First, we present our main results as follows.
Theorem 3.1.
Let
for any
. Put
for any
and assume that there exist four positive constants
with
and
such that
(A1) 
(A2)
.
Furthermore, put
and for each
,
Then, for each
the problem (1.1) admits at least three solutions in
and, moreover, for each
, there exist an open interval
and a positive real number
such that, for each
, the problem (1.1) admits at least three solutions in
whose norms in
are less than
.
Remark 3.2.
By the condition (A1), we have
That is,
Thus, we get
Namely, we obtain the fact that
.
Proof of Theorem 3.1.
Let
be the Hilbert space
. Thanks to Remark 2.1, we can apply Theorem 1.1 to the two functionals
and
. We know from the definitions in (2.5) that
is a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on
, and
is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Now, put
for any
, it is easy to see that
and
.
Next, in view of the assumption (A2) and the relation (2.4), we know that for any
and
,
Taking into account the fact that
, we obtain, for all
,
The condition (i) of Theorem 1.1 is satisfied.
Now, we let
It is clear that
,
In view of
, we get
So, the assumption (ii) of Theorem 1.1 is obtained. Next, we verify that the assumption (iii) of Theorem 1.1 holds. From Lemma 2.2, the estimate
implies that
for any
. From the definition of
, it follows that
Thus, for any
, we have
On the other hand, we get
Therefore, it follows from the assumption (A1) that
that is, the condition (iii) of Theorem 1.1 is satisfied.
Note that
By a simple computation, it follows from the condition (A1) that
. Applying Theorem 1.1, for each
, the problem (1.1) admits at least three solutions in
.
For each
, we easily see that
Taking the condition (A1) into account, it forces that
. Then from Theorem 1.1, for each
, there exist an open interval
and a positive real number
, such that, for
, the problem (1.1) admits at least three solutions in
whose norms in
are less than
. The proof of Theorem 3.1is complete.
As a special case of the problem (1.1), we consider the following systems:
where
and
are nonnegative. Define
Then Theorem 3.1 takes the following simple form.
Corollary 3.3.
Let
and
be two nonnegative functions. Assume that there exist four positive constants
with
and
such that
(A′1) 
(A′2)
for any
.
Furthermore, put
and for each
,
Then, for each
the problem (3.19) admits at least three solutions in
and, moreover, for each
, there exist an open interval
and a positive real number
such that, for each
, the problem (3.19) admits at least three solutions in
whose norms in
are less than
.
Proof.
Note that from fact
for any
, we have
On the other hand, we take
. Obviously, all assumptions of Theorem 3.1 are satisfied.
To the end of this paper, we give an example to illustrate our main results.
Example 3.4.
We consider (1.1) with
, where
We have that
and
It can be easily shown that, when
, and
, all conditions of Corollary 3.3 are satisfied.