Under general hypotheses, we prove the existence of a nontrivial solution for the equation
L
u
=
N
(
u
)
Lu = N(u)
, where
u
u
belongs to a Hilbert space
H
H
,
L
L
is an invertible continuous selfadjoint operator, and
N
N
is superlinear. We are particularly interested in the case where
L
L
is strongly indefinite and
N
N
is not compact. We apply the result to the Choquard-Pekar equation
\[
−
Δ
u
(
x
)
+
p
(
x
)
u
(
x
)
=
u
(
x
)
∫
R
3
u
2
(
y
)
|
x
−
y
|
d
y
,
u
∈
H
1
(
R
3
)
,
u
≠
0
,
- \Delta u(x) + p(x)u(x) = u(x)\int _{{\mathbb {R}^3}} {\frac {{{u^2}(y)}} {{|x - y|}}dy,\qquad u \in {H^1}({\mathbb {R}^3}),\quad u \ne 0,}
\]
where
p
∈
L
∞
(
R
3
)
p \in {L^\infty }({\mathbb {R}^3})
is a periodic function.