Nonlocal eigenvalue problems with variable exponent

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.