We consider the following classical autonomous variational problem: Minimize
\left\{F(u)=\int_a^b f(u(x),u'(x))\,dx\,:\,u\in AC([a,b]), u(a)=\alpha, u(b)=\beta,\,u([a,b]) \subseteq I \right\}
{
F
(
u
)
=
∫
a
b
f
(
u
(
x
)
,
u
′
(
x
)
)
d
x
:
u
∈
A
C
(
[
a
,
b
]
)
,
u
(
a
)
=
α
,
u
(
b
)
=
β
,
u
(
[
a
,
b
]
)
⊆
I
}
where
I
I
is a real interval,
\alpha, \beta\in I
α
,
β
∈
I
, and
f:I\times \mathbb{R}\to [0,+\infty)
f
:
I
×
R
→
[
0
,
+
∞
)
is possibly neither continuous, nor coercive, nor convex; in particular
f(s,\cdot)
f
(
s
,
⋅
)
may be not convex at
0
0
. Assuming the solvability of the relaxed problem, we prove under mild assumptions that the above variational problem has a solution, too.