We consider the Dirichlet problem for the equation − u = λu ± f (x, u) + h(x) in a bounded domain, where f has a sublinear growth and h ∈ L 2 . We find suitable conditions on f and h in order to have at least two solutions for λ near to an eigenvalue of − . A typical example to which our results apply is when f (x, u) behaves at infinity like a(x)|u| q−2 u, with M > a(x) > δ > 0, and 1 < q < 2.
We study the Schrödinger equations
−
Δ
u
+
V
(
x
)
u
=
f
(
x
,
u
)
-\Delta u + V(x)u = f(x,u)
in
R
N
\mathbb {R}^N
and
−
Δ
u
−
λ
u
=
f
(
x
,
u
)
-\Delta u - \lambda u = f(x,u)
in a bounded domain
Ω
⊂
R
N
\Omega \subset \mathbb {R}^N
. We assume that
f
f
is superlinear but of subcritical growth and
u
↦
f
(
x
,
u
)
/
|
u
|
u\mapsto f(x,u)/|u|
is nondecreasing. In
R
N
\mathbb {R}^N
we also assume that
V
V
and
f
f
are periodic in
x
1
,
…
,
x
N
x_1,\ldots ,x_N
. We show that these equations have a ground state and that there exist infinitely many solutions if
f
f
is odd in
u
u
. Our results generalize those by Szulkin and Weth [J. Funct. Anal. 257 (2009), 3802–3822], where
u
↦
f
(
x
,
u
)
/
|
u
|
u\mapsto f(x,u)/|u|
was assumed to be strictly increasing. This seemingly small change forces us to go beyond methods of smooth analysis.
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