On a class of semilinear fractional elliptic equations involving outside Dirac data

0.23 MB·English

by  Huyuan Chen

On a class of semilinear fractional elliptic equations
involving outside Dirac data
Huyuan Chen1 Hichem Hajaiej2 Ying Wang3
Abstract
Thepurposeofthisarticleis togiveacompletestudy oftheweaksolutionsofthe fractional
elliptic equation
(−∆)αu+up =0 in B1(eN),
(0.1)
u=δ0 in RN \B1(eN),
where p ≥ 0, (−∆)α with α ∈ (0,1) denotes the fractional Laplacian operator in the principle
5 value sense, B1(eN) is the unit ball centered at eN = (0,···,0,1) in RN with N ≥ 2 and δ0 is
1
the Dirac mass concentrated at the origin. We prove that problem (0.1) admits a unique weak
0
2 solution when p > 1+ 2Nα. Moreover, if in addition p ≥ NN+−22, the weak solution vanishes as
α→1−. We also show that problem (0.1) doesn’t have any weak solution when p∈[0,1+ 2α].
y N
a Theseresults areverysurprisingsince thereareintotalcontradictionwiththe classicalsetting,
M i.e.
−∆u+up =0 in B1(eN),

5
2 u=δ0 in RN \B1(eN),
for which it has been proved that there are no solutions for p≥ N+1.
] N−1
P
A
Key words: Fractional Laplacian; Dirac mass; Weak solution; Existence; Uniqueness.
.
h MSC2010: 35R06, 35A01, 35J66.
t
a
m
1 Introduction
[

2
Fractional PDEs have gained tremendous interest, not only from mathematicians but also from
v
2 physicistsandengineering,duringthelastyears. Thisisessentiallyduetotheirwidespreaddomains
4 of applications. In fact the fractional Laplacian arises in many areas including medicine [12], bio-
2
engineering [19, 20, 21, 22], relativistic physics[1, 17, 18], Modeling populations [29], flood flow,
6
0 material viscoelastic theory, biology and earthquakes. It is also particularly relevant to study some
. situations, in which the fractional Laplacian is involved in PDEs, featuring irregular data such that
1
0 those phenomena describing source terms which are concentrated at points. In our context, the
5
source is placed outside the unit ball B (e ). This generates long-term interactions and short-
1 1 N
: term interactions, described by the nonlocal operator (−∆)α and the nonlinear absorption up
v
respectively. (−∆)α has also a probabilistic interpretation, related to the above one. It is the
i
X α−stable subordinated infinitesimal killed Brownian motion.
r Let B (e ) be the unit ball in RN(N ≥ 2) with center e = (0,···,0,1) and δ be the Dirac
a 1 N N 0
mass concentrated at the origin. Our main objective in this article is to investigate the existence,
nonexistence and uniqueness of positive weak solutions of the semilinear fractional equation
(−∆)αu+up = 0 in B (e ),
1 N
(1.1)
u = δ in RN \B (e ),
0 1 N
where p ≥ 0 and the fractional Laplacian (−∆)α with α ∈(0,1) is defined by
(−∆)αu(x) = c lim (−∆)αu(x),
N,α ǫ
ǫ→0+
[email protected]
[email protected]
[email protected]
1
where
−1
1−cos(z )
1
c = dz (1.2)
N,α RN |z|N+2α
(cid:18)Z (cid:19)
with z = (z ,···,z )∈ RN and
1 N
u(z)−u(x)
(−∆)αu(x) = − dz.
ǫ ZRN\Bǫ(x) |z−x|N+2α
In 1991, a fundamental contribution to semilinear elliptic equations involving measures as
boundary data is due to Gmira and V´eron [15], where they studied the existence and uniqueness
of weak solutions for
−∆u+h(u) = 0 in Ω,
(1.3)
u = µ on ∂Ω,
where Ω is a bounded C2 domain and µ is a bounded Radon measure defined in ∂Ω. A function u
is said to be a weak solution of (1.3) if u ∈L1(Ω), h(u) ∈ L1(Ω,ρdx) and
∂ξ(x)
[u(−∆)ξ+h(u)ξ]dx = dµ(x), ∀ξ ∈C1.1(Ω), (1.4)
∂~n 0
ZΩ Z∂Ω x
where ρ(x) = dist(x,∂Ω) and ~n denotes the unit inward normal vector at a point x. Gmira and
x
V´eron proved that the problem (1.3) admits a unique weak solution when h is a continuous and
nondecreasing function satisfying
∞
[h(s)−h(−s)]s−1−NN−+11ds < +∞. (1.5)
Z1
The weak solution of (1.3) is approached by the classical solutions of (1.3) when µ is replaced by
a sequence of regular functions {µ }, which converge to µ in the distribution sense. Furthermore,
n
they showed that there is no weak solution of (1.3) when µ = δ with x ∈ ∂Ω and h(s) = |s|p−1s
x0 0
with p ≥ N+1. Later on, this subject has been vastly expanded in recent works, see the papers of
N−1
Marcus and V´eron [23, 24, 25, 26], Bidaut-V´eron and Vivier [3] and references therein.
In the fractional setting, the equivalent of (1.3) when µ = δ has been considered in [7], where
x0
the authors proved that the weak solution of
(−∆)αu+g(u) = k∂αδx0 in Ω¯,
∂~nα
(1.6)
u = 0 in Ω¯c
is approximated by the weak solutions, as t → 0+, of
(−∆)αu+g(u) = kt−αδ in Ω¯,
x0+t~nx0
u = 0 in Ω¯c.
More precisely, in the fractional setting, ∂αδx0 plays the same role of u = δ on ∂Ω in (1.3). Our
∂~nα x0
purpose in this article is to study the solution of
(−∆)αu+up = 0 in B (e )
1 N
when the exact Dirac mass concentrated at the origin is considered. Our main idea is to make use
of nonlocal properties of the fractional Laplacian to move the Dirac mass at −te when t → 0+
N
and we then proceed by approximation techniques. Before giving our main results, we must first
give an appropriate definition of weak solution of (1.1). It is then worth to mention two important
results. The equation
(−∆)αu+up = 0 in B (e ),
1 N
(1.7)
u = f in RN \B (e ),
1 N
2
where f ∈ C (RN \B (e )), admits a unique classical solution u , see [6, Theorem 2.5]. Further-
0 1 N f
more, let u˜ = u in B (e ) and u˜ = 0 in RN \B (e ), then u˜ is the unique classical solution
f f 1 N f 1 N f
of
(−∆)αu+up = c f(y) dy in B (e ),
N,α RN\B1(eN) |x−y|N+2α 1 N (1.8)
u= 0 R in RN \B1(eN)
and satisfies the identity:
ξ(x)f(y)
[u(x)(−∆)αξ(x)+up(x)ξ(x)]dx = c dydx,
ZB1(eN) N,αZB1(eN)ZRN\B1(eN) |x−y|N+2α
for any ξ ∈ C∞(B (e )). Let us mention that C∞(B (e )) is the space of test functions ξ ∈
0 1 N 0 1 N
C∞(RN) with support in B (e ).
1 N
Inspired by above identity, we give the definition of weak solution to (1.1) as follows.
Definition 1.1 We say that u is a weak solution of (1.1) if u ∈ L1(B (e )), up ∈ L1 (B (e ))
1 N loc 1 N
and
[u(x)(−∆)αξ(x)+up(x)ξ(x)]dx = ξ(x)Γ (x)dx, ∀ξ ∈ C∞(B (e )), (1.9)
0 0 1 N
ZB1(eN) ZB1(eN)
where
c
Γ (x)= N,α , ∀x ∈RN \{0}. (1.10)
0 |x|N+2α
It is well known that the definition of the weak solution heavily depends on the test functions
space and the best function space is the one which enables us to get the ”strongest” weak solution.
In [9, 10], semilinear fractional equations with measures has been studied via the test functions
space X ⊂ C(RN) for a C2 bounded open domain Ω, where X is the space of functions ξ
α,Ω α,Ω
satisfying:
(1) supp(ξ)⊂ Ω¯;
(2) (−∆)αξ(x) exists for all x ∈ Ω and |(−∆)αξ(x)| ≤ C for some C > 0;
(3) there exist ϕ ∈ L1(Ω,ραdx) and ǫ > 0 such that |(−∆)αξ|≤ ϕ a.e. in Ω, for all ǫ ∈ (0,ǫ ],
0 ǫ 0
where ρ(x)= dist(x,∂Ω).
Thetest functionsspace C∞(B (e )) has stronger topology than X does, the weak solution
0 1 N α,B1(eN)
in Definition 1.1 with test functions space X would be stronger than the one with test
α,B1(eN)
functionsspaceC∞(B (e )). ItisthenworthtomentionthatthetestfunctionsspaceC∞(B (e ))
0 1 N 0 1 N
could not be replaced to the test functions space X in our setting. For example, ξ :=
α,B1(eN) 0
G [1] ∈ X , but (1.9) does not hold for ξ , where G denotes the Green kernel of (−∆)α in
α α,B1(eN) 0 α
B (e )×B (e ) and G is the Green operator defined as
1 N 1 N α
G [f](x) = G (x,y)f(y)dy, f ∈ L1(B (e ),ραdx). (1.11)
α α 1 N
ZB1(eN)
Let us state our existence result.
Theorem 1.1 Assume that α ∈ (0,1) and p > 1+ 2α. Then there exists a unique nonnegative
N
weak solution u of (1.1) such that for some c > 1, we have
α,p 1
−N+2α
0 < uα,p(x)≤ c1|x| p , ∀x∈ B1(eN) (1.12)
and
1 −N+2α −N+2α
t p ≤ uα,p(teN) ≤ c1t p , ∀t ∈ (0,1). (1.13)
c
1
3
Remark 1.1 (i) The existence results are very surprising as they are in total different from the
Laplacian case, where (1.3) with µ = δ has a weak solution only when p < N+1.
0 N−1
(ii) From (1.13), the singularity is only near the origin. We also notice that
1
up (te ) ≥ t−(N+2α), ∀t ∈ (0,1),
α,p N c
1
which implies that the absorption nonlinearity up plays a primary role in the equation (1.1). While
the absorption nonlinearity always plays a second role in a measure framework.
(iii) The uniqueness cannot directly follow Kato’s inequality [9, Proposition 2.4] since it has
been built in the framework of the test functions space X . In this paper, as mentioned
α,B1(eN)
above, C∞(B (e )) is the appropriate test functions space. This will give birth to a lot of technical
0 1 N
difficulties to prove the existence, nonexistence and uniqueness of weak solutions of (1.1).
If p > 1 + 2α, the weak solution u of (1.1) is approximated by the unique solution u
N α,p s
(s ∈ (0,1)) of
(−∆)αu+up = 0 in B (e ),
1 N
(1.14)
u= δ in RN \B (e ).
−seN 1 N
When p ∈ [0,1 + 2α], we will prove that {u } blows up everywhere in B (e ) as s → 0+,
N s 1 N
therefore, we can deduce the nonexistence of weak solutions of (1.1) when p ≤ 1 + 2α. More
N
precisely, we have the following results.
Theorem 1.2 Assume that α ∈(0,1) and 0≤ p ≤ 1+ 2α. Then problem (1.1) does not have any
N
weak solution.
In the proof of Theorem 1.2, we will first need to prove the crucial estimate
up (x) ≥ c |x|−(N+2α), ∀x ∈ C,
α,p 2
whereC = {x ∈ RN : ∃t∈ (0,1) s.t. |x−te | < t} is a cone in B (e ). We combine the symmetry
N 8 1 N
property and decreasing monotonicity in our proof of the nonexistence. This phenomena is due to
the nonlocal characteristic of fractional Laplacian that requires the functions to be in L1 (RN).
loc
Finally, our interest is to study the asymptotic behavior of u as α goes to 1−.
α,p
Theorem 1.3 Assume that α ∈ (0,1), p ≥ N+1 and u is the unique weak solution of problem
N−1 α,p
(1.1). Then {u } vanishes as α→ 1−.
α,p α
Remark 1.2 (i) for p ≥ N+2, a sequence of barrier functions, which converge to 0 locally in B (0),
N−2 1
could be constructed directly to control {u } ;
α,p α
(ii) for N+1 ≤ p < N+2, Theorem 3.1 in [15] is involved to control {u } ;
N−1 N−2 α,p α
(iii) for p < 1+ 2, there exists α ∈ (0,1) such that p ≤ 1+ 2α for α ∈ (α ,1), there is no weak
N p N p
solution for problem (1.1) from Theorem 1.2;
(iv) for p ∈ [1+ 2, N+1), it is still open for the limit of {u } as α → 1−.
N N−1 α,p α
In Section 2, we treat the problem (1.14). When the Dirac mass concentrates at point −se
N
away from Ω¯, we buildthe existence, uniqueness weak solution u of (1.14) and show how the Dirac
s
mass is transformed into the nonhomogeneous term. In this case, the test functions space could be
improved into X , since the solution has no singularity in Ω¯.
α,B1(eN)
InSection 3, wegive adetailed account oftheprocedureenablingustomove thesingularpoints
{−se } to the origin. The first difficulty arises from the fact that G [Γ ] blows up everywhere as
N α s
s → 0+ [see Lemma 3.1], that is, there is no solution of
(−∆)αu= 0 in B (e ),
1 N
(1.15)
u= δ in RN \B (e ).
0 1 N
4
Therefore, we have to resort a barrier function, that is the minimal classical solution of
(−∆)αu+up = Γ in B (e ),
0 1 N
u = 0 in Bc(e )\{0}.
1 N
In order to control the limit of {u } near ∂B , especially near the origin, some typical truncated
s 1
functions have to be constructed carefully. The second difficulty comes from the proof of the
uniqueness. We proceed by contradiction, assuming that there is two solutions and we will show
their difference could be improved the test function from C∞ into X , this enables us to use
0 α,B1(eN)
Kato’s inequality [9, Proposition 2.4] and to conclude.
Section 4 is devoted to blow-up case. The difficulty is to obtain the blow-up everywhere in
B (e ) just from a lower bounds of u , see Lemma 4.1. To overcome it, we combine the symmetric
1 N s
of the domain and resort the symmetry result of u and then one point blowing up leads to blowing
s
up every where in B (e ).
1 N
Finally, we analyse decay approximation of the weak solution for problem (1.1) when α → 1−.
For p ≥ N+2, thefirstchallenge is to construct a sequence upperboundsthat converges to zero. To
N−2
thisend,wehavetostudylim (−∆)αΦ , whereΦ (x) = |x|−σ andthenuseproperparameters
α→1− σ σ
to construct the bounds. For N+1 ≤ p ≤ N+2, Φ could not be used to construct properly the
N−1 N−2 σ
upper bounds, and then we use some argument of [15].
2 Dirac mass concentrated at {−se } with s ∈ (0,1)
N
The purpose of this section is to introduce some preliminaries. First let us state an important
Comparison Principle.
Theorem 2.1 [6, Theorem 2.3] Let u and v be super-solution and sub-solution, respectively, of
(−∆)αu+h(u) = f in O,
where O is an open, bounded and connected domain of class C2, the function f : O → R is
continuous and h :R → R is increasing.
Suppose that v(x) ≤ u(x), ∀x∈ Oc, u and v are continuous in O¯. Then
u(x) ≥ v(x), ∀x ∈ O.
Now we investigate the weak solution of
(−∆)αu+up = 0 in B (e ),
1 N
(2.1)
u= δ in RN \B (e ),
−seN 1 N
where s ∈ (0,1). To this end, we construct a sequence of C2 functions to approximate the Dirac
measure. Let g : RN → [0,1] be a radially symmetric decreasing C2 function with the support in
0
B (0) such that g (x)dx = 1. For any n ∈ N and s ∈ (0,1), we denote
1 RN 0
2
R g (x) = nNg (n(x+se )), ∀x ∈RN.
n 0 N
Then we certainly have that
g ⇀ δ as n → +∞,
n −seN
in the distribution sense and for any s > 0, there exists N > 0 such that for any n ≥ N ,
s s
supp(gn)⊂ Bs(−seN).
2
In order to investigate the solution of (2.1), we consider the approximating solution w of
n
(−∆)αu+up = 0 in B (e ),
1 N
(2.2)
u= g in RN \B (e ).
n 1 N
5
Lemma 2.1 Assume that p > 0 and {g } is a sequence of C2 functions converging to δ with
n −seN
supports in Bs(−seN). Denote that
2
g (y)
n
g˜ (x) := c dy, ∀x ∈B (e ). (2.3)
n N,α RN |x−y|N+2α 1 N
Z
Then problem (2.2) admits a unique solution w such that
n
0 < w ≤ G [g˜ ] in B (e ).
n α n 1 N
Moreover, the function w˜ := w χ is the unique solution of
n n B1(eN)
(−∆)αu+up = g˜ in B (e ),
n 1 N
(2.4)
u = 0 in RN \B (e ).
1 N
Proof. The existence and uniqueness of solution to problem (2.2) refers to [6, Theorem 2.5]. For
n ≥Ns, we have that supp(gn) ⊂ Bs(−seN) and then g˜n ∈ C1(B1(eN)) and
2
w˜ = w −g in RN.
n n n
By the definition of fractional Laplacian, it implies that
(−∆)αw˜ (x)+w˜ (x)p = (−∆)αw (x)−(−∆)αg (x)+w (x)p
n n n n n
g (z)
n
= c dz = g˜ (x).
N,α RN |z−x|N+2α n
Z
Therefore, w˜ is a classical solution of (2.4) and
n
w˜ ≤ G [g˜ ] in B (e ),
n α n 1 N
which implies that
w ≤ G [g˜ ] in B (e ).
n α n 1 N
The proof ends. (cid:3)
We remark that w˜ is the classical solution of (2.4), then by Lemma 2.1 and Lemma 2.2 in [9],
n
we have that
[w (−∆)αξ+wpξ]dx = ξg˜ dx, ∀ξ ∈ C∞(B (e )). (2.5)
n n n 0 1 N
ZB1(eN) ZB1(eN)
Here (2.5) holds even for ξ ∈ X .
α,B1(eN)
Lemma 2.2 Let {g˜ } be defined in (2.3), then g˜ converges to Γ uniformly in B (e ) and in
n n s 1 N
Cθ(B (e )) for θ ∈ (0,1), where
1 N
c
Γ (x) = N,α , ∀x ∈ RN \{−se }. (2.6)
s |x+se |N+2α N
N
Proof. It is obvious that g˜ converges to Γ every point in B (e ). For x,y ∈ B (e ) and any
n s 1 N 1 N
n ∈N, we have that
1 1
|g˜ (x)−g˜ (y)| = c | [ − ]g (z)dz|
n n N,α |x−z|N+2α |y−z|N+2α n
ZBs(−seN)
2
||x−z|N+2α−|y−z|N+2α|
≤ c g (z)dz
N,α |x−z|N+2α|y−z|N+2α n
ZBs(−seN)
2
|x−z|N+2α−1 +|y−z|N+2α−1
≤ c (N +2α)|x−y| g (z)dz
N,α |x−z|N+2α|y−z|N+2α n
ZBs(−seN)
2
≤ c |x−y| g (z)dz
3 n
ZBs(−seN)
2
= c |x−y|,
3
6
where c > 0 independent of n. So {g˜ } is uniformly bounded in C0,1(B (e )). Combining the
3 n n 1 N
converging
g˜ → Γ every point in B (e ).
n s 1 N
We conclude that g˜ converges to Γ uniformly in B (e ) and in Cθ(B (e )) for θ ∈ (0,1). (cid:3)
n s 1 N 1 N
Proposition 2.1 Assume that p > 0, s ∈ (0,1) and Γ is given by (2.6). Then problem (2.1)
s
admits a unique weak solution u such that
s
0 ≤ u (x)≤ G [Γ ], x ∈ B (e ). (2.7)
s α s 1 N
Moreover, u˜ := u χ is the unique classical solution of
s s B1(eN)
(−∆)αu+up = Γ in B (e ),
s 1 N
(2.8)
u= 0 in RN \B (e ).
1 N
Proof. Existence. It infers by Lemma 2.1 that the solution w of (2.2) satisfies that
n
0 < w ≤ G [g˜ ] in B (e ). (2.9)
n α n 1 N
By Lemma 2.2 we have that g˜ converges to Γ uniformly in B (e ) and in Cθ(B (e )) with
n s 1 N 1 N
θ ∈ (0,1). Therefore, there exists some constant c > 0 independent of n such that
4
c c
G [g˜ ](x) ≤ 4 N,α ≤ c c s−N−2α, ∀x ∈ B (e ).
α n |x+se |N+2α 4 N,α 1 N
N
Thus,
kw k ≤ c c s−N−2α, kw k ≤ c c s−N−2α|B (e )|.
n L∞(B1(eN)) 4 N,α n L1(B1(eN)) 4 N,α 1 N
By [28, Theorem 1.2], we have that
kwnk ≤ c [kwpk +kΓ k ]
ρα Cα(B1(eN)) 5 n L∞(B1(eN)) s L∞(B1(eN))
(2.10)
≤ c [s−N−2α+s−(N+2α)p]
6
for some c ,c > 0.
5 6
In order to see the inner regularity, we denote O the open sets with i= 1,2,3 such that
i
O ⊂ O¯ ⊂ O ⊂ O¯ ⊂ O ⊂ O¯ ⊂ B (e ).
1 1 2 2 3 3 1 N
By [11, Lemma 3.1], for β ∈ (0,α), there exists c ,c > 0 independent of n such that
7 8
p
kw k ≤c [kw k +kw k +kw k ]
n Cβ(O2) 7 n L1(B1(eN)) n L∞(O3) n L∞(O3)
≤c [s−N−2α+s−(N+2α)p].
8
It follows by [28, Corollary 2.4] that there exist c ,c > 0 such that
9 10
p
kw k ≤ c [kw k +kw k +kw k ]
n C2α+β(O1) 9 n L1(B1(eN)) n Cβ(O2) n Cβ(O2)
(2.11)
≤ c [s−N−2α+s−(N+2α)p].
10
Therefore, by the Arzela-Ascoli Theorem, there exist u ∈ C2α+ǫ in B (e ) for some ǫ ∈ (0,β)
s loc 1 N
and a subsequence {w } such that
nk
w → u locally in C2α+ǫ as n → ∞. (2.12)
nk s k
7
Passing the limit of (2.5) with ξ ∈ X as n → ∞, we have that
α,B1(eN) k
[u (−∆)αξ+upξ]dx = ξ(x)Γ (x). (2.13)
s s s
ZB1(eN) ZB1(eN)
Moreover, since w → u and g˜ → Γ uniformly in B (e ) as n → ∞, then it infers that
n s n s 1 N
0 ≤ u ≤ G [Γ ] in B (e ).
s α s 1 N
Uniqueness. Let v be a weak solution of (2.2) and then ϕ := u −v is a weak solution to
s s s s
(−∆)αϕ +up−vp = 0 in B (e ),
s s s 1 N
ϕ = 0 in RN \B (e ).
s 1 N
By Kato’s inequality [9, Proposition 2.4],
|ϕ |(−∆)αξ+ [up−vp]sign(u −v )ξdx = 0.
s s s s s
ZB1(eN) ZB1(eN)
Taking ξ = G [1], we have that
α
[up−vp]sign(u −v )ξdx ≥ 0 and |ϕ |dx = 0,
s s s s s
ZB1(eN) ZB1(eN)
then ϕ = 0 a.e. in B (e ). Then the uniqueness is proved.
s 1 N
Furthermore, we see that w˜ = w −g is the unique classical solution of
n n n
(−∆)αu(x)+up(x) = g˜ (x), ∀x ∈ B (e ),
n 1 N
(2.14)
u(x) = 0, ∀x ∈ B (e )c
1 N
and w˜ converges to u˜ uniformly in B (e ). By Stability Theorem [5, Lemma 4.5] and (2.12),
n s 1 N
u χ is the classical solution of (2.8). (cid:3)
s B1(eN)
3 Proof of Theorem 1.1.
In this section, we prove Theorem 1.1 by moving the points {−se } to the origin. To this end, we
N
need derive more properties for u , where u is the unique weak solution of (2.1).
s s
Lemma 3.1 Let p > 0, s ∈ (0,1) and u be the unique weak solution of (2.1). Then the mapping
s
s 7→ u is decreasing, that is,
s
u ≥ u if s ≤ s .
s1 s2 1 2
Proof. By Proposition 2.1, u˜ := u χ is the unique classical solution of (2.8).
s s B1(eN)
We claim that the mapping: s 7→ Γ is decreasing. For x ∈ B (e ) and s ≤ s , we observe
s 1 N 1 2
that |x+s e |≤ |x+s e |, then Γ (x) ≥ Γ (x). The claim is proved.
1 N 2 N s1 s2
Therefore, for s ≤ s , u and u are super solution and solution of (2.8) replaced Γ by Γ ,
1 2 s1 s2 s s2
then it infers by the Comparison Principle that u ≥ u in B (e ). (cid:3)
s1 s2 1 N
Lemma 3.2 Let s ∈ (0,1) and denote
G [Γ ](x) = G (x,y)Γ (y)dy, ∀x∈ B (e ).
α s α s 1 N
ZB1(eN)
Then
lim G [Γ ](x) = +∞, ∀x ∈ B (e ).
α s 1 N
s→0+
8
Proof. Using [8, Theorem 1.2], it follows that
c c ρα(x)ρα(y)
11 11
G (x,y) ≥ min , , x,y ∈ B (e ),
α |x−y|N−2α |x−y|N 1 N
(cid:26) (cid:27)
where c > 0 dependent of N,α. Now for x ∈ B (e ) and y ∈ B (e )∩B (0), we have that
11 1 N 1 N |x|
4
c ρα(x)ρα(y)
11
G (x,y) ≥
α |x−y|N
and
c c ρα(x)ρα(y) c
G [Γ ](x) ≥ min 11 , 11 N,α dy
α s |x−y|N−2α |x−y|N |y+se |N+2α
ZB1(eN) (cid:26) (cid:27) N
c ρα(x)ρα(y) c
11 N,α
≥ dy
|x−y|N |y+se |N+2α
ZB1(eN)∩B|x|(0) N
4
4 c ρα(y)
≥ c ρα(x)|x|−N N,α dy
5 11 |y+se |N+2α
ZB1(eN)∩B|x|(0) N
4
→ +∞ as s→ 0+.
The proof ends. (cid:3)
From Lemma 3.2, it is informed that the limit of G [Γ ] as s → 0+ can’t be used as a barrier
α s
function to control the sequence {u }. So we have to find new upper bound for sequence {u }.
s s
Proposition 3.1 Let Γ be defined in (1.10) and
0
2α
p > 1+ , (3.1)
N
then problem
(−∆)αu+up = Γ in B (e ),
0 1 N
(3.2)
u= 0 in Bc(e )\{0}.
1 N
admits a minimum positive solution u , that is, u ≤ u for any nonnegative solution u of (3.2).
0 0
Moreover,
lim u (x) =0 (3.3)
0
x∈B1(eN), x→∂B1(eN)\{0}
and
1 −N+2α −N+2α
t p ≤ u0(teN)≤ c12t p , t ∈ (0,1), (3.4)
c
12
where c > 1 is independent of α.
12
Proof. The existence of solution to (3.2). It implies by Lemma 3.2 that the mapping s 7→ u is
s
decreasing in B (e ), where u χ is the solution of (2.8). So what we have to do is just to
1 N s B1(eN)
find a super solution U of (3.2) such that u ≤ U in B (e ). To this end, we consider the radial
0 1 N
function
1
Φ (x) = , ∀x ∈ RN \{0}, (3.5)
σ |x|σ
where σ ∈ [0, N). By scaling property of Φ , (also see [13]) we know that
σ
c(σ,α)
(−∆)αΦ (x) = , (3.6)
σ |x|σ+2α
9
where c(σ,α) ∈ R. Now we choose σ = σ = N+2α, then σ ∈ (0, N) if p > 1+ 2α. Therefore,
0 p 0 N
there exist some k > 1 dependent of |c(σ ,α)| and c but independent of n such that
0 N,α
U(x) = kΦ (x) (3.7)
σ0
is a super solution of (3.2). Thus, U ∈ L1(B (e )) and by Theorem 2.1, we have that
1 N
0 ≤ u ≤ U for any s ∈ (0,1). (3.8)
s
For any x ∈ RN \ {0}, u (x) := lim u (x) ≤ U(x) < +∞. Following the same argument
0 s→0+ s
of Proposition 2.1, we can prove that u is a classical solution of (3.2). Furthermore, u is the
0 0
minimum solution of (3.2).
Proof of (3.3). Let x¯ ∈ ∂B (e )\{0}, K = ∂B (e )∩B (x¯) and K = ∂B (e )∩Bc (x¯).
1 N 1 1 N |x¯|/8 2 1 N |x¯|/2
Let O be an open and C2 set such that
B (e )∩B (x¯)⊂ O ⊂ B (e )∩B (x¯).
1 N |x¯|/4 1 N |x¯|/2
Then we see that
K ⊂ ∂O and ∂O∩K = ∅.
1 2
We would like to find a super solution of (3.2) in O with vanishing boundary value in K for
1
any n. Denote
V = Uη+λV ,
λ O
where U is defined (3.7), η is a C2 function such that
1 if x ∈ Bc (x¯),
|x¯|/4
η(x) =
(0 if x ∈ B (x¯).
|x¯|/8
and V is the solution of
O
(−∆)αu= 1 in O,
u= 0 in Oc.
Since U(1−η) is C2 in RN, then there exists c > 0 such that
13
|(−∆)αU(1−η)| ≤ c in O¯.
13
Thus, there exists c > 0 such that |(−∆)αUη| ≤ c in O¯. Choosing λ > 0 suitable, we have
14 14 0
that for λ ≥ λ ,
0
c
(−∆)αV +Vp ≥ N,α in O.
λ λ |x|N+2α
Moreover, since η = 1 in RN \O, then V ≥ U ≥ u in RN \O. By the Comparison Principle, we
λ s
have that for any s ∈ (0,1),
u ≤ V in B (e ). (3.9)
s λ 1 N
which implies (3.3).
Proof of (3.4). For t ∈ (0,1), denote that
Vt(x) = cNp1,αt−N+p2αVB x−tteN , x ∈ RN,
(cid:18) (cid:19)
where V is the solution of
B
(−∆)αu= 1 in B (0),
1
(3.10)
u= 0 in RN \B (0).
1
10