Affiliation:
1. College of Mathematics, Changchun Normal University , Changchun 130032 , Jilin , PR China
Abstract
Abstract
In this article, we are concerned with the following critical nonlocal equation with variable exponents:
(
−
Δ
)
p
(
x
,
y
)
s
u
=
λ
f
(
x
,
u
)
+
∣
u
∣
q
(
x
)
−
2
u
in
Ω
,
u
=
0
in
R
N
\
Ω
,
\left\{\begin{array}{ll}{\left(-\Delta )}_{p\left(x,y)}^{s}u=\lambda f\left(x,u)+{| u| }^{q\left(x)-2}u& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right,\end{array}\right.
where
Ω
⊂
R
N
\Omega \subset {{\mathbb{R}}}^{N}
is a bounded domain with Lipschitz boundary,
N
≥
2
N\ge 2
,
p
∈
C
(
Ω
×
Ω
)
p\in C(\Omega \times \Omega )
is symmetric,
f
:
C
(
Ω
×
R
)
→
R
f:C\left(\Omega \times {\mathbb{R}})\to {\mathbb{R}}
is a continuous function, and
λ
\lambda
is a real positive parameter. We also assume that
{
x
∈
R
N
:
q
(
x
)
=
p
s
∗
(
x
)
}
≠
∅
\left\{x\in {{\mathbb{R}}}^{N}:q\left(x)={p}_{s}^{\ast }\left(x)\right\}\ne \varnothing
, and
p
s
∗
(
x
)
=
N
p
˜
(
x
)
⁄
(
N
−
s
p
˜
(
x
)
)
{p}_{s}^{\ast }\left(x)=N\tilde{p}\left(x)/\left(N-s\tilde{p}\left(x))
is the critical Sobolev exponent for variable exponents. We prove the existence of non-trivial solutions in the case of low perturbations (
λ
\lambda
small enough) by using the mountain pass theorem, the concentration-compactness principles for fractional Sobolev spaces with variable exponents, and the Moser iteration method. The features of this article are the following: (1) the function
f
f
does not satisfy the usual Ambrosetti-Rabinowitz condition and (2) this article contains the presence of critical terms, which can be viewed as a partial extension of the previous results concerning the the existence of solutions to this problem in the case of
s
=
1
s=1
and subcritical case.