In this section we present that problem (1.5) has countably many solutions if and satisfy some suitable conditions.

For convenience, we denote

Theorem 4.1.

Suppose conditions and hold, let be such that Let and be such that

where , . Furthermore, for each natural number , assume that satisfies the following two growth conditions:

for all ,

for all .

Then problem (1.5) has countably many positive solutions such that for each

Proof.

Consider the sequences and of open subsets of defined by

Let be as in the hypothesis and note that for all . For each , define the cone by

Fixed and let . For , we have

By condition , we get

Now let , then for all . By condition , we get

It is obvious that . Therefore, by Theorem 2.4, the operator has at least one fixed point such that . Since was arbitrary, Theorem 4.1 is completed.

Let is defined by Theorem 4.1. We define the nonnegative continuous concave functionals on by

We observe here that, for each , .

For convenience, we denote

Theorem 4.2.

Suppose conditions and hold, let be such that Let , , and be such that

where , . Furthermore, for each natural number , assume that satisfies the following growth conditions:

for all ,

for all ,

for all .

Then problem (1.5) has three infinite families of solutions , and such that

for each

Proof.

We note first that is completely continuous operator. If , then from properties of , , and by Lemma 3.7, . Consequently, .

If , then , and by condition , we have

Therefore, . Standard applications of Arzela-Ascoli theorem imply that is completely continuous operator.

In a completely analogous argument, condition implies that condition of Theorem 2.5 is satisfied.

We now show that condition of Theorem 2.5 is satisfied. Clearly,

If , then , for . By condition , we get

Therefore, condition of Theorem 2.5 is satisfied.

Finally, we show that condition of Theorem 2.5 is also satisfied.

If and , then

Therefore, condition is also satisfied. By Theorem 2.5, There exist three infinite families of solutions , , and for problem (1.5) such that

for each Thus, Theorem 4.2 is completed.