DOI: 10.1515/dema-2025-0291 ISSN: 2391-4661
Liouville results for double-phase problems involving Δ
λ
-Laplacian
Yunfeng Wei, Caisheng Chen Abstract
In this paper, we examine the following double-phase problem with weights
−
div
λ
∇
λ
u
r
−
2
∇
λ
u
+
w
(
x
)
∇
λ
u
p
−
2
∇
λ
u
=
f
(
x
)
|
u
|
q
−
1
u
∫
R
N
f
(
y
)
|
u
(
y
)
|
q
+
1
|
x
−
y
|
λ
τ
d
y
i
n
R
N
,
$$-{\text{div}}_{\lambda }\left({\left\vert {\nabla }_{\lambda }u\right\vert }^{r-2}{\nabla }_{\lambda }u+w\left(x\right){\left\vert {\nabla }_{\lambda }u\right\vert }^{p-2}{\nabla }_{\lambda }u\right)=f\left(x\right)\vert u{\vert }^{q-1}u\underset{{\mathbb{R}}^{N}}{\int }\frac{f\left(y\right)\vert u\left(y\right){\vert }^{q+1}}{\vert x-y{\vert }_{\lambda }^{\tau }}\mathrm{d}y \mathrm{i}\mathrm{n} {\mathbb{R}}^{N},$$
where
p
≥
r
≥ 2,
q
>
p
− 1,
τ
> 0. Let
w
(
x
)
,
f
(
x
)
∈
L
loc
1
(
R
N
)
$w\left(x\right),f\left(x\right)\in {L}_{\text{loc}}^{1}\left({\mathbb{R}}^{N}\right)$
be nonnegative functions such that
w
(
x
)
≤
C
1
|
x
|
λ
a
$w\left(x\right)\le {C}_{1}\vert x{\vert }_{\lambda }^{a}$
and
f
(
x
)
≥
C
2
|
x
|
λ
b
$f\left(x\right)\ge {C}_{2}\vert x{\vert }_{\lambda }^{b}$
for large |
x
|
λ
. Here,
C
i
(
i
= 1, 2) are positive constants,
a
,
b
∈
R
$a,b\in \mathbb{R}$
with
b
>
a
−
p
, the functions
λ
=
(
λ
1
,
λ
2
,
…
,
λ
k
)
:
R
N
→
R
k
$\lambda =\left({\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{k}\right) : {\mathbb{R}}^{N}\to {\mathbb{R}}^{k}$
satisfy some suitable conditions, and |⋅|
λ
the homogenous norm associated to the Δ
λ
-Laplacian. First, we establish Liouville results for stable weak solutions when
p
− 1 <
q
<
q
c
, where
q
c
is explicitly given. Next, we prove Liouville type theorems for finite Morse index solutions under some certain assumptions. The main tools we use are energy method and a Pohožaev type identity.