Affiliation:
1. Sidi Mohamed Ben Abdellah University , Faculty of Sciences Dhar Al Mahraz , Laboratory of Mathematical Analysis and Applications , Fez , Morocco .
Abstract
Abstract
We consider the nonlocal eigenvalue problem of the following form
(
𝒫
k
)
{
K
p
(
x
)
u
(
x
)
+
|
u
(
x
)
|
p
¯
(
x
)
-
2
u
(
x
)
=
λ
|
u
(
x
)
|
r
(
x
)
-
2
u
(
x
)
i
n
Ω
,
u
=
0
i
n
N
\
Ω
,
$$(\mathcal{P}k)\left\{ {\matrix{ {\mathcal{L}_K^{p(x)}u(x) + {{\left| {u(x)} \right|}^{\bar p(x) - 2}}u(x)} \hfill & = \hfill & {\lambda {{\left| {u(x)} \right|}^{r(x) - 2}}u(x)} \hfill & {in} \hfill & {\Omega ,} \hfill \cr u \hfill & = \hfill & 0 \hfill & {in} \hfill & {{{\rm\mathbb{R}}^N}\backslash \Omega ,} \hfill \cr } } \right.$$
where Ω is a smooth open and bounded set in
N
(N ⩾ 3), λ > 0 is a real number, K is a suitable kernel and p, r are two bounded continuous functions on ̄Ω. The main result of this paper establishes that any λ > 0 sufficiently small is an eigenvalue of the above nonhomogeneous nonlocal problem. The proof relies on some variational arguments based on Ekeland's variational principle.
Subject
Applied Mathematics,Control and Optimization,Numerical Analysis,Analysis
Reference22 articles.
1. [1] E. Azroul, A. Benkirane, M. Shimi, An introduction to generalized fractional Sobolev Space with variable exponent, Preprint. https://arxiv.org/abs/1901.05687
2. [2] A. Bahrouni, V. Rădulescu, On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. 11 (2018), 379-389.
3. [3] G. M. Bisci, V. Rădulescu, R. Servadi, Variational methods for nonlocal fractional problems,Encyclopedia of mathematics and its applications, Cambridge University Press, Cambridge (2016).
4. [4] L. Caffarelli, Nonlocal diffusions, drifts and games, in nonlinear partial differential equations, Abel Symposia 7, Springer, Berlin (2012), 37-52.
5. [5] D.V. Cruz-Uribe , A. Fiorenza , Variable Lebesgue spaces. Foundations and harmonic analysis. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Heidelberg, 2013.
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