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

• Published in: Calculus of variations and partial differential equations Vol. 56; no. 2; pp. 1 - 16
• Main Authors: Barrios, B., Del Pezzo, L., García-Melián, J., Quaas, A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-04-2017; Springer Nature B.V
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Half spaces; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Systems Theory; Theoretical; 45M20; 47G10; 35S15
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-017-1133-9

In this paper we consider classical solutions
u
of the semilinear fractional problem
(
-
Δ
)
s
u
=
f
(
u
)
in
R
+
N
with
u
=
0
in
R
N
\
R
+
N
, where
(
-
Δ
)
s
,
0
<
s
<
1
, stands for the fractional laplacian,
N
≥
2
,
R
+
N
=
{
x
=
(
x
′
,
x
N
)
∈
R
N
:
x
N
>
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
∂
x
N
>
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
.