On a class of fractional Laplacian problems with variable exponents and indefinite weights
Let Ω ⊂ R N , N ≥ 2 , be a bounded smooth domain. In this paper, we consider a class of fractional Laplacian problems of the form ( Δ ) p 1 ( x , . ) s u ( x ) + ( Δ ) p 2 ( x , . ) s u ( x ) + | u | q ( x ) - 2 u = λ V 1 ( x ) | u ( x ) | r 1 ( x ) - 2 u ( x ) - μ V 2 ( x ) | u ( x ) | r 2 ( x ) -...
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| Published in | Collectanea mathematica (Barcelona) Vol. 71; no. 2; pp. 223 - 237 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Milan
Springer Milan
01.05.2020
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Summary: | Let
Ω
⊂
R
N
,
N
≥
2
, be a bounded smooth domain. In this paper, we consider a class of fractional Laplacian problems of the form
(
Δ
)
p
1
(
x
,
.
)
s
u
(
x
)
+
(
Δ
)
p
2
(
x
,
.
)
s
u
(
x
)
+
|
u
|
q
(
x
)
-
2
u
=
λ
V
1
(
x
)
|
u
(
x
)
|
r
1
(
x
)
-
2
u
(
x
)
-
μ
V
2
(
x
)
|
u
(
x
)
|
r
2
(
x
)
-
2
u
(
x
)
in
Ω
,
u
(
x
)
=
0
in
∂
Ω
,
where
(
Δ
)
p
i
(
.
,
.
)
s
(
0
<
s
<
1
)
,
i
=
1
,
2
, are the fractional
p
i
(
.
,
.
)
-Laplacians,
p
i
∈
C
(
Ω
¯
×
Ω
¯
)
,
q
,
r
i
∈
C
(
Ω
¯
)
,
i
=
1
,
2
while
λ
,
μ
are two positive parameters,
V
1
,
V
2
are weight functions in generalized Lebesgue spaces
L
α
1
(
.
)
(
Ω
)
and
L
α
2
(
.
)
(
Ω
)
respectively such that
V
1
may change sign in
Ω
and
V
2
(
x
)
≥
0
for all
x
∈
Ω
. Using variational techniques and Ekeland’s variational principle, we establish some existence results for the problem in an appropriate space of functions. |
|---|---|
| ISSN: | 0010-0757 2038-4815 |
| DOI: | 10.1007/s13348-019-00254-5 |