Abstract
This paper deals with maximum principles depending on the domain and ABP estimates associated to the following Lane–Emden system
involving fractional Laplace operators:
{
(
-
Δ
)
s
u
=
λ
ρ
(
x
)
|
v
|
α
-
1
v
in
Ω
,
(
-
Δ
)
t
v
=
μ
τ
(
x
)
|
u
|
β
-
1
u
in
Ω
,
u
=
v
=
0
in
ℝ
n
∖
Ω
,
\left\{\begin{aligned} \displaystyle(-\Delta)^{s}u&\displaystyle=\lambda\rho(x%
)\lvert v\rvert^{\alpha-1}v&&\displaystyle\phantom{}\text{in }\Omega,\\
\displaystyle(-\Delta)^{t}v&\displaystyle=\mu\tau(x)\lvert u\rvert^{\beta-1}u&%
&\displaystyle\phantom{}\text{in }\Omega,\\
\displaystyle u&\displaystyle=v=0&&\displaystyle\phantom{}\text{in }\mathbb{R}%
^{n}\setminus\Omega,\end{aligned}\right.
where
s
,
t
∈
(
0
,
1
)
{s,t\in(0,1)}
,
α
,
β
>
0
{\alpha,\beta>0}
satisfy
α
β
=
1
{\alpha\beta=1}
, Ω is a smooth bounded domain in
ℝ
n
{\mathbb{R}^{n}}
,
n
≥
1
{n\geq 1}
, and ρ and τ are
continuous functions on
Ω
¯
{\overline{\Omega}}
and positive in Ω. We establish some maximum principles depending on Ω. In particular, we explicitly characterize the measure of Ω for which the maximum principles corresponding to this problem hold in Ω. For this,
we derived an explicit lower estimate of principal eigenvalues in terms of the measure of Ω. Aleksandrov–Bakelman–Pucci (ABP) type estimates for the above systems are also proved. We also show the existence of a viscosity solution for a nonlinear perturbation of the nonhomogeneous counterpart of the above problem with
polynomial and exponential growths. As an application of the maximum principles, we measure explicitly how small
|
Ω
|
{\lvert\Omega\rvert}
has to be to ensure the positivity of the obtained solutions.
Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems