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

(A_{1})

(A_{2}) .

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 (A_{1}), 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 (A_{2}) 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 (A_{1}) that

that is, the condition (iii) of Theorem 1.1 is satisfied.

Note that

By a simple computation, it follows from the condition (A_{1}) 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 (A_{1}) 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.