s o
− |x − z|2 + R2 2
dzdx ≤
u2 (z) |x − z|2 2
|x|≤R,|x−z|<2R
ˆ
ˆ
¡
¢− n +1−s
u2 (z) |x − z|2 2
dzdx ≤ CR2(1−s)
u2 (z) dz ≤
n+1
B3R
|x|≤R,|x−z|<2R
µˆ
CR
2(1−s)
2
|u|p+1 ηR
2
¶ p+1
È
n+1
B3R
µˆ
p−1
CR2(1−s)+n p+1
−
4
! p−1
p+1
ηR p−1
u2 (z)ρR (z) dz
2
¶ p+1
≤
4s
≤ CRn+2(1−s)− p−1 ,
where we used Hölder’sin equality, then Lemma 2.1 and then Lemma 2.4. For the
region |x − z| ≥ 2R, the mean-value inequality implies that
ˆ
n¡
¢− n +1−s ¡
¢− n +1−s o
u2 (z) |x − z|2 2
− |x − z|2 + R2 2
dzdx ≤
|x|≤R,|x−z|≥2R
ˆ
ˆ
CR2
u2 (z)|x − z|−(n+2s) dzdx ≤ CRn+2
u2 (z)|z|−(n+2s) dz
|x|≤R,|x−z|≥2R
|z|≥R
ˆ
4s
≤ CR2
u2 ρ dz ≤ CRn+2(1−s)− p−1 ,
|z|≥R
where we used again Corollary 2.3 in the penultimate inequality and Lemma 2.4 in
the last one.
¤
n
Lemma 2.6. Let u be a solution to (1.3) which is stable outside a ball BR
and
0
ū its extension, solving (1.10). Then, there exists a constant C = C(n, p, s, u) > 0
such that
ˆ
ˆ
p+1
1−2s
2
|u|p+1 dx ≤ CRn−2s p−1
t
|∇ū| dx dt +
n+1
∩∂Rn+1
BR
+
n+1
n+1
∩R+
BR
Proof. The Lp+1 estimate follows from Lemmata 2.1 and 2.4. Now take a cut-off
n+1
n+1
n+1
function η ∈ Cc1 (Rn+1
∩ (BR
\ B2R
) and η = 0 on
+ ) such that η = 1 on R+
0
n+1
n+1
n+1
BR0 ∪ (R+ \ B2R ), and multiply equation (1.10) by ūη 2 . Then,
ˆ
ˆ
©
ª
κs
|ū|p+1 η 2 dx =
t1−2s ∇ū · ∇(ūη 2 ) dx dt
Rn+1
+
∂Rn+1
+
(2.6)
ˆ
=
Rn+1
+
©
ª
t1−2s |∇(ūη)|2 − ū2 |∇η|2 dx dt.
n+1
Since u is stable outside BR
, so is ū and we deduce that
0
ˆ
ˆ
©
ª
1
t1−2s |∇(ūη)|2 dx dt ≥
t1−2s |∇(ūη)|2 − ū2 |∇η|2 dx dt.
p Rn+1
Rn+1
+
+
In other words,
p0
(2.7)
where
1
p0
+
1
p
ˆ
Rn+1
+
ˆ
t1−2s ū2 |∇η|2 dx dt ≥
Rn+1
+
= 1. We then apply Lemma 2.5.
t1−2s |∇(ūη)|2 dx dt,
¤
ON THE FRACTIONAL LANE-EMDEN EQUATION
9
3. The subcritical case
In this section, we prove Theorem 1.1 for 1 < p ≤ pS (n).
n
Proof. Take a solution u which is stable outside some ball BR
. Apply Lemma
0
s
2.4 and let R → +∞. Since p ≤ pS (n), we deduce that u ∈ Ḣ (Rn ) ∩ Lp+1 (Rn ).
Multiplying the equation (1.3) by u and integrating, we deduce that
ˆ
|u|p+1 = kuk2Ḣ s (Rn ) ,
(3.1)
Rn
while multiplying by uλ given for λ > 0 and x ∈ Rn by
uλ (x) = u(λx)
yields
ˆ
ˆ
ˆ
(−∆)s/2 u(−∆)s/2 uλ = λs
|u|p−1 uλ =
Rn
wwλ ,
Rn
Rn
s/2
where w = (−∆)
√ u. Following Ros-Oton and Serra [2727, 27], we use the change
of variable y = λ x to deduce that
ˆ
ˆ
√
√
2s−n
s
λ
2
λ
ww dx = λ
w λ w1/ λ dy
Rn
Rn
Hence,
−
n
p+1
ˆ
ˆ
ˆ
|u|p+1
|u|p+1 =
x·∇
=
(|u|p−1 u)x · ∇u =
p+1
Rn
Rn
Rn
¯
¯
ˆ
ˆ
√
√
2s−n
d ¯¯
d ¯¯
λ 1/ λ
p−1
λ
2
w
w
dy =
|u|
uu
=
λ
dλ ¯λ=1 Rn
dλ ¯λ=1
Rn
¯
ˆ
ˆ
√
√
d ¯¯
2s − n
2s − n
w2 +
kuk2Ḣ s (Rn )
w λ w1/ λ dy =
¯
2
dλ λ=1 Rn
2
Rn
In the last equality, we have used the fact that w ∈ C 1 (Rn ), as follows by elliptic
regularity. We have just proved the following Pohozaev identity
ˆ
n
n − 2s
|u|p+1 =
kuk2Ḣ s (Rn )
p + 1 Rn
2
For p < pS (n), the above identity together with (3.1) force u ≡ 0. For p = pS (n), we
are left with proving that there is no stable nontrivial solution. Since u ∈ Ḣ s (Rn ),
we may apply the stability inequlatiy (1.5) with test function ϕ = u, so that
ˆ
p
|u|p+1 ≤ kuk2Ḣ s (Rn ) .
Rn
This contradicts (3.1) unless u ≡ 0.
¤
In the following sections, we present several tools to study the supercritical case.
4. The monotonicity formula
In this section, we prove Theorem 1.4.
10
J. DÁVILA, L. DUPAIGNE, AND J. WEI
Proof. Since the equation is invariant under translation, it suffices to consider the
case where the center of the considered ball is the origin x0 = 0. Let
(4.1)
E1 (ū; λ) =
È
λ
p+1
2s p−1
−n
n+1
Rn+1
∩Bλ
+
t
2
1−2s |∇ū|
2
ˆ
dx dt −
n+1
∂Rn+1
∩Bλ
+
κs
|ū|p+1 dx
p+1
!
For X ∈ Rn+1
+ , let also
2s
U (X; λ) = λ p−1 ū(λX).
(4.2)
Then, U satisfies the three following properties: U solves (1.10),
(4.3)
E1 (ū; λ) = E1 (U ; 1),
and, using subscripts to denote partial derivatives,
(4.4)
λUλ =
2s
U + rUr .
p−1
Differentiating the right-hand side of (4.3), we find
ˆ
ˆ
dE1
1−2s
|U |p−1 Uλ dx.
t
∇U · ∇Uλ dx dt − κs
(ū; λ) =
n+1
n+1
n+1
dλ
∂R
∩B
Rn+1
∩B
1
1
+
+
Integrating by parts and then using (4.4),
ˆ
dE1
t1−2s Ur Uλ dσ
(ū; λ) =
dλ
∂B1n+1 ∩Rn+1
+
ˆ
ˆ
2s
1−2s 2
t
Uλ dσ −
=λ
t1−2s U Uλ dσ
n+1
n+1
n+1
n+1
p
−
1
∂B1 ∩R+
∂B1 ∩R+
È
!
ˆ
s
1−2s 2
1−2s 2
t
Uλ dσ −
=λ
t
U dσ
p−1
∂B1n+1 ∩Rn+1
∂B1n+1 ∩Rn+1
+
+
Scaling back, the theorem follows.
λ
¤
5. Homogeneous solutions
Theorem 5.1. Let ū be a stable homogeneous solution of (1.10). Assume that
n+2s
p > n−2s
and
(5.1)
p
Γ( n2 −
s
s
p−1 )Γ(s + p−1 )
s
s
Γ( p−1
)Γ( n−2s
− p−1
)
2
>
2
Γ( n+2s
4 )
.
2
Γ( n−2s
4 )
Then, ū ≡ 0.
Proof. Take standard polar coordinates in Rn+1
+ : X = (x, t) = rθ, where r = |X|
t
X
. Let θ1 = |X|
denote the component of θ in the t direction and
and θ = |X|
n
S+
= {X ∈ Rn+1
: r = 1, θ1 > 0} denote the upper unit half-sphere.
+
Step 1. Let ū be a homogeneous solution of (1.10) i.e. assume that for some
n
ψ ∈ C 2 (S+
),
2s
ū(X) = r− p−1 ψ(θ).
ON THE FRACTIONAL LANE-EMDEN EQUATION
Then,
ˆ
ˆ
(5.2)
n
S+
θ11−2s |∇ψ|2 + β
ˆ
n
S+
where κs is given by (1.9) and
2s
β=
p−1
11
θ11−2s ψ 2 = κs
|ψ|p+1 ,
n
∂S+
µ
¶
2s
n − 2s −
.
p−1
Indeed, since ū solves (1.10) and is homogeneous, ψ solves

n
 − div(θ11−2s ∇ψ) + βθ11−2s ψ = 0 on S+
(5.3)
n
,
 − lim θ11−2s ∂θ1 ψ = κs |ψ|p−1 ψ on ∂S+
θ1 →0
Multiplying (5.3) by ψ and integrating, (5.2) follows.
n
Step 2. For all ϕ ∈ C 1 (S+
),
µ
¶2 ˆ
ˆ
ˆ
n − 2s
1−2s
p−1 2
2
(5.4)
κs p
|ψ| ϕ ≤
θ1 |∇ϕ| +
θ11−2s ϕ2
n
n
n
2
∂S+
S+
S+
By definition, ū is stable if for all φ ∈ Cc1 (Rn+1
+ ),
ˆ
ˆ
t1−2s |∇φ|2 dxdt
(5.5)
|ū|p−1 φ2 dx ≤
κs p
Rn+1
+
∂Rn+1
+
Choose a standard cut-off function η² ∈ Cc1 (R∗+ ) at the origin and at infinity i.e.
n
χ(²,1/²) (r) ≤ η² (r) ≤ χ(²/2,2/²) (r). Let also ϕ ∈ C 1 (S+
), apply (5.5) with
φ(X) = r−
n−2s
2
for X ∈ Rn+1
+ ,
η² (r)ϕ(θ)
and let ² → 0. Inequality (5.4) follows.
n
Step 3. For α ∈ (0, n−2s
2 ), x ∈ R \ {0}, let
vα (x) = |x|−
n−2s
+α
2
and v̄α its extension, as defined in Theorem 1.3. Then, v̄α is homogeneous i.e. there
n
exists φα ∈ C 2 (S+
) such that for X ∈ Rn+1
\ {0},
+
v̄α (X) = r−
In addition, for all ϕ ∈ C
n
S+
θ11−2s |∇ϕ|2
+
φα (θ).
n
(S+
),
õ
ˆ
(5.6)
1
n−2s
+α
2
n − 2s
2
!ˆ
¶2
−α
2
ˆ
= κs λ(α)
n
S+
θ11−2s ϕ2
ˆ
2
ϕ +
n
∂S+
n
S+
θ11−2s φ2α
¯ µ ¶¯2
¯
¯
¯∇ ϕ ¯
¯
φα ¯
Indeed, according to Fall [1616; 16, Lemma 3.1], v̄α is homogeneous. Using the
calculus identity stated by Fall-Felli in [1717; 17, Lemma 2.1], we get
!
õ

¶2

n − 2s
2
n
 − div(θ1−2s ∇φ ) +
− α θ11−2s φα = 0 on S+
α
1
2
(5.7)


n
φα = 1 on ∂S+
.
12
J. DÁVILA, L. DUPAIGNE, AND J. WEI
Multiply equation (5.7) by ϕ2 /φα , integrate by parts, apply the calculus identity
¯
¯
¯ ϕ ¯2 2
ϕ2
2
¯
∇φα · ∇
= |∇ϕ| − ¯∇ ¯¯ φα
φα
φα
and recall from Fall [1616; 16, Lemma 3.1] that
− lim t1−2s ∂t v α = κs λ(α)|x|−
n−2s
+α−2s
2
t→0
,
where λ(α) is given by (1.8).
Step 4. For α ∈ (0, n−2s
2 )
(5.8)
n
on S+
.
φ0 ≤ φα
n
Indeed, on S+
,
µ
div(θ11−2s ∇φ0 )
=
n − 2s
2
õ
¶2
θ11−2s φ0
≥
n − 2s
2
!
¶2
−α
2
θ11−2s φ0
so φ0 is a sub-solution of (5.7). By the maximum principle, the conclusion follows.
Step 5. End of proof. Fix α ∈ (0, n−2s
2 ) given by
α=
so that
µ
n − 2s
2
¶2
n − 2s
2s
−
2
p−1
2s
−α =
p−1
2
µ
¶
2s
n − 2s −
= β,
p−1
where β is the constant appearing in (5.3).
0
Use the stability inequality (5.4) with ϕ = ψφ
φα :
¯ µ
¶¯2 µ
¶2 ˆ
µ
¶2
ˆ
ˆ
¯
ψφ0 ¯¯
ψφ0
n − 2s
1−2s
θ11−2s ¯¯∇
|ψ|p+1 ≤
θ
(5.9) κs p
.
+
1
n
n
n
φα ¯
2
φα
S+
∂S+
S+
Note that a particular case of the identity (5.6) is
(5.10)
¯ µ ¶¯2
¶2 ˆ
µ
ˆ
ˆ
ˆ
¯
ϕ ¯¯
n − 2s
1−2s
1−2s 2
1−2s 2 ¯
2
2
θ1 |∇ϕ| +
θ1 ϕ = κs Λn,s
ϕ +
θ1 φ0 ¯∇
n
n
n
n
2
φ0 ¯
S+
∂S+
S+
S+
0
Using (5.10) (with ϕ = ψφ
φα ), (5.9) becomes
ˆ
ˆ
ˆ
κs p
|ψ|p+1 ≤ κs Λn,s
ψ2 +
n
∂S+
n
∂S+
By (5.8), we deduce that
ˆ
ˆ
κs p
|ψ|p+1 ≤ κs Λn,s
n
∂S+
n
S+
ˆ
ψ2 +
n
∂S+
n
∂S+
¯ µ ¶¯2
¯
¯
¯∇ ψ ¯ .
¯
φα ¯
¯ µ ¶¯2
¯
ψ ¯¯
θ11−2s φ2α ¯¯∇
.
n
φα ¯
S+
Using again the identity (5.6), we deduce that
ˆ
ˆ
ˆ
κs p
|ψ|p+1 ≤ κs (Λn,s − λ(α))
ψ2 +
n
∂S+
θ11−2s φ20
n
S+
ˆ
θ11−2s |∇ψ|2 + β
n
S+
θ11−2s ψ 2
ON THE FRACTIONAL LANE-EMDEN EQUATION
Comparing with (5.2), it follows that
ˆ
ˆ
(5.11)
(p − 1)
|ψ|p+1 ≤ (Λn,s − λ(α))
n
∂S+
But from (5.2) and (5.6)
13
ψ2 .
n
∂S+
ˆ
ˆ
|ψ|p+1 ≥ λ(α)
ψ2
n
∂S+
n
∂S+
Combined with (5.11), we find that
λ(α)p ≤ Λn,s
unless ψ ≡ 0.
¤
6. Blow-down analysis
Proof of Theorem 1.1. Assume that p > pS (n). Take a solution u of (1.3) which is
stable outside the ball of radius R0 and let ū be its extension solving (1.10).
Step 1. limλ→+∞ E(ū, 0; λ) < +∞.
Since E is nondecreasing, it suffices to show that E(ū, 0; λ) is bounded. Write
E = E1 + E2 , where E1 is given by (4.1) and
ˆ
p+1
s
t1−2s ū2 dσ
E2 (ū; λ) = λ2s p−1 −n−1
p + 1 ∂B n+1 (0,λ)∩Rn+1
+
By Lemma 2.6, E1 is bounded. Since E is nondecreasing,
ˆ
ˆ
p+1
1 2λ
t1−2s ū2 .
E(ū; λ) ≤
E(u; t) dt ≤ C + λ2s p−1 −n−1
n+1
n+1
λ λ
B2λ ∩R+
Applying Lemma 2.5, we deduce that E is bounded.
Step 2. There exists a sequence λi → +∞ such that (ūλi ) converges weakly in
1
1−2s
Hloc
(Rn+1
dxdt) to a function ū∞ .
+ ;t
1
1−2s
This follows from the fact that (ūλi ) is bounded in Hloc
(Rn+1
dxdt) by
+ ;t
Lemma 2.6.
Step 3. ū∞ is homogeneous
To see this, apply the scale invariance of E, its finiteness and the monotonicity
formula: given R2 > R1 > 0,
0
=
=
≥
≥
lim E(ū; λi R2 ) − E(ū; λi R1 )
i→+∞
lim E(ūλi ; R2 ) − E(ūλi ; R1 )
i→+∞
µ
¶2
∂ ūλi
2s ūλi
+
dx dt
lim inf
t
r
i→+∞ (B n+1 \B n+1 )∩Rn+1
p−1 r
∂r
+
R2
R1
µ
¶2
ˆ
4s
2s ū∞
∂ ū∞
t1−2s r2−n+ p−1
+
dx dt
n+1
n+1
p−1 r
∂r
(BR
\BR
)∩Rn+1
+
ˆ
4s
1−2s 2−n+ p−1
2
1
Note that in the last inequality we only used the weak convergence of (ūλi ) to ū∞
1
1−2s
in Hloc
(Rn+1
dxdt). So,
+ ;t
2s ū∞
∂ ū∞
+
=0
p−1 r
∂r
a.e. in Rn+1
+ .
14
J. DÁVILA, L. DUPAIGNE, AND J. WEI
And so, u∞ is homogeneous.
Step 4. ū∞ ≡ 0
Simply apply Theorem 5.1.
n+1
Step 5. (ūλi ) converges strongly to zero in H 1 (BR
\Bεn+1 ; t1−2s dxdt) and (uλi )
n+1
converges strongly to zero in Lp+1 (BR \ Bεn+1 ) for all R > ² > 0. Indeed, by
1
1−2s
Steps 2 and 3, (ūλi ) is bounded in Hloc
(Rn+1
dxdt) and converges weakly to
+ ;t
λi
1−2s
0. It follows that (ū ) converges strongly to 0 in L2loc (Rn+1
dxdt). Indeed,
+ ;t
by the standard Rellich-Kondrachov theorem and a diagonal argument, passing to
a subsequence we obtain
ˆ
t1−2s |ūλi |2 dxdt → 0,
n+1
Rn+1
∩(BR
\A)
+
n+1
n+1
as i → ∞, for any BR
= BR
(0) ⊂ Rn+1 and A of the form A = {(x, t) ∈ Rn+1
:
+
0 < t < r/2}, where R, r > 0. By [1515; 15, Theorem 1.2],
ˆ
ˆ
t1−2s |ūλi |2 dxdt ≤ Cr2
t1−2s |∇ūλi |2 dxdt
Rn+1
∩Brn+1 (x)
+
Rn+1
∩Brn+1 (x)
+
n+1
for any x ∈ ∂Rn+1
∩ A with
+ , |x| ≤ R, with a uniform constant C. Covering BR
n+1
n+1
n+1
half balls Br (x) ∩ R+ , x ∈ ∂R+ with finite overlap, we see that
ˆ
ˆ
t1−2s |∇ūλi |2 dxdt ≤ Cr2 ,
t1−2s |ūλi |2 dxdt ≤ Cr2
n+1
∩A
BR
n+1
∩A
BR
1−2s
and from this we conclude that (ūλi ) converges strongly to 0 in L2loc (Rn+1
dxdt).
+ ;t
n+1
λi
1
1−2s
Now, using (2.7), (ū ) converges strongly to 0 in Hloc (R+ \ {0}; t
dxdt)
n
and by (2.6), the convergence also holds in Lp+1
(R
\
{0}).
loc
Step 6.
ū ≡ 0.
Indeed,
ˆ
ˆ
|∇ūλ |2
κs
dx dt −
|ūλ |p+1 dx
n+1
n+1
n+1
n+1
2
p+1
R+ ∩B1
∂R+ ∩B1
ˆ
ˆ
λ 2
κs
1−2s |∇ū |
=
t
dx dt −
|ūλ |p+1 dx+
n+1
n+1
n+1
n+1
2
p+1
R+ ∩B²
∂R+ ∩B²
ˆ
ˆ
λ 2
κs
1−2s |∇ū |
t
dx dt −
|ūλ |p+1 dx
n+1
n+1
n+1
n+1
n+1
n+1 p + 1
2
R+ ∩B1 \B²
∂R+ ∩B1 \B²
ˆ
ˆ
λ 2
p+1
κs
n−2s p−1
1−2s |∇ū |
=ε
E1 (ū; λε) +
t
dx dt −
|ūλ |p+1 dx
n+1
n+1
n+1
n+1
n+1
n+1 p + 1
2
R+ ∩B1 \B²
∂R+ ∩B1 \B²
ˆ
ˆ
λ 2
p+1
κs
|∇ū |
dx dt −
|ūλ |p+1 dx
≤ Cεn−2s p−1 +
t1−2s
n+1
n+1
n+1 p + 1
n+1
n+1
n+1
2
∂R+ ∩B1 \B²
R+ ∩B1 \B²
E1 (ū; λ) = E1 (ūλ ; 1) =
t1−2s
Letting λ → +∞ and then ε → 0, we deduce that limλ→+∞ E1 (ū; λ) = 0. Using
the monotonicity of E,
ˆ
ˆ
p+1
1 2λ
E(t) dt ≤ sup E1 + Cλ−n−1+2s p−1
ū2
E(ū; λ) ≤
n+1
n+1
λ λ
[λ,2λ]
B2λ \Bλ
ON THE FRACTIONAL LANE-EMDEN EQUATION
15
and so limλ→+∞ E(ū; λ) = 0. Since u is smooth, we also have E(ū; 0) = 0. Since
E is monotone, E ≡ 0 and so ū must be homogeneous, a contradiction unless
u ≡ 0.
¤
7. Construction of radial entire stable solutions
defined by
Let ūs denote the extension of the singular solution us (1.7) to Rn+1
+
ˆ
P (X, y)u(y) dy.
ūs (X) =
Rn
Let
B1n+1
(7.1)
n+1
denote the unit ball in R
and for λ ≥ 0, consider

div (t1−2s ∇u) = 0
in B1n+1 ∩ Rn+1

+


n+1
u = λūs
on ∂B1 ∩ Rn+1
+


 − lim (t1−2s ut ) = κs up on B n+1 ∩ {t = 0}.
1
t→0
Take λ ∈ (0, 1). Since us is a positive supersolution of (7.1), there exists a minimal
solution u = uλ . By minimality, the family (uλ ) is nondecreasing and uλ is axially
symmetric, that is, uλ (x, t) = uλ (r, t) with r = |x| ∈ [0, 1]. In addition, for a fixed
value λ ∈ (0, 1), uλ is bounded, as can be proved by the truncation method of
[11, 1], see also [1010, 10] and radially decreasing by the moving plane method (see
[77, 7] for a similar setting). From now on let us assume that pS (n) < p and
p
Γ( n2 −
s
s
p−1 )Γ(s + p−1 )
s
s
Γ( p−1
)Γ( n−2s
− p−1
)
2
≤
2
Γ( n+2s
4 )
,
2
Γ( n−2s
4 )
which means that the singular solution us is stable. Then, uλ ↑ us as λ ↑ 1, using
the classical convexity argument in [22, 2] (see also Section 3.2.2 in [1414, 14]). Let
λj ↑ 1 and
mj = kuλj kL∞ = uλj (0),
p−1
Rj = mj 2s ,
so that mj , Rj → ∞ as j → ∞. Set
vj (x) = m−1
j uλj (x/Rj ).
Then 0 ≤ vj ≤ 1 is a bounded solution of

div (t1−2s ∇vj ) = 0



vj = λj ūs


 − lim (t1−2s (v ) ) = κ v p
j t
t→0
s j
n+1
in BR
∩ Rn+1
+
j
n+1
on ∂BR
∩ Rn+1
+
j
n+1
on BR
∩ {t = 0}.
j
n+1
Moreover vj ≤ ūs in BR
∩ Rn+1
and vj (0) = 1. Using elliptic estimates we find
+
j
n+1
(for a subsequence) that (vj ) converges uniformly on compact sets of R+
function v that is axially symmetric and solves
(
div (t1−2s ∇v) = 0
in Rn+1
+
− lim (t1−2s vt ) = κs v p
t→0
to a
on Rn × {0}.
Moreover 0 ≤ v ≤ 1, v(0) = 1 and v ≤ ūs . This v restricted to Rn × {0} is a radial,
bounded, smooth solution of (1.3) and from v ≤ ūs we deduce that v is stable.
16
J. DÁVILA, L. DUPAIGNE, AND J. WEI
Acknowledgments: J. Dávila is supported by Fondecyt 1130360, CAPDE Millennium Nucleus NC130017 and Fondo Basal CMM, Chile. The research of J. Wei
is partially supported by NSERC of Canada.
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Departamento de Ingenierı́a Matemática and CMM, Universidad de Chile, Casilla
170 Correo 3, Santiago, Chile
E-mail address: jdavila@dim.uchile.cn
LAMFA, UMR CNRS 7352, Université de Picardie Jules Verne, 33 rue St Leu, 80039,
Amiens Cedex, France
E-mail address: louis.dupaigne@math.cnrs.fr
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada,
V6T 1Z2.
E-mail address: jcwei@math.ubc.ca