Monotonicity of solutions for some nonlocal elliptic problems in half-spaces

Abstract

In this paper we consider classical solutions u of the semi-linear fractional problem (−∆) s u = f (u) in R N + with u = 0 in R N \ R N + , where (−∆) s , 0 < s < 1, stands for the fractional laplacian, N ≥ 2, R N + = {x = (x , xN) ∈ R N : xN > 0} is the half-space and f ∈ C 1 is a given function. With no additional restriction on the function f , we show that bounded, nonnegative, nontrivial classical solutions are indeed positive in R N + and verify ∂u ∂xN > 0 in R N +. This is in contrast with previously known results for the local case s = 1, where nonnegative solutions which are not positive do exist and the monotonicity property above is not known to hold in general even for positive solutions when f (0) < 0.

Full text

MONOTONICITY OF SOLUTIONS FOR SOME NONLOCAL
ELLIPTIC PROBLEMS IN HALF-SPACES
B. BARRIOS, L. DEL PEZZO, J. GARC´
IA-MELI ´
AN
AND A. QUAAS
Abstract. In this paper we consider classical solutions uof the semi-
linear fractional problem (−∆)su=f(u) in RN
+with u= 0 in RN\RN
+,
where (−∆)s, 0 <s<1, stands for the fractional laplacian, N≥2,
RN
+={x= (x0, xN)∈RN:xN>0}is the half-space and f∈C1
is a given function. With no additional restriction on the function f,
we show that bounded, nonnegative, nontrivial classical solutions are
indeed positive in RN
+and verify
∂u
∂xN
>0 in RN
+.
This is in contrast with previously known results for the local case s= 1,
where nonnegative solutions which are not positive do exist and the
monotonicity property above is not known to hold in general even for
positive solutions when f(0) <0.
1. Introduction
The objective of the present paper is to deal with the semilinear problem
(1.1) ((−∆)su=f(u) in RN
+,
u= 0 in RN\RN
+,
where N≥2, RN
+={x= (x0, xN)∈RN:xN>0}is the half-space and fis
aC1function. The operator (−∆)s, 0 <s<1, is the well-known fractional
laplacian, which is defined on smooth functions as
(1.2) (−∆)su(x) = ZRN
u(x)−u(y)
|x−y|N+2sdy,
up to a normalization constant which will be omitted for brevity. The in-
tegral in (1.2) has to be understood in the principal value sense, that is, as
the limit as ε→0 of the same integral taken in the ball Bε(x) of center
xand radius ε. Alternatively, this operator can be defined (omitting again
the normalization constant) as
(1.3) (−∆)su(x) = 1
2ZRN
2u(x)−u(x+y)−u(x−y)
|y|N+2sdy,
where now the integral is absolutely convergent for sufficiently smooth func-
tions.
Problems with nonlocal diffusion related to (1.1) have been intensively
studied in the last years, after their appearance when modeling different
situations. For instance, anomalous diffusion and quasi-geostrophic flows,
turbulence and water waves, molecular dynamics and relativistic quantum
1

2 B. BARRIOS, L. DEL PEZZO, J. GARC´
IA-MELI ´
AN AND A. QUAAS
mechanics of stars (see [9, 15, 17, 37] and references); or mathematical fi-
nance (cf. [2, 7, 18]), elasticity problems [34], thin obstacle problem [11],
phase transition [1, 10, 36], crystal dislocation [22, 38] and stratified mate-
rials [32].
Our inspiration to study problem (1.1) comes from the local case s= 1,
that is
(1.4) (−∆u=f(u) in RN
+,
u= 0 on ∂RN
+.
In a seminal series of papers (cf. [3, 4, 5, 6]), Berestycki, Caffarelli and
Nirenberg obtained interesting qualitative properties for positive solutions
of (1.4) and Lipschitz nonlinearities f. Among other results, they showed
that if f(0) ≥0, then any positive solution of (1.4) verifies
(1.5) ∂u
∂xN
>0 in RN
+
(see [4] or [5]). This property had been shown initially with additional
assumptions on both the solutions and the nonlinearities by Dancer in [20]
and [21].
The case f(0) <0 is, however, more subtle, and only partial results are
known for the moment. See [5], [26] for the case N= 2, [27] for N= 2,3
and [19] for dimensions N≥2. The main reason for this difference is the
existence of nonnegative (periodic) solutions which are not strictly positive.
With regard to a similar property as (1.5) for solutions of the nonlocal
problem (1.1), only some partial results have been achieved so far, at the best
of our knowledge. Let us mention [25] and [30] where monotone, positive
nonlinearities where considered, and [16] for the particular case f(t) = tp,
p > 1. On the other hand, the very recent preprint [23] analyzes the same
question in more general domains, but with a very restricted class of non-
linearities.
Our intention is to show that actually property (1.5) continues to be true
for bounded, nonegative, nontrivial solutions of (1.1) with no additional
asumptions placed on the nonlinearity faside its regularity. This is in
striking contrast with problem (1.4), where, as we have remarked, the case
f(0) <0 remains unsolved in its full generality for the moment.
Throughout this work, we will deal with bounded, classical solutions of
(1.1). However, this will not cause a loss in generality, since it is well-known
from the regularity theory developed in [35, 13, 14] and bootstrapping that
bounded, viscosity solutions of (1.1) are automatically classical. Observe
that classical solutions verify u∈C2s+β(RN
+) for every β∈(0,1), and in
particular they are seen to be in C1(RN
+).
Our main result is the following:
Theorem 1. Assume f∈C1(R)and let ube a bounded, nonnegative,
nontrivial classical solution of (1.1). Then uis positive and
(1.6) ∂u
∂xN
>0in RN
+.

MONOTONICITY OF SOLUTIONS IN HALF-SPACES 3
As a consequence of Theorem 1, we can also obtain some Liouville theo-
rems for problem (1.1) with some special nonlinearities.
Theorem 2. Assume f∈C1(R)is such that f0(t)>0for t > 0, and one
of the following holds:
(a) f(0) 6= 0;
(b) f(0) = 0 and f0(0) >0.
Then problem (1.1) does not admit bounded, nonnegative, nontrivial solu-
tions.
An interesting particular case in Theorem 2 is obtained when we set
f(t) = t−1. In this case the differences between the local version (1.4)
and its nonlocal counterpart (1.1) become more evident, since in the former
there exists a unique nonnegative solution given by u(x) = 1 −cos xN(see
[19]), while for the latter we have:
Corollary 3. The problem
((−∆)su=u−1in RN
+,
u= 0 in RN\RN
+
does not admit any bounded, nonnegative, nontrivial solution.
It is interesting to remark that Theorem 1 is an important tool to prove
other Liouville theorems for bounded solutions of (1.1). Indeed, passing to
the limit as xN→+∞, we find that such solutions converge to a stable
solution of (−∆)su=f(u) in RN−1. Then one can use the nonexistence
theorems already available in that situation (cf. for instance [24]).
We conclude the introduction with a couple of comments on our proofs.
We use moving planes to show that any nonnegative, bounded, classical
solution of (1.1) is monotone in the xNdirection. To deal with the moving
planes method, we mainly follow the approach in [25]. However, instead
of representing the solutions of (1.1) with the aid of Green’s function in
RN
+at the onset, we use it for an adequate truncation related to uand its
reflections. This allows us to avoid any monotonicity or sign restriction on
f.
It is to be noted that at one point in the argument, when it is assumed
that the moving of the planes stop somewhere, we need to rule out the
existence of solutions which are symmetric with respect to a hyperplane
contained in RN
+. In the local case, this is only possible under the additional
restriction f(0) ≥0, since such solutions do exist if f(0) <0. However, we
show in the present work that symmetric solutions can not exist with no
additional restriction on f(see Theorem 8 below). In our opinion, this is a
result of independent interest. Its proof is based on the regularity inherited
by symmetry in the strip, which allows to evaluate the equation on the
boundary of the half space. Then the nonlocality of the operator implies,
loosely speaking, that the interactions between points which are far away
in RN
+is too strong and the solution must vanish. This is a remarkable
difference with respect to the case s= 1, where this interaction is not
present.

4 B. BARRIOS, L. DEL PEZZO, J. GARC´
IA-MELI ´
AN AND A. QUAAS
The rest of the paper is organized as follows: in Section 2 we give some
preliminaries and introduce the notation to be used for the moving planes.
Section 3 deals with some properties of the Green’s function in a half-space
taken from [25] and with a different representation in terms of this function.
In Section 4, we obtain a nonexistence result for bounded, nonnegative,
nontrivial solutions which are symmetric with respect to a hyperplane, and
in Section 5 we perform the proof of our main results, Theorems 1 and 2.
2. Some preliminaries
In this section, we gather some preliminary properties which will be useful
in the forthcoming sections. We notice that, although we are mostly con-
cerned with solutions in the classical sense, other more general concepts of
solutions have to be considered at some places in the present work, mainly
due to the fact that we work with truncations of the original functions.
Thus, throughout this section we will consider inequalities in the viscosity
sense (see [13] for a definition). We begin by considering a version of the
maximum principle for the operator (−∆)sin unbounded domains, which
will be needed below. We believe that this result is new.
Lemma 4. Assume D⊂RN
+is a domain and let u∈C(RN)be a bounded
function verifying (−∆)su≥0in Din the viscosity sense, with u≥0in
RN\D. Then u≥0in D.
Proof. First of all observe that the function ϕ(x)=(xN)s
+is s−harmonic in
RN
+, where (xN)+is the function which coincides with xNin (0,+∞) and
vanishes in (−∞,0]. Indeed, when x∈RN
+:
(−∆)sϕ(x) = ZR
(xN)s−(yN)s
+
|xN−yN|1+2s ZRN−1
|xN−yN|1+2s
((xN−yN)2+|x0−y0|2)N+2s
2
dy0
!dyN
=ZR
(xN)s−(yN)s
+
|xN−yN|1+2s ZRN−1
dz
(1 + z2)N+2s
2!dyN
=CZR
(xN)s−(yN)s
+
|xN−yN|1+2sdyN= 0
(see for instance the introduction in [12] or Proposition 3.1 in [31]).
Next take ε > 0 and consider the function
vε(x) = u(x) + ε(xN)s
+, x ∈RN
+.
Since uis bounded, there exists M > 0 such that vε≥0 if xN≥M. Define
the set DM=D∩ {x∈RN: 0 < xN< M }. Then, in the viscosity sense,
(2.1) ((−∆)svε≥0 in DM,
vε≥0 in RN\DM.
We are in a position to apply Theorem 2.3 in [30] to conclude that vε≥0
in RN. Letting ε→0, we obtain that u≥0 in RN.

MONOTONICITY OF SOLUTIONS IN HALF-SPACES 5
Before giving our next result, let us introduce some notation related to
the method of moving planes. For λ > 0 we denote, as customary:
Σλ:= {x∈RN
+: 0 < xN< λ}
Tλ:= {x∈RN:xN=λ}
xλ:= (x0,2λ−xN) (the reflection of xwith respect to Tλ).
If fis a given nonlinearity and ustands for a a bounded, classical nonneg-
ative solution of our problem (1.1) we also set
uλ(x) = u(xλ)
wλ(x) = uλ(x)−u(x)x∈Σλ.
Since our ultimate objective is to show that wλis always nonnegative in Σλ,
the following will also be relevant:
Dλ={x∈Σλ:wλ(x)<0}
Wλ={x∈Dλ:f(u(x)) > f(uλ(x))}
vλ=wλχDλ,
where χwill stand throughout the paper for the characteristic function of
a set. It is plain that the function vλwill only be meaningful when wλis
negative somewhere in Σλ. We next state one of its important properties.
Lemma 5. Assume wλ<0somewhere in Σλ, for some λ > 0. Then,
(2.2) (−∆)svλ≥(f(uλ)−f(u))χDλin RN
+,
in the viscosity sense.
Proof. Let us prove first that, when x∈Dλ, (2.2) holds in the classical sense
(cf. the proof of Theorem 1.1 in [28]). Denote
zλ=wλ−vλ.
It is clear that in Dλ(2.2) is equivalent to (−∆)szλ≤0. To prove this
last inequality, denote by Eλthe reflection trough the hyperplane Tλof Dλ.
Using that zλ≡0 in Dλand zλ≥0 in Σλ\Dλ, we have for every x∈Dλ:
(−∆)szλ(x) = − ZΣλ\Dλ
+ZEλ
+ZΣc
λ\Eλ!zλ(y)
|x−y|N+2sdy
≤ − ZΣλ\Dλ
+ZΣc
λ\Eλ!zλ(y)
|x−y|N+2sdy
=−ZΣλ\Dλ
zλ(y)1
|x−y|N+2s−1
|x−yλ|N+2sdy ≤0.
Here we have used that |x−y|≤|x−yλ|when x∈Dλ,y∈Σλ, which can
be easily checked. Thus (2.2) is proved in Dλ.
On the other hand, when x∈RN
+\Dλ, it is immediate that
(−∆)svλ(x) = −ZDλ
vλ(y)
|x−y|N+2sdy ≥0,

6 B. BARRIOS, L. DEL PEZZO, J. GARC´
IA-MELI ´
AN AND A. QUAAS
since vλ= 0 in RN
+\Dλand vλ<0 in Dλ. Therefore (2.2) also holds in the
classical sense in RN
+\Dλ.
However, the function vλneeds not be smooth on ∂Dλ, so that it is not to
be expected that its fractional laplacian is even well-defined there. But the
inequality can be checked in the viscosity sense. To prove this, take x0∈∂Dλ
and let ϕ∈C∞(RN) be such that ϕ<vλin a reduced neighborhood
U \ {x0}of x0, with ϕ(x0) = vλ(x0) = 0. Then (−∆)svλ(x0)≥0 means
(−∆)sψ(x0)≥0, where
ψ(x) = ϕ(x)x∈ U
vλ(x)x∈RN\ U
(cf. [13]). The inequality (−∆)sψ(x0)≥0 is easily checked since, taking
into account that vλ≤0 in RN, so that ϕ≤0 in U, we deduce
(−∆)sψ(x0) = −ZU
ϕ(y)
|x−y|N+2sdy −ZRN\U
vλ(y)
|x−y|N+2sdy ≥0,
as was to be shown. 
3. A representation in the half-space
In this section, we will show that the function vλverifies an inequality
which involves the Green’s function in the half-space. As we have already
remarked in the Introduction, the representation is rather general and does
not impose any additional properties on the nonlinear term f. Recall that
we are always assuming 0 <s<1.
We introduce the Green’s function for RN
+(see [25]). If x, y ∈RN
+, we let
(3.1) G+
∞(x, y) = ks
N
2|x−y|2s−NZψ+
∞(x,y)
0
ts−1
(t+ 1)N
2
dt,
where
ψ+
∞(x, y) = 4xNyN
|x−y|2.
In (3.1), kNis a positive constant whose actual value is immaterial for us.
It is shown in [25] that if u∈L∞(RN) vanishes outside RN
+and (−∆)su∈
L∞(RN
+) is nonnegative, then
u(x) = ZRN
+
G+
∞(x, y)(−∆)su(y)dy, x ∈RN
+.
To avoid the sign restriction just mentioned, we follow a different approach.
The information we obtain is slightly weaker, but it suffices for our argu-
ments in the proofs of Section 5. Here is the main result of this section:
Lemma 6. Assume fis locally bounded and let ube a nonnegative, bounded
solution of (1.1). Suppose wλ<0somewhere in Σλ, for some λ > 0. Then,
for every x∈Wλ={x∈Dλ:f(u(x)) > f(uλ(x))},
vλ(x)≥ZWλ
G+
∞(x, y)(f(uλ(y)) −f(u(y)))dy,
where the integral is absolutely convergent.

MONOTONICITY OF SOLUTIONS IN HALF-SPACES 7
In order to prove Lemma 6, we borrow some notation and results from
[25]. For R > 0, define B+
R:= BR(ReN)⊂RN
+, where eNstands for the last
vector in the canonical basis, and let
G+
R(x, y) = ks
N
2|x−y|2s−NZψ+
R(x,y)
0
ts−1
(t+ 1)N
2
dt,
with
ψ+
R(x, y) = (R2− |x−ReN|2)(R2− |y−ReN|2)
R2|x−y|2,
be the Green’s function in the ball B+
R. We also introduce
Γ+
R(x, y) = CN,s R2− |x−ReN|2
|y−ReN|2−R2s
|x−y|−N,
the Poisson kernel for the same ball (cf. [8] for some properties of both
functions). According to Corollary 2.9 in [25], if hRis the unique solution
of the Dirichlet problem
((−∆)sh=g1in B+
R,
h=g2in RN\B+
R,
where g1∈L∞(B+
R) and g2∈L∞(RN\B+
R), then we can write:
(3.2) hR(x) = ZB+
R
G+
R(x, y)g1(y)dy +ZRN\B+
R
Γ+
R(x, y)g2(y)dy.
Regarding this representation, it is to be noted that, when g2∈L∞(RN),
as a consequence of equation (3.7) in [25], then
(3.3) lim
R→+∞ZRN\B+
R
Γ+
R(x, y)g2(y)dy = 0
for every x∈RN
+. The following properties of Green’s function will be used
in our proof of Lemma 6 and in the proof of Theorem 1 in Section 5.
Lemma 7. Fix R0>0. Then the functions G+
R(x, y)are nondecreasing
with respect to Rin B+
R0×B+
R0if R > R0and verify
G+
R→G+
∞in B+
R0×B+
R0as R→+∞.
Moreover, for every λ > 0, there exists C=C(N, s, λ)such that
(3.4) G+
∞(x, y)≤Cmin{|x−y|2s−N,|x−y|−N}for x, y ∈Σλ.
In addition, the function G+
∞(x, y)enjoys the following properties:
(a) If {xn}is a bounded sequence, then for every λ > 0
lim
R→+∞ZΣλ∩Bc
R
G+
∞(xn, y)dy = 0,
uniformly in n.
(b) If λ > 0and {xn}is a bounded sequence, then for every R > 0there
exists a positive constant Csuch that
ZΣλ∩BR
G+
∞(xn, y)dy ≤Cfor every n∈N.

8 B. BARRIOS, L. DEL PEZZO, J. GARC´
IA-MELI ´
AN AND A. QUAAS
(c) For every λ > 0and x0∈∂RN
+, we have
lim
x→x0ZΣλ
G+
∞(x, y)dy = 0.
Sketch of proof. The statements about monotonicity and convergence of G+
R
are a consequence of Lemma 3.2 in [25]. The estimates (3.4) follow because
of Lemma 4.1 there.
Parts (a) and (b) are a direct consequence of (3.4), while for the proof of
(c), we only have to notice that, if h∈C(RN) is the unique solution of
((−∆)sh=χΣλin RN
+,
h= 0 in RN\RN
+,
then by Theorem 3.1 in [25] we have
h(x) = ZΣλ
G+
∞(x, y)dy,
and the proof follows because of the continuity of hup to the boundary of
RN
+.
We can now proceed to the proof of Lemma 6.
Proof of Lemma 6. We start by observing that, by Lemma 5
(3.5) (−∆)svλ≥(f(uλ)−f(u))χDλin RN
+.
Consider the balls B+
Rintroduced before and denote by hRthe unique solu-
tion of the problem
(3.6) ((−∆)sh= (f(uλ)−f(u))χDλin B+
R,
h=vλin RN\B+
R.
It is clear by (3.5) and the maximum principle that vλ≥hRin B+
R. There-
fore, according to (3.2), we may write
(3.7)
vλ(x)≥hR(x) = ZB+
R∩Dλ
G+
R(x, y)(f(uλ(y)) −f(u(y)))dy
+ZRN
+\B+
R
Γ+
R(x, y)vλ(y)dy.
Our intention is to pass to the limit in (3.7). Notice that, since vλis bounded,
we have by (3.3) that the last integral converges to zero as R→+∞.
On the other hand, we obtain from Lemma 7 that G+
Ris nondecreasing
as a function of Rand, for fixed x∈Σλ
G+
R(x, y)≤G+
∞(x, y)≤Cmin{|x−y|−N+2s,|x−y|−N} ∈ L1(Σλ),
as a function of y. Therefore, letting R→+∞in (3.7) and using dominated
convergence we arrive at
vλ(x)≥ZDλ
G+
∞(x, y)(f(uλ(y)) −f(u(y)))dy
≥ZWλ
G+
∞(x, y)(f(uλ(y)) −f(u(y)))dy,

MONOTONICITY OF SOLUTIONS IN HALF-SPACES 9
as was to be proved. 
Remark 1.Similar results as the ones given in this section follow easily in
other half-spaces by means of a simple change of variables. For instance, in
H:= {x∈RN:xN< λ}, the Green’s function is given by
G(x;y) = G+
∞(x0, λ −xN;y0, λ −yN), x, y ∈H,
and similar properties as those given in Lemma 6 are obtained at once.
4. A nonexistence theorem
In this section we will state and prove a nonexistence result for nonneg-
ative solutions of (1.1) which are symmetric with respect to a hyperplane.
This result is fairly important in the moving planes argument, and is the
responsible that the case f(0) <0 can be included in our proofs, in contrast
with the local case s= 1. We believe it is interesting in its own right.
It is to be noted that, when the nonlinearity verifies f(0) ≥0, the nonex-
istence of these symmetric solutions is a direct consequence of the strong
maximum principle. The proof we give, however, covers also this case. Ob-
serve that next theorem holds with minimal hypotheses on f.
Theorem 8. Assume fis continuous at zero and let u∈C2s+β(RN
+)
(0< β < 1) be a bounded, nonnegative, classical solution of (1.1) which
is symmetric with respect to Tλin Σ2λfor some λ > 0, that is
u(x0,2λ−xN) = u(x0, xN), x ∈Σ2λ.
Then f(0) = 0 and u≡0in RN.
Proof. We begin by showing that uverifies the equation at x= 0, that is,
(4.1) ZRN
+
u(y)
|y|N+2sdy =−f(0).
To see this, we first observe that with no loss of generality we may assume
that βis restricted to satisfy 2s+β < 2. Thus the symmetry of uimplies
that the same regularity holds up to the boundary of RN
+, since necessarily
uand ∇uvanish there. Therefore u∈C2s+β(RN).
Take an arbitrary sequence {xn} ⊂ Σ2λwith xn→0. Evaluating the
equation at xn, but using expression (1.3) for (−∆)s, we see that
(4.2) f(u(xn)) = 1
2ZRN
2u(xn)−u(xn+y)−u(xn−y)
|y|N+2sdy.
Now we have to distinguish between the cases 0 <s<1
2and 1
2≤s < 1. In
the former case, assuming βis such that 2s+β < 1, we deduce from the
regularity of uthat for sufficiently large n:
(4.3) |u(xn)−u(xn+y)| ≤ C|y|2s+βwhenever |y| ≤ 1,
for some positive constant C. In the latter, if βis such that 2s+β < 2, the
regularity implies, also for large enough n
(4.4) |u(xn)−u(xn+y)−u(xn−y)| ≤ C|y|2s+βif |y| ≤ 1.

10 B. BARRIOS, L. DEL PEZZO, J. GARC´
IA-MELI ´
AN AND A. QUAAS
On the other hand, for |y| ≥ 1,
(4.5) 
u(xn)−u(xn+y)−u(xn−y)
|y|N+2s≤3kukL∞(RN
+)|y|−N+2s.
Inequalities (4.3), (4.4) and (4.5) show that the integrand in (4.2) is bounded
in absolute value by a function which is in L1(RN). Therefore, we may pass
to the limit in (4.2) with the aid of dominated convergence to arrive at
f(0) = −1
2ZRN
u(y) + u(−y)
|y|N+2sdy,
which is equivalent to (4.1).
By evaluating the equation at x= 2λeN, using the fact that u(2λeN)=0
by symmetry, we deduce from (4.1) that
(4.6) ZRN
+
u(y)
|y|N+2sdy =ZRN
+
u(y)
|2λeN−y|N+2sdy.
Next, we split the second integral in two parts and use the symmetry of u
in Σ2λto have:
ZRN
+
u(y)
|2λeN−y|N+2sdy = ZΣ2λ
+ZRN
+\Σ2λ!u(y)
|2λeN−y|N+2sdy
=ZΣ2λ
u(z)
|z|N+2sdz +ZRN
+\Σ2λ
u(y)
|2λeN−y|N+2sdy.
Hence, from (4.6) we see that
ZRN
+\Σ2λ
u(y)
|y|N+2sdy =ZRN
+\Σ2λ
u(y)
|2λeN−y|N+2sdy.
Taking into account that u≥0 and |2λeN−y| ≤ |y|for y∈RN
+\Σ2λ, we
deduce u≡0 in RN
+\Σ2λ.
Using this information and evaluating the equation at points x∈RN
+\Σ2λ,
we obtain
(4.7) ZΣ2λ
u(y)
|x−y|N+2sdy =−f(0).
Now observe that the integral above is a smooth function of xif, say, xN≥
2λ+1, since the integrand is uniformly bounded and the integral is uniformly
convergent at infinity when xbelongs to a compact set. Therefore, we are
allowed to differentiate (4.7) with respect to xNto get:
ZΣ2λ
u(y)(xN−yN)
|x−y|N+2s+2 dy = 0.
However, xN−yN≥xN−2λ≥1 for y∈Σ2λand the chosen values of x,
so that the integrand is nonnegative and this gives u≡0 in Σ2λ, therefore
in RN. It is clear that this can only happen when f(0) = 0, and the proof
is concluded. 

MONOTONICITY OF SOLUTIONS IN HALF-SPACES 11
5. Proof of the main results
In this final section we will prove our main contributions, Theorems 1
and 2. The proof of Corollary 3 will not be given, since it is an immediate
consequence of Theorem 2.
Proof of Theorem 1. We have already said that the proof is an application
of the method of moving planes, as used in [30] and [25], but with some
significant changes. In particular, we remark that we work with some trun-
cations of the original functions, so we are led to the use of inequalities in the
viscosity sense and Lemma 6. We also need at some point the nonexistence
result given by Theorem 8.
We follow the notation introduced in Section 2.
Step 1.wλ≥0 in Σλif λ > 0 is small enough.
To prove this, assume for a contradiction that Dλis not empty if λis
small. Since uis bounded and fis C1, there exists a constant Lsuch that
f(uλ)−f(u)≥ −L|uλ−u|. Therefore, using Lemma 5 we have
(−∆)svλ≥f(uλ)−f(u)≥ −L|uλ−u|=Lvλin Dλ.
By Theorem 2.4 in [30] we obtain vλ≥0 in Dλwhen λis small enough,
which is a contradiction. Therefore, Dλ=∅for small λand this shows the
claim.
Step 2. Setting
λ∗= sup{λ > 0 : wµ≥0 in Σµfor every µ∈(0, λ)},
we have λ∗= +∞.
Assume again for a contradiction that λ∗<+∞. Then there exists a
sequence {λj}of values such that λj> λ∗for every jand λj→λ∗as
j→+∞, with wλjnegative somewhere in Σλj. Consider the sets
Dj={x∈Σλj:wλj(x)<0}
and
Wj={x∈Dj:f(u(x)) > f(uλj(x))}.
By the choice of λj, the sets Djare nonempty for every j. We claim that
the same is true for Wj. Indeed, if we had Wj=∅, then f(uλj)≥f(u) in
Dj. Hence
(−∆)svλj≥0 in Dj.
By Lemma 4 we obtain vλj≥0 in Dj, which is not possible. Therefore
Wj6=∅. Thus it is possible to choose points xj∈Wjsuch that
(5.1) vλj(xj)≤ −1
2kvλjkL∞(Wj),
and we can define the functions
euj(x) = u(x0+x0
j, xN), x ∈RN
+.

12 B. BARRIOS, L. DEL PEZZO, J. GARC´
IA-MELI ´
AN AND A. QUAAS
It is easily seen that eujis a solution of (1.1), verifying keujkL∞(RN
+)=
kukL∞(RN
+). It is then standard, with the use of regularity theory, Ascoli-
Arzel´a’s theorem and a diagonal argument, that for some subsequence
euj→¯u
locally uniformly in RN, where ¯uis a nonnegative solution of (1.1). We may
also assume that xj,N →x0∈[0, λ∗]. Now three cases are possible:
(a) ¯u6≡ 0, x0∈(0, λ∗);
(b) ¯u6≡ 0, x0= 0 or x0=λ∗;
(c) ¯u≡0.
Before dealing with each of this cases, let us introduce some notation related
to the functions euj. Let:
ewλj(x) = euj(xλj)−euj(x), x ∈Σλj,
e
Dj={x∈Σλj:ewj(x)<0},
evλj(x) = ewλj(x)χe
Dj(x), x ∈Σλj,
f
Wj={x∈e
Dj:f(euj(x)) > f(euj(xλ
j))}
(observe that e
Djand f
Wjare nothing more than translations of Djand Wj,
respectively). Denote also zj= (0, xj,N ), z0= (0, x0). By our choice of xj
in (5.1) above, since it follows that ewλj(zj)<0 and f(euj(zj)) > f(euj(zλj
j)),
we deduce that zj∈f
Wj. Moreover we also get
(5.2) evλj(zj)≤ −1
2kevλjkL∞(
f
Wj).
Now consider in turn each one of the cases (a), (b) and (c).
In case (a), we see from euj(xλ∗)≥euj(x) in Σλ∗that ¯u(xλ∗)≥¯u(x) in
Σλ∗. Moreover, since eu(zλj
j)<eu(zj), we also have ¯u(zλ∗
0) = ¯u(z0). Let us
see that this implies ¯u(xλ∗)≡¯u(x) in RN. Indeed, arguing as in the proof
of Lemma 5 and denoting ¯wλ∗= ¯uλ∗−¯u, we obtain
0 = f(¯uλ∗(z0)) −f(¯u(z0)) = (−∆)s¯wλ∗(z0)
=−ZΣλ∗∪RN
−
¯wλ∗(y)1
|z0−y|N+2s−1
|z0−yλ|N+2sdy,
which implies ¯wλ∗≡0 in RN, since ¯wλ∗≥0 in Σλ∗∪RN
−and |z0−y| ≤
|z0−yλ|for every y∈Σλ∗∪RN
−.
This means that ¯uis symmetric with respect to Tλ∗, so that Theorem 8
implies ¯u≡0, a contradiction.

MONOTONICITY OF SOLUTIONS IN HALF-SPACES 13
As for case (b), assume x0= 0. We deduce from Lemma 6 and the choice
of the points zj:
1
2kevλjkL∞(
f
Wj)≤ −evλj(zj)≤LkevλjkL∞(
f
Wj)Zf
Wj
G+
∞(zj, y)dy
≤LkevλjkL∞(
f
Wj)ZΣλ∗+1
G+
∞(zj, y)dy,
where Ldenotes a bound for the derivate of fin [0,kukL∞(RN
+)]. By Lemma
7, part (c), the last integral converges to zero since zj→0∈∂RN
+. We
deduce that evλj= 0 in Wjwhen jis large enough, a contradiction.
When x0=λ∗, a similar contradiction is reached. The only difference is
that one now works with the Green’s function in the half-space {x∈RN:
xN< λj}(see Remark 1).
Finally, we consider case (c). This case can only arise when f(0) = 0 and
the proof is different depending on the sign of f0(0). We begin by assuming
that f0(0) >0.
In what follows, we denote by λ1(Ω) the principal eigenvalue of (−∆)s
in Ω under Dirichlet boundary conditions (cf. Proposition 9 in [33]). If we
take a ball BRwith arbitrary center and radius Rthen it can be seen by
means of a simple scaling that
λ1(BR) = λ1(B1)
R2s→0 as R→+∞.
Thus it is possible to select a ball B⊂⊂ RN
+with the property that
(5.3) λ1(B)<1
2f0(0).
Since euj→0 uniformly in B, we deduce
(5.4) (−∆)seuj=f(euj)
eujeuj≥1
2f0(0) eujin B
if jis large enough. By Theorem 1.1 in [29], (5.4) implies the opposite
inequality in (5.3), which is a contradiction.
Hence to conclude the proof only the case f0(0) ≤0 needs to be dealt
with. Using the mean value theorem, we may write, for y∈f
Wj:
f(euλj
j(y)) −f(euj(y)) = f0(ξj(y))evj(y),
where ξj(y) is an intermediate value between euj(y) and euλj
j(y). Observe
that ξj→0 uniformly on compact sets of RN
+while f0(ξj)≥0. By Lemma
6 and (5.2), we see that
(5.5) 1
2kevλjkL∞(
f
Wj)≤ kevλjkL∞(
f
Wj)ZΣλ∗+1
G+
∞(zj, y)f0(ξj(y))dy.

14 B. BARRIOS, L. DEL PEZZO, J. GARC´
IA-MELI ´
AN AND A. QUAAS
Now choose R > 0 and split the integral in (5.5) in BRand Bc
R. If Lstands
again for a bound for the derivate of fin [0,kukL∞(RN
+)], we have
1
2kevλjkL∞(
f
Wj)≤ kevλjkL∞(
f
Wj) ZΣλ∗+1∩BR
G+
∞(zj, y)f0(ξj(y))dy
+LZΣλ∗+1∩Bc
R
G+
∞(zj, y)dy!.
Observe that the integral in Σλ∗+1 ∩Bc
Rcan be made as small as desired by
taking Rlarge enough, thanks to the fact that {zj}is a bounded sequence
and Lemma 7, part (a). Therefore the last term in the above inequality can
be made less than 1
4, say, if Ris chosen large. After we have fixed such a
value of R, we observe that ξj→0 uniformly in BR, so that f0(ξj)≤o(1),
since we are assuming f0(0) ≤0. Thefore, using Lemma 7, part (b), we
arrive at 1
4kevλjkL∞(
f
Wj)≤o(1)kevλjkL∞(
f
Wj)
which, as above, is a contradiction.
Step 3. Proof of (1.6).
As a consequence of steps 1 and 2, we have shown that uλ≥uin Σλfor
every λ > 0, that is, uis nondecreasing as a function of the variable xN.
Since u∈C1(RN
+), this implies
(5.6) ∂u
∂xN
≥0 in RN
+.
To conclude the proof of our theorem, we only have to show that the in-
equality is strict in (5.6). This is a consequence of the strong maximum
principle for the derivative with respect to xN. However, this function does
not directly verify an equation in RN
+, since uis not expected to be C1on
∂RN
+. We overcome this difficulty by localizing the problem and working
with incremental quotients.
Assume there exists x0∈RN
+such that
(5.7) ∂u
∂xN
(x0)=0.
Choose δ > 0 such that B2δ(x0)⊂⊂ RN
+, and let φ∈C∞
0(B2δ(x0)) be a
cutoff function with the usual properties: 0 ≤φ≤1 and φ= 1 in Bδ(x0).
Choose a small τ > 0 and define for x∈RN:
zτ(x) = u(x+τ eN)−u(x)
τφ(x).
Since uis nondecreasing we have zτ≥0, and we obtain
(−∆)szτ≥f(u(x+τeN)) −f(u(x))
τin Bδ(x0),
with zτ≥0 in RN. Letting τ→0, it is clear that
zt→z:= ∂u
∂xN
φuniformly in RN,

MONOTONICITY OF SOLUTIONS IN HALF-SPACES 15
and using Lemma 4.5 in [13] we see that
(−∆)sz≥f0(u)zin Bδ(x0),
in the viscosity sense. The strong maximum principle and (5.7) imply z= 0
in Bδ(x0), so that ∂u
∂xN= 0 in Bδ(x0). A standard connectedness argument
then implies that
∂u
∂xN
= 0 in RN
+,
which is impossible. This concludes the proof of (1.6). Observe by the way
that u > 0 in RN
+is a direct consequence of (1.6). 
Proof of Theorem 2. Assume uis a nonnegative, bounded, nontrivial solu-
tion of (1.1). By Theorem 1, we have u > 0 in RN
+. We first claim that for
every δ > 0, there exists c(δ)>0 such that
u(x)≥c(δ) if xN≥δ.
Suppose for a contradiction that this is not true. Then there exists δ > 0
and a sequence {x0
n} ⊂ RN−1such that u(x0
n, δ)→0. Define
un(x) = u(x0+x0
n, xN), x ∈RN
+.
Since {un}is uniformly bounded, we obtain after passing to a subsequence
that un→¯ulocally uniformly in RN, where ¯uis a nonnegative, bounded
solution of (1.1) wich verifies ¯u(0, δ) = 0. Again by Theorem 1 we see that
¯u≡0.
This is impossible if fverifies (a) in the statement. When fverifies (b),
a similar argument as in case (c) in step 2 of the proof of Theorem 1 also
leads to a contradiction. This contradiction proves the claim.
Now fix any δ > 0 and let
θ= inf
c(δ)≤t≤Mf0(t)>0,
where M=kukL∞(RN
+). If we choose any function φ∈C∞
0(RN
+\Σδ) such
that 0 ≤φ≤1 and φ= 1 in RN\Σ2δ, we see as in step 3 in the proof of
Theorem 1 that the function
z=∂u
∂xN
φ
verifies
(−∆)sz≥θz in RN\Σ2δ.
Arguing again with the principal eigenvalue λ1(B) in a sufficiently large
ball Bcontained in RN\Σ2δwe reach a contradiction. This shows that
no bounded, nonnegative, nontrivial solution to (1.1) may exist under our
hypotheses. 
Acknowledgements. B. B. was partially supported by a MEC-Juan de
la Cierva postdoctoral fellowship (Spain). L. D. P. was partially supported
by PICT2012 0153 from ANPCyT (Argentina). J. G-M. and A. Q. were par-
tially supported by Ministerio de Ciencia e Innovaci´on under grant MTM2014-
52822-P (Spain). A. Q. was also partially supported by Fondecyt Grant No.

16 B. BARRIOS, L. DEL PEZZO, J. GARC´
IA-MELI ´
AN AND A. QUAAS
1151180 Programa Basal, CMM. U. de Chile and Millennium Nucleus Cen-
ter for Analysis of PDE NC130017. B. B. and J. G-M. would like to thank
the Mathematics Department of Universidad T´ecnica Federico Santa Mar´ıa
where part of this work has been done for its kind hospitality.
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B. Barrios
Departamento de An´
alisis Matem´
atico, Universidad de La Laguna
C/. Astrof
´
ısico Francisco S´
anchez s/n, 38200 – La Laguna, SPAIN
E-mail address:bbarrios@ull.es
L. Del Pezzo
CONICET
Departamento de Matem´
atica, FCEyN UBA
Ciudad Universitaria, Pab I (1428)
Buenos Aires, ARGENTINA.
E-mail address:ldpezzo@dm.uba.ar

18 B. BARRIOS, L. DEL PEZZO, J. GARC´
IA-MELI ´
AN AND A. QUAAS
J. Garc
´
ıa-Meli´
an
Departamento de An´
alisis Matem´
atico, Universidad de La Laguna
C/. Astrof
´
ısico Francisco S´
anchez s/n, 38200 – La Laguna, SPAIN
and
Instituto Universitario de Estudios Avanzados (IUdEA) en F´
ısica At´
omica,
Molecular y Fot´
onica, Universidad de La Laguna
C/. Astrof
´
ısico Francisco S´
anchez s/n, 38200 – La Laguna, SPAIN.
E-mail address:jjgarmel@ull.es
A. Quaas
Departamento de Matem´
atica, Universidad T´
ecnica Federico Santa Mar
´
ıa
Casilla V-110, Avda. Espa˜
na, 1680 – Valpara
´
ıso, CHILE.
E-mail address:alexander.quaas@usm.cl