The Brezis-Nirenberg problem for fractional elliptic operators

0.27 MB·

by  Ko-Shin Chen

THE BREZIS-NIRENBERG PROBLEM FOR FRACTIONAL
ELLIPTIC OPERATORS
6
1 KO-SHINCHEN,MARCOSMONTENEGRO,ANDXIAODONGYAN
0
2
Abstract. LetL=div(A(x)∇)beauniformlyellipticoperatorindivergence

n forminaboundedopensubsetΩofRn. WestudytheeffectoftheoperatorL
a ontheexistenceandnonexistenceofpositivesolutionsofthenonlocalBrezis-
J Nirenbergproblem

8
 (−L)su = unn−+22ss +λu inΩ,
]  u = 0 onRn\Ω
P
A where(−L)sdenotesthefractionalpowerof−LwithzeroDirichletboundary
. values on ∂Ω, 0 < s < 1, n > 2s and λ is a real parameter. By assuming
h A(x)≥A(x0)forallx∈Ωand A(x)≤A(x0)+|x−x0|σIn near somepoint
t
a x0∈Ω,weproveexistencetheoremsforanyλ∈(0,λ1,s(−L)),whereλ1,s(−L)
m denotesthefirstDirichleteigenvalueof(−L)s. Ourexistenceresultholdstrue
[ forσ>2sandn≥4sintheinteriorcase(x0∈Ω)andforσ> 2sn(n−−42ss) and
n>4sintheboundarycase(x0∈∂Ω). Nonexistenceforstar-shapeddomains
1 isobtained foranyλ≤0.
v
6
6
7
1
0 1. Introduction and statements
.
1 A lot of attention has been paid to a number of counterparts of the Brezis-
0 Nirenberg problem since the pioneer paper [5] which consists in determining all
6
values of λ for which the problem
1
:
v
i −∆u = unn−+22 +λu in Ω,
X (1.1)
u = 0 on ∂Ω
r (cid:26)
a
admits a positive solution, where Ω is a bounded open subset of Rn, n ≥ 3 and λ
is a real parameter.
According to [5], the problem (1.1) admits a positive solution for any λ ∈
(0,λ (−∆)) provided that n ≥ 4, where λ (−∆) denotes the first Dirichlet eigen-
1 1
value of −∆ on Ω. Moreover, the problem has no such solution for n≥ 3 if either
λ≥ λ (−∆) or λ ≤ 0 and Ω is a star-shaped C1 domain. When n = 3 and Ω is a
1
ball, a positive solution of (1.1) exists if, and only if, λ∈(1λ (−∆),λ (−∆)).
4 1 1
TheBrezis-Nirenbergproblemforuniformlyellipticoperatorsindivergenceform
has been studied in the works [14,16,18]. Precisely, consider the problem
Date:January11,2016.
2000 Mathematics Subject Classification. Primary 35A01,35J20.
Keywordsandphrases. fractionalellipticoperator,criticalexponent,existence,nonexistence.
1
2 KO-SHINCHEN,MARCOSMONTENEGRO,ANDXIAODONGYAN
n+2
(1.2) −Lu = un−2 +λu in Ω,
u = 0 on ∂Ω
(cid:26)
where L = div(A(x)∇). Assume that A(x) = (a (x)) is a positive definite sym-
ij
metricmatrixforeachx∈ΩwithcontinuousentriesonΩ,sothatLisaselfadjoint
uniformly elliptic operator. Assume also there exist a point x ∈Ω and a constant
0
C >0 such that
0
(1.3) A(x)≥A(x ) for every x∈Ω
0
and
(1.4) A(x)≤A(x )+C |x−x |σI locally around x
0 0 0 n 0
both in the sense of bilinear forms, where I denotes the n×n identity matrix.
n
In [16], Egnell focused on the interior case (x ∈Ω) and provedthat problem (1.2)
0
admits a positive solution for any λ∈(0,λ (−L)) provided that n≥4 and σ >2,
1
where λ (−L) denotes the first Dirichlet eigenvalue of −L on Ω. The boundary
1
case (x ∈ ∂Ω) has recently been treated in [18], which proves the existence of a
0
positive solution for any λ ∈ (0,λ (−L)) provided that n > 4, σ > 2n−4 and the
1 n−4
boundary of Ω is α-singular at x , with α∈[1,σ n−4 ), in the following sense:
0 2n−4
Definition 1. The boundary of an open subset Ω of Rn is said to be α-singular at
the point x , with α ≥ 1, if there exist a constant δ > 0 and a sequence (x ) ⊂ Ω
0 j
such that x →x as j →+∞ and B(x ,δ|x −x |α)⊆Ω.
j 0 j j 0
The nonexistence ofpositive solutionsof(1.2)for λ≤0 onstar-shapeddomains
has been proved in [16] (see also [14]) by assuming a ∈ C1(Ω\{x }) such that
ij 0
a′ (x) := ∇a (x)·(x−x ) extends continuously to x and A′(x) = (a′ (x)) is
ij ij 0 0 ij
positivesemi-definite foreveryx∈Ω. Nonexistence ofpositive solutionin the case
λ≥λ (−L) follows from a standard argument.
1
When 0 < s < 1, Caffarelli and Silvestre [8] introduced the characterization
of the fractional power of the Laplace operator (−∆)s in terms of a Dirichlet-to-
Neumannmapassociatedto a suitable extensionproblem. Since then, a greatdeal
of attention has been dedicated in the last years to nonlinear nonlocal problems
involving this operator. See for example [2,3,7,10,12,13,25,26], among others.
Two of them (see [25] for s = 1 and [2] for other values of s ∈(0,1)) consider the
2
following counterpart of the Brezis-Nirenberg problem
(1.5) (−∆)su = unn−+22ss +λu in Ω,
u = 0 in Rn\Ω
(cid:26)
In particular, it has been proved that problem (1.5) admits a positive viscosity
solution for any λ ∈(0,λ (−∆)) provided that n≥ 4s, where λ (−∆) denotes
1,s 1,s
thefirstDirichleteigenvalueof(−∆)s onΩ. Moreover,thereexistsnosuchsolution
in C1(Ω) for n > 2s if either λ ≤ 0 and Ω is a star-shaped C1 domain or λ ≥
λ (−∆).
1,s
This work dedicates special attention to the effect of the elliptic operator L (or
ofthematrixA(x))ontheexistenceandnonexistenceofpositiveviscositysolutions
of the following Brezis-Nirenberg problem involving the fractional power of −L,
THE BREZIS-NIRENBERG PROBLEM FOR FRACTIONAL ELLIPTIC OPERATORS 3
(−L)su = unn−+22ss +λu in Ω,
(1.6)
u = 0 in Rn\Ω
(cid:26)
Denote by λ (−L) the first Dirichlet eigenvalue of (−L)s on Ω.
1,s
The main existence theorems are:
Theorem 1. (Interior case) Let 0 < s < 1 and Ω ⊂ Rn be a bounded open set.
Assume entries of the matrix A are continuous in Ω and A satisfies (1.3), (1.4)
for some x ∈ Ω. Then (1.6) admits at least one positive weak solution for any
0
λ ∈ (0,λ (−L)) provided n ≥ 4s and σ > 2s. If ∂Ω is of C1,1 class and each
1,s
entry of A(x) belongs to C1(Ω), then our weak solution u belongs to C0,α(Ω) if
0<s<1/2 and to C1,α(Ω) if 1/2≤s<1 for any 0<α<1.
Theorem 2. (Boundary case) Let 0 < s < 1 and Ω ⊂ Rn be a bounded open set.
Assume entries of the matrix A are continuous in Ω and A satisfies (1.3), (1.4)
for some x on ∂Ω. Suppose ∂Ω is α-singular at x . Then (1.6) admits at least
0 0
one positive weak solution for any λ ∈ (0,λ (−L)) if n > 4s, σ > 2s(n−2s) and
1,s n−4s
1≤α< σ(n−4s). If ∂Ω is of C1,1 class and each entry of A(x) belongs to C1(Ω),
2s(n−2s)
then our weak solution u belongs to C0,α(Ω) if 0 < s < 1/2 and to C1,α(Ω) if
1/2≤s<1 for any 0<α<1.
The nonexistence theorem states that
Theorem 3. Let 0 < s < 1, n > 2s, and Ω ⊂ Rn be a bounded open set which
is star-shaped with respect to some point x ∈ Ω and its boundary is of C1 class.
0
Assume matrix A satisfies (1.3)-(1.4) for some x in Ω, moreover, assume a ∈
0 ij
C1(Ω \ {x }) and a′ (x) := ∇a (x) · (x − x ) extends continuously to x and
0 ij ij 0 0
A′(x) = (a′ (x)) is positive semi-definite for every x ∈ Ω. Then (1.6) admits no
ij
positive solution in C1(Ω) for any λ ≤ 0. Furthermore, if ∂Ω is of C1,1 class and
each entry of A(x) belongs to C1(Ω), then (1.6) admits no positive weak solution
for any λ≤0 provided that s≥1/2.
Theorems 1 and 3 extend the existence and nonexistence results of [2] and [25],
since the constant matrix A(x) = I clearly fulfills our assumptions. All results
n
in [16,18] for s = 1 are also extended fully to any 0 < s < 1. One prototype
example of operator L is A(x)=A +|x−x |σI .
0 0 n
A natural approach for solving (1.6) consists in searching for minimizers of the
functional
u7→ |(−L)s/2u|2−λ u2dx
ZRn ZΩ
subject to
2n
|u|n−2sdx=1.
ZΩ
However, fractional integrals of this type are generally difficult to be handled
directly. On the other hand, fractional powers of elliptic operators in divergence
formwererecentlydescribedin[23]asDirichlet-to-Neumannmapsforanextension
problem in the spirit of the extension problem for the fractional Laplace operator
on Rn of [8]. In section 2, we take some advantage of this description and provide
4 KO-SHINCHEN,MARCOSMONTENEGRO,ANDXIAODONGYAN
an equivalent variational formulation which will be used in our proof of existence.
We also present an existence tool and a regularity result of weak solutions.
The existence of minimizers for the new constrained functional often relies on
the construction and estimates of suitable bubbles involving extremal functions
of Sobolev type inequalities. Although, this is a well known strategy, new and
important difficulties arise in present context. Indeed, the most delicate part in
the proof of Theorem 1 (the interior case) is caused by the term |x−x |σ in the
0
inequality (1.4), which essentially involves estimates of the multiple integral
∞
|x|σy1−2s|∇ w (x,y)|2dydx
x 1
ZBR(0)Z0
on the whole ball of radius R for R>0 large enough, where w (x,y) is given by
1
u (ξ)
w (x,y):=c y2s 1 dξ,
1 s ZRn (|x−ξ|2+y2)n+22s
with c being an appropriate normalization constant and
s
1
u (ξ)= .
1 (1+|ξ|2)n−22s
In Section 3, we estimate the integral mentioned above and, as a byproduct, we
prove Theorem 1. The bubbles used in the proof of Theorem 1 do not work in
the boundary case because we need to compare the least energy level to the cor-
responding best trace constant in Rn+1. The idea for overcoming this difficulty is
+
to consider suitable bubbles concentrated in interior points converging fast to the
boundary point x in an appropriate way. The construction depends on the order
0
α of the singularity of the boundary at x . In Section 4 we introduce such bubbles
0
and derive the necessary estimates in the boundary case. Proofof Theorem2 then
follows. Finally, in Section 5, we establish a Pohozaev identity for C1 solutions of
(1.6) and use it to prove Theorem 3.
2. The variational framework and main tools
For the precise definition of the fractional power of the selfadjoint elliptic oper-
ator −L, we consider an orthonormal basis of L2(Ω) consisting of eigenfunctions
φ ∈ H1(Ω), k = 1,2,···, that correspond to eigenvalues λ < λ ≤ ···. The
k 0 1 2
domain of (−L)s, denoted here by Hs, 0 < s < 1, is defined as the Hilbert space
of functions u = ∞ c φ ∈ L2(Ω) such that ∞ λsc2 < ∞ endowed with
k=1 k k k=1 k k
the inner product hu,vi := ∞ λsc d , where v = ∞ d φ . For each
u = ∞ c φ ∈PHs, wHesdefine (−k=L1)skuk=k cPλsϕ . Withk=t1hiskdkefinition, we
k=1 k k P k k k k P
have
P P
s s
hu,viHs =h(−L)2 u,(−L)2 viL2(Ω).
It is well known that (see details in [9,21])
Hs(Ω), if 0<s<1/2,
Hs = H1/2(Ω), if s=1/2,
 00
 H0s(Ω), if 1/2<s<1

THE BREZIS-NIRENBERG PROBLEM FOR FRACTIONAL ELLIPTIC OPERATORS 5
The spaces Hs(Ω) and Hs(Ω), s 6= 1/2, are the classical fractional Sobolev spaces
0
given as the completion of C∞(Ω) under the norm
0
kuk2 =kuk2 +[u]2 ,
Hs(Ω) L2(Ω) Hs(Ω)
where
(u(x)−u(y))2
[u]2 = dxdy.
Hs(Ω) |x−y|n+2s
ZΩZΩ
The space H1/2(Ω) is the Lions-Magenes space which consists of functions u ∈
00
L2(Ω) such that [u]2 <∞ and
H1/2(Ω)
u(x)2
dx<∞.
dist(x,∂Ω)
ZΩ
The Hilbert space Hs is compactly embedded in Lq(Ω) for any 1 ≤ q < 2n
n−2s
2n
andcontinuouslyinLn−2s(Ω)providedthatn>2s. So,anaturalstrategytosolve
(1.6) in a weak sense consists in searching minimizers of
(2.1) IA(u)= (−L)2s u 2dx−λ u2dx
λ
ZΩ(cid:12) (cid:12) ZΩ
constrained to the Nehari manifold(cid:12) (cid:12)
(cid:12) (cid:12)
E = u∈Hs : |u|n2−n2sdx=1 .
(cid:26) ZΩ (cid:27)
In fact, the functional IA is well defined on Hs and its least energy level on E,
λ
denoted by
SA := inf IA(u) ,
λ λ
u∈E
is finite. Moreover,a minimizer u∈Hs satisfies
(2.2) (−L)2s u(−L)2s ζdx= SλA|u|n−4s2su+λu ζdx
ZΩ ZΩ(cid:16) (cid:17)
for every ζ ∈ Hs. In particular, if SA is positive and u is nonnegative in Ω, then
λ
u is a nonnegativeweak solutionof (1.6). On the other hand, using the variational
characterizationof the first Dirichlet eigenvalue of (−L)s given by
s 2
(−L)2 u dx
Ω
λ (−L)= inf ,
1,s u∈H\{0}R (cid:12)(cid:12) Ωu2dx(cid:12)(cid:12)
(cid:12) (cid:12)
one can easily check that the positivity of SA is equivalent to λ < λ (−L). So,
λ R 1,s
the conclusion of Theorems 1 and 2 follows if we are able to prove the existence
and regularity of nonnegative minimizers of IA in E for any λ>0.
λ
Inspired by the recent work in [23], an equivalent definition for the operator
(−L)s in Ω with zero Dirichlet boundary condition can be formulated as an exten-
sion problem in a cylinder. Let C = Ω×(0,∞) ⊂ Rn+1. We denote the points
Ω +
in C by z = (x,y) with x ∈ Ω and the lateral boundary ∂Ω×[0,∞) by ∂ C .
Ω L Ω
Then for u∈Hs, we define the (s,A)-extension w =EA(u) as the solution to the
s
problem
6 KO-SHINCHEN,MARCOSMONTENEGRO,ANDXIAODONGYAN
div y1−2sB(x)∇w(x,y) = 0 in C ,
Ω
(2.3) w = 0 on ∂ C ,
 (cid:0) (cid:1) L Ω
w = u on Ω×{y =0}.

A(x) 0
HereB(x)= isan(n+1)×(n+1)matrix. Theextensionfunction
0 1
(cid:18) (cid:19)
belongs to the space
H0s,,LA(CΩ)=C0∞(Ω×[0,∞))k·kH0s,,LA
with
1/2
kwk = c y1−2s(∇w)T B(x)∇w dxdy .
Hs,A s
0,L (cid:18) ZZCΩ (cid:19)
Here c is a normalization constant such that EA : Hs(Ω) → Hs,A(C ) is an
s s 0,L Ω
isometry between Hilbert spaces. In particular,
EAu =kuk
s Hs,A Hs
0,L
for every u∈Hs. (cid:13) (cid:13)
(cid:13) (cid:13)
For A(x) = I , EA(u) is the canonical s-harmonic extension of u, see [8]. In
n s
this case, we denote EA(u) by E (u) and the extension function space Hs,A(C )
s s 0,L Ω
is denoted by Hs (C ).
0,L Ω
It is known from [23] that the extension function w satisfies
∂w
(2.4) −c lim y1−2s (x,y)=(−L)su(x)
s
y→0+ ∂y
for every x∈Ω.
Using the extension map EA, we can reformulate the problem (1.6) as
s
−div y1−2sB(x)∇w = 0 in C
Ω,
(2.5) w = 0 on ∂CΩ,
 (cid:0) ∂νsw(cid:1) = wnn−+22ss +λw in Ω×{y =0}.
Here

∂w
∂sw :=−c lim y1−2s .
ν sy→0+ ∂y
Using that the trace of functions in Hs,A(C ) is compactly embedded in Lq(Ω)
0,L Ω
for any 1 ≤ q < 2n and continuously in Ln−2n2s(Ω) for n > 2s, we can consider
n−2s
the minimization of functional
(2.6) JA(w)=c y1−2s(∇w)T B(x)∇w dxdy−λ w2(x,0)dx
λ s
ZZCΩ ZΩ
in the admissible set
F = w ∈H0s,,LA(CΩ): |w(x,0)|n−2n2sdx=1 .
(cid:26) ZΩ (cid:27)
Clearly, we have
SA = inf JA(w) .
λ λ
w∈F
THE BREZIS-NIRENBERG PROBLEM FOR FRACTIONAL ELLIPTIC OPERATORS 7
Moreover, u ∈ E is a minimizer of IA on E if, and only if, w = EA(u) ∈ F is a
λ s
minimizer of JA on F.
λ
There are essentially two advantages in considering the minimization problem
for JA. Firstly, it follows directly that |w| ∈ F and JA(|w|) = JA(w) for every
λ λ λ
w∈F,sothatminimizersofJA onF canbe assumednonnegativeinΩ. Secondly,
λ
the integral
y1−2s(∇w)T B(x)∇w dxdy
ZZCΩ
is more easily to be handled comparing to the one in (2.1). Therefore, from now
on we will concentrate on the existence of minimizers of JA in F.
λ
One of the tool used in our existence proof is the following trace inequality
2
r
(2.7) |f(x,0)|rdx ≤A y1−2s|∇f(x,y)|2dxdy
(cid:18)ZΩ (cid:19) ZCΩ
for 1 ≤ r ≤ 2n , n > 2s and every f ∈ Hs(C ). When r = 2n , we denote the
n−2s 0 Ω n−2s
best constantin (2.7)by K (n). This constantis not achievedin bounded domain
s
and achieved when Ω=Rn and f =E (u) with
s
εn−22s
(2.8) u(x)= .
n−2s
|x|2+ε2 2
(cid:16) (cid:17)
By a change of variable argument, we have
n−2s
2n n
(2.9) |f(x,0)dx|n−2s
(cid:18)ZΩ (cid:19)
≤det(A(x0))−ns Ks(n) y1−2s(∇f(x,y))T B(x0)∇f(x,y)dxdy
ZCΩ
for every f ∈Hs(C ).
0 Ω
Thefollowingpropositionstatesanecessaryconditionforexistenceofminimizer
of IA in E.
λ
Proposition 1. Let n>2s. Assume there exists a point x ∈Ω such that (1.3) is
0
satisfied and
(2.10) SλA <csdet(A(x0))ns Ks(n)−1.
Then, the infimum SA of IA in E is achieved by some nonnegative function u.
λ λ
Furthermore, if λ < λ (−L), then u is a nonnegative weak solution of (1.6),
1,s
module a suitable scaling.
Proof. As noted above,it suffices to prove that the infimum of JA in F, given also
λ
by SA, is assumed by some nonnegative function w.
λ
Let {w } ⊂ Hs,A(C ) be a minimizing sequence of JA on F. Clearly, {w } is
m 0,L Ω λ m
bounded in Hs,A(C ). Siince Ω is bounded, up to a subsequence, we have
0,L Ω
8 KO-SHINCHEN,MARCOSMONTENEGRO,ANDXIAODONGYAN
w ⇀w weakly in Hs,A(C ),
m 0,L Ω
2n
w (·,0)→w(·.0) strongly in Lq(Ω) for 1≤q < ,
m
n−2s
w (·,0)→w(·,0) a.e. in Ω.
m
A direct calculation, taking into account of the weak convergence, gives
kw k2 = kw −wk2 +kwk2
m Hs,A m Hs,A Hs,A
0,L 0,L 0,L
+c y1−2s ∇wTB(x)∇(w −w)+∇T (w −w)B(x)∇w dxdy
s m m
ZZCΩ
= kw −wk2 +(cid:0) kwk2 +o(1). (cid:1)
m Hs,A Hs,A
0,L 0,L
Now using Brezis-Lieb Lemma (see [4]), (1.3) and (2.7), we obtain
SA = kw k2 −λkw (·,0)k2 +o(1)
λ m Hs,A m L2
0,L
= kw −wk2 +kwk2 −λkw(·,0)k2 +o(1)
m Hs,A Hs,A L2
0,L 0,L
≥ c y1−2s(∇(w −w))T B(x )∇(w −w)dxdy+JA(w)+o(1)
s m 0 m λ
ZZCΩ
≥ csdet(A(x0))ns Ks(n)−1k(wm−w)(·,0)k2 2n +JλA(w)+o(1)
Ln−2s
n−2s
≥ csdet(A(x0))ns Ks(n)−1 1− |w(x,0)|n−2n2sdx n
(cid:18) ZΩ (cid:19)
n−2s
+SλA |w(x,0)|n−2n2sdx n +o(1)
(cid:18)ZΩ (cid:19)
so that
n−2s n−2s
SλA 1− |w(x,0)|n2−n2sdx n ≥csdet(A(x0))ns Ks(n)−1 1− |w(x,0)|n−2n2sdx n .
(cid:18)ZΩ (cid:19) ! (cid:18) ZΩ (cid:19)
So, using the assumption (2.10) and the fact that
2n
|w(x,0)|n−2sdx≤1,
ZΩ
we derive
2n
|w(x,0)|n−2sdx=1.
ZΩ
But this implies that
2n
wm(·,0)→w(·.0) strongly in Ln−2s (Ω).
Then, w ∈F and by lower semicontinuity of JA,
λ
JA(w)≤lim inf JA(w ) ,
λ λ m
m→∞
so that JA(w)=SA. Therefore, |w| is a nonnegative minimizer of JA in F.
λ λ λ
THE BREZIS-NIRENBERG PROBLEM FOR FRACTIONAL ELLIPTIC OPERATORS 9
The remainder of the proof is direct because u=|w|(·,0) is a nonnegative min-
imizer of IA on E and the inequality λ < λ (−L) is equivalent to the positivity
λ 1,s
of SA. (cid:3)
λ
The discussionmade sofar aboutthe existence ofnonnegativeweaksolutions of
(1.6) can be resumed in the following remark:
Remark 1. Proposition 1 provides the existence of a nonnegative weak solution
of (1.6) by assuming the conditions (1.3), (2.10) and λ < λ (−L). Up to a
1,s
scaling this solution can be seen as the trace of a nonnegative minimizer of QA on
λ
Hs,A(C )\{0}, where QA denotes the Rayleigh quotient
0,L Ω λ
kwk2 −λkw(·,0)k2
Hs,A L2
QA(w)= 0,L .
λ kw(·,0)k2
2n
Ln−2s
Note that (2.10) is equivalent to existence of a function w ∈ Hs,A(C )\{0} so
0 0,L Ω
that
(2.11) QAλ (w0)<csdet(A(x0))ns Ks(n)−1.
So, in light of Proposition 1, Sections 3 and 4 are dedicated to the construction of
w by using the remaining assumptions assumed in Theorems 1 and 2, respectively.
0
The next proposition shows further regularity of weak solution of (1.6).
Proposition 2. Let u∈Hs\{0}be a nonnegative weak solution of (1.6). Assume
that A(x) is a positive definite symmetric matrix for each x ∈ Ω with continuous
entries on Ω. Then u∈Lp(Ω) for every p≥1. Furthermore, if ∂Ω is of C1,1 class
and each entry of A(x) ∈ C1 Ω then u belongs to C0,α(Ω) if 0 < s < 1/2 and to
C1,α(Ω) if 1/2≤s<1 for any 0<α<1.
(cid:0) (cid:1)
Proof. The function w =EA(u) satisfies (2.5). Since u is assumed to be nonnega-
s
tive, w is also nonnegative. For each k ≥1, we define w by
k
w (x,y):=min{w(x,y),k}.
k
Since ww2β is in Hs,A(C ) for all β ≥ 0, using it as a test function in (2.5), we
k 0,L Ω
obtain
(2.12) f(u)uu2β dx
k
ZΩ
= y1−2s(∇w)TB(x)∇(ww2β)dxdy
k
ZZCΩ
= y1−2sw2β(∇w)TB(x)∇w+2β y1−2sw2β(∇w )TB(x)∇w dxdy,
k k k k
ZZCΩ
n+2s
where f(u) := un−2s +λu. Note that the last equality comes from the fact the
w=w in the set where w ≤k and w is a constant otherwise.
k k
On the other hand, we have
10 KO-SHINCHEN,MARCOSMONTENEGRO,ANDXIAODONGYAN
(2.13) y1−2s(∇(wwβ))TB(x)∇(wwβ)dxdy
k k
ZZCΩ
= y1−2sw2β(∇w)TB(x)∇w+(2β+β2)y1−2sw2β(∇w )TB(x)∇w dxdy.
k k k k
ZZCΩ
Combining (2.12) and (2.13), we derive
(2.14) y1−2s|∇(wwβ)|2 dxdy ≤C |f(u)|uu2β dx
k 1 k
ZZCΩ ZΩ
for some constant C > 0 which depends only on β and the matrix A. Let h =
1
|f(u)|/(1+u) and Ω ={x∈Ω:u(x)>m}.
m
Since h∈L2ns(Ω), there exists m∈N large enough such that
2s
|h|2ns dx n ≤ Ks(n),
4C
(cid:18)ZΩm (cid:19) 1
where K (n) is the best constant with respect to the embedding Hs (C ) ֒→
s 0,L Ω
2n
Ln−2s(Ω). Since u is bounded on Ω\Ωm, there exists a constant C2 > 0 which
depends only on m and Ω such that
|f(u)|uu2β dx= |f(u)|uu2β dx+ |f(u)|uu2β dx
k k k
ZΩ ZΩ\Ωm ZΩm
(2.15) ≤C +2 hu2u2β dx
2 k
ZΩm
Using (2.15), we deduce that
2s n−2s
|f(u)|uu2kβ dx≤C2+2 h2ns dx n (uuβk)n2−n2s dx n
ZΩ (cid:18)ZΩm (cid:19) (cid:18)ZΩm (cid:19)
1
≤C + y1−2s|∇(wwβ)|2 dxdy.
2 2C k
1 ZZCΩ
Plugging this in (2.14) and taking k →∞ we have uβ+1 ∈Ln2−n2s(Ω) for all β ≥0.
Thusf(u)∈Lp(Ω)foreveryp≥1andtherestoftheprooffollowsfromTheorems
1.1, 1.2, 1.3 and 1.5 of [9]. (cid:3)
3. Proof of Theorem 1 - Interior case
Throughout this section, we assume x is an interior point of Ω. According to
0
Remark 1 and Proposition 2 of the previous section, our main task in this section
is to prove the following proposition:
Proposition 3. Assume (1.4) for some x ∈ Ω. If n ≥ 4s and σ > 2s, then for
0
any λ>0 there exists w ∈Hs,A(C )\{0} such that
0 0,L Ω
QAλ (w0)<csdet(A(x0))ns Ks(n)−1.
We first derive some necessary estimate. For simplicity of notations, we first
assume x = 0 and A(0) = I . Choose a smooth nonincreasing cut-off function
0 n
φ(t)∈C∞(R ) such that
+
1
φ(t)=1 for 0≤t≤ and φ(t)=0 if t≥1.
2