We study the following class of double-phase nonlinear eigenvalue problems
−
div
[
ϕ
(
x
,
|
∇
u
|
)
∇
u
+
ψ
(
x
,
|
∇
u
|
)
∇
u
]
=
λ
f
(
x
,
u
)
in
Ω
,
u
=
0
on
∂
Ω
, where Ω
is a bounded domain from
R
N
and the potential functions
ϕ
and
ψ
have
(
p
1
(
x
)
;
p
2
(
x
)
)
variable growth. The primitive of the reaction term of the problem (the right-hand side) has indefinite sign in the variable u and allows us to study functions with slower growth near
+
∞
, that is, it does not satisfy the Ambrosetti–Rabinowitz condition. Under these hypotheses we prove that for every parameter
λ
∈
R
+
∗
, the problem has an unbounded sequence of weak solutions. The proofs rely on variational arguments based on energy estimates and the use of Fountain Theorem.
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