We consider the nonlinear eigenvalue problem
\[
−
div
(
|
∇
u
|
p
(
x
)
−
2
∇
u
)
=
λ
|
u
|
q
(
x
)
−
2
u
-\textrm {div}\left (|\nabla u|^{p(x)-2}\nabla u\right )=\lambda |u|^{q(x)-2}u
\]
in
Ω
\Omega
,
u
=
0
u=0
on
∂
Ω
\partial \Omega
, where
Ω
\Omega
is a bounded open set in
R
N
\mathbb R^N
with smooth boundary and
p
p
,
q
q
are continuous functions on
Ω
¯
\overline \Omega
such that
1
>
inf
Ω
q
>
inf
Ω
p
>
sup
Ω
q
1>\inf _\Omega q> \inf _\Omega p>\sup _\Omega q
,
sup
Ω
p
>
N
\sup _\Omega p>N
, and
q
(
x
)
>
N
p
(
x
)
/
(
N
−
p
(
x
)
)
q(x)>Np(x)/\left (N-p(x)\right )
for all
x
∈
Ω
¯
x\in \overline \Omega
. The main result of this paper establishes that any
λ
>
0
\lambda >0
sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland’s variational principle.