1. Introduction
In this paper, we consider stationary Schrödinger–Poisson systems of the form
We assume the following conditions for the nonlinearity
f and the potential
V:
The condition (
2) means that the nonlinearity
f is sublinear at the origin and subcritical at infinity. Under these mild conditions, we have the following multiplicity results.
Theorem 1. Suppose and are satisfied, then the problem (1) has a sequence of weak solutions . Example 1. Let and be defined byThen, f and V satisfy our assumptions and with , respectively. Remark 1. Note that in the assumption , we have assumed that V is positive. This condition can be replaced by the following weaker one which allows V to be negative somewhere:
To see this, set , withThen, and satisfy the assumptions and , and hence we can apply Theorem 1 to the equivalent problem The problem (
1) is variational. Let
be ths solution of the second equation in (
1). As proposed by Benci et al. [
1,
2], it is well known that if
u is a critical point of
being
, then
is a solution of (
1). Using this idea, many results on (
1) have been obtained assuming that
in the last two decades; see [
3,
4,
5] for a case where the Schrödinger operator
is positive and [
6] for a case where the Schrödinger operator
S is indefinite.
In all these papers, and many papers on Schrödinger–Poisson systems, some conditions on the nonlinearity f, like the Ambrosetti–Rabinowitz condition, are needed to ensure that the variational functional satisfies the Palais–Smale condition.
Unlike all these papers, our assumption
on the nonlinearity
f implies that the limit in (
4) is infinity, and our assumptions on
f are not sufficient for ensuring the boundedness of
sequences. Motivated by He and Wu [
7], who studied the semilinear elliptic boundary value problem
on a bounded domain
, we apply the truncation method and a version of Clark’s theorem created by Liu and Wang [
8] to overcome this difficulty.
Using the same idea, we obtained a similar result for Schrödinger–Kirchhoff equations of the form (
)
We make the following assumptions:
Note that is the critical Sobolev exponent. If , then and and reduce to and , respectively.
Theorem 2. Suppose and are satisfied, then the problem (5) has a sequence of weak solutions . For problem (
5), we make a remark similar to Remark 1. This paper is organized as follows. In
Section 2, we present the functional spaces as our frameworks for studying problems (
1) and (
5), and Clark’s theorem as altered by Liu and Wang [
8] is also recalled in this section. The proofs of Theorems 1 and 2 will be given in
Section 3 and
Section 4, respectively.
2. Preliminaries
Equip the subspace (
)
with the norm
and the corresponding inner product
given by
Then,
is a Hilbert space. Since
, to prove Theorem 1, it suffices to find a sequence of critical points of the
-functional
given in (
3). Note that by the definition of
,
Obviously, the embedding
is continuous. According to [
9] [Lemma 2.1], we obtain the following result.
Proposition 1. Under the assumption , can be continuously embedded into for and the embedding is compact for .
Corollary 1. Under the assumptions and , if in , then up to a subsequence Proof. Given
, there is
such that
Thus, by Hölder inequality
Because
is bounded in
and
in
by Proposition 1, we obtain
for some
, which implies (
6). □
To find the critical points of
, the properties of
and the functional
are crucial. Similar to [
10] [Lemma 2.2 (1)], we have the following proposition.
Proposition 2. Let be the unique solution of for . Then, there is a constant such that To conclude this section, we recall Clark’s theorem, which will be needed for proving our main results.
Proposition 3 ([
8] [Theorem 1.1]).
Let X be a Banach space and be an even coercive functional satisfying the condition for and . If for any there is a k-dimensional subspace and such thatwhere for , , then Ψ has a sequence of critical points such that , . 3. Proof of Theorem 1
Let
be a decreasing
-function such that
,
We consider the truncated functional
,
The derivative of
is given by
for
.
Lemma 1. The functional Ψ is coercive and satisfies for .
Proof. Let
be a sequence such that
. Then, for
n large, we have
and
. Thus
Hence
is coercive.
For
, let
be a
sequence of
. That is
,
. Then for
n large we have
We claim that
We consider two cases:
If
, then
; hence (
11) is true.
If
, then the right-hand side of (
10) is negative or
n large. Thus
which implies (
11) because
.
Because
is coercive, the
sequence
is bounded in
X. Thus, up to a subsequence, we have
By Hölder inequality and Proposition 2, we obtain
where
S is the best constant of the embedding
. Consequently, from (
9), (
11), Corollary 1 and
we deduce
We conclude that
. Noting
in
, we obtain
in
. □
Now we are ready to prove Theorem 1.
Proof of Theorem 1. Since for (the unit ball in ), it suffices to show that has a sequence of critical points satisfying . For this purpose, we shall apply Proposition 3.
Firstly, we remark that by condition
, there is
such that
Let
and
be a
k-dimensional subspace of
. Since all norms on
are equivalent, there is
such that
implies
; thus
for
. Hence, for
, using Proposition 2 we obtain
being
for some constant
. Because
it is clear that there is
such that
. Hence,
As we have seen, the -functional is even, coercive and satisfies for . Since is trivially true, by Proposition 3 has a sequence of critical points converging to the zero function. This completes the proof of Theorem 1. □