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.