Affiliation:
1. School of Mathematics, Iran University of Science and Technology , P.O. Box 16844-13114 , Narmak , Tehran , Iran
2. Department of Mathematics, Aalto University , P.O. Box 11100 , FI-00076 Aalto , Finland
Abstract
Abstract
We consider the nonlocal quasilinear elliptic problem:
−
Δ
m
u
(
x
)
=
H
(
x
)
(
(
I
α
*
(
Q
f
(
u
)
)
)
(
x
)
)
β
g
(
u
(
x
)
)
in
Ω
,
-{\Delta }_{m}u\left(x)=H\left(x){(\left({I}_{\alpha }* \left(Qf\left(u)))\left(x))}^{\beta }g\left(u\left(x))\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,
where
Ω
\Omega
is a smooth domain in
R
N
{{\mathbb{R}}}^{N}
,
β
≥
0
\beta \ge 0
,
I
α
{I}_{\alpha }
,
0
<
α
<
N
0\lt \alpha \lt N
, stands for the Riesz potential,
f
,
g
:
[
0
,
a
)
→
[
0
,
∞
)
f,g:\left[0,a)\to \left[0,\infty )
,
0
<
a
≤
∞
0\lt a\le \infty
, are monotone nondecreasing functions with
f
(
s
)
,
g
(
s
)
>
0
f\left(s),g\left(s)\gt 0
for
s
>
0
s\gt 0
, and
H
,
Q
:
Ω
→
R
H,Q:\Omega \to {\mathbb{R}}
are nonnegative measurable functions. We provide explicit quantitative pointwise estimates on positive weak supersolutions. As an application, we obtain bounds on extremal parameters of the related nonlinear eigenvalue problems in bounded domains for various nonlinearities
f
f
and
g
g
such as
e
u
,
(
1
+
u
)
p
{e}^{u},{\left(1+u)}^{p}
, and
(
1
−
u
)
−
p
{\left(1-u)}^{-p}
,
p
>
1
p\gt 1
. We also discuss the Liouville-type results in unbounded domains.
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