Semi-linear elliptic inequalities on weighted graphs

• Published in: Calculus of variations and partial differential equations Vol. 62; no. 2
• Main Authors: Gu, Qingsong, Huang, Xueping, Sun, Yuhua
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-03-2023; Springer Nature B.V
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Systems Theory; Theoretical; Secondary 58J05; Primary 35J61; 42B37; 31B10
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-022-02384-4

Let
(
V
,
μ
)
be an infinite, connected, locally finite weighted graph. We study the problem of existence or non-existence of positive solutions to a semi-linear elliptic inequality
Δ
u
+
u
σ
≤
0
in
V
,
where
Δ
is the standard graph Laplacian on
V
and
σ
>
0
. For
σ
∈
(
0
,
1
]
, the inequality admits no nontrivial positive solution. For
σ
>
1
, assuming condition
(
p
0
)
on
(
V
,
μ
)
, we obtain a sharp condition for nonexistence of positive solutions in terms of the volume growth of the graph, that is
μ
(
B
(
o
,
n
)
)
≲
n
2
σ
σ
-
1
(
ln
n
)
1
σ
-
1
for some
o
∈
V
and all large enough
n
. For any
ε
>
0
, we can construct an example on a homogeneous tree
T
N
with
μ
(
B
(
o
,
n
)
)
≍
n
2
σ
σ
-
1
(
ln
n
)
1
σ
-
1
+
ε
, and a solution to the inequality on
(
T
N
,
μ
)
to illustrate the sharpness of
2
σ
σ
-
1
and
1
σ
-
1
.