In this section we state our main results and proofs.

Definition 3.1.

where

is the first eigenvalue of the corresponding linear problem

Remark 3.2.

It is well known that the eigenvalues of (3.2) are given by

For each
, algebraic multiplicity of
is equal to 1, and the corresponding eigenfunction
has exactly
simple zeros in
.

Definition 3.3 (see [7]).

Let
. Then
is said to be *superlinear* (resp., *sublinear*) on
if
(resp.,
) on
. And
is said to be *sup-sub* (resp., *sub-sup*) on
if there exists
such that
is superlinear (resp., sublinear) on
, and superlinear (resp., sublinear) on
.

Proof.

The proof is similar to that of Proposition
in [7], so we omit it.

Lemma 3.5.

Let (H1)–(H4) hold, let
be a bounded and closed interval, and let
. Suppose that
are positive solutions of (1.1). Then,

(i)
, if
,

(ii)
, if
.

Proof.

Let

be such that

as a bifurcation problem from
. Note that (3.6) is the same as to (1.1). From Remark 3.2 and the standard bifurcation theorem from simple eigenvalues [17], we have (i).

as a bifurcation problem from infinity. Note that (3.7) is also the same as to (1.1). The proof of Theorem
in [5] ensures that (ii) is correct.

Lemma 3.6.

Let (H1), (H4) hold. Suppose that

is a positive solution of (1.1). Then,

Proof.

Suppose to the contrary that

By (1.1) and (H1), we have

. Note that

. By the uniqueness of solutions of initial value problem, the problem

has a unique solution
. This contradicts
.

The following Lemma is an interesting and important result.

Lemma 3.7.

Let (H1)–(H4) hold. Suppose that
is a positive solution of (1.1), then
is nondegenerate.

Proof.

From conditions (H1)–(H3), we can check easily that

In fact, let

, then

since
. Note that
, if
and
, if
. This together with (3.12) implies that
and (3.11).

Now, we give the proof in two cases.

Case I (
).

On the contrary, suppose that

is a degenerate solution with

, then

for all

. By (3.11), we get

Multiplying (1.1) by

and (2.7) by

, subtracting, and integrating, we have

By Lemma 2.2, (3.13), and
for all
, the right side of (3.14) is negative. This is a contradiction.

Case II (
).

On the contrary, suppose that
is a degenerate solution with
. According to Lemmas 2.2 and 2.3, we know that all solutions of (1.1) near
satisfy
for
and some
, where
. It follows that for
close to
we have two solutions
and
with
strictly increasing in
and
with strictly decreasing in
. We will show that the lower branch
is strictly increasing for all
.

Note that

for

close to

and all

. Let

be the largest

where this inequality is violated; that is,

and

for some

. Differentiating (1.1) with respect to

,

We can extend evenly

, and

on

, then we obtain

By the strong maximum principle, we conclude that
for all
. This contradicts that
.

By Lemma 2.3, we get
at every degenerate positive solution. Hence, there is no degenerate positive solution on the lower branch
. However, the lower branch has no place to go. In fact, there must exist some positive constant
such that
for any
lying on
. Hence, the lower branch cannot go to the
axis. And it also cannot go to the
axis, since (1.1) has only the trivial solution at
.

So,
is nondegenerate.

Our main result is the following.

Theorem 3.8.

Let (H1)–(H4) hold. Then the following are considered.

(i) All positive solutions of (1.1) lie on two continuous curves
and
without intersection.
bifurcates from
to infinity and
;
bifurcates from
to infinity and
. There is no degenerate positive solution on these curves. For any
,
, and for any
,
.

(ii) Equation (1.1) has no positive solution for
has exactly one positive solution for
but and has exactly two positive solutions for
(see Figure 1).

Proof.

- (i)
Since

and

, then

. From Lemma 3.5(i) and the standard Crandall and Rabinowitz theorem on local bifurcation from simple eigenvalues [

17],

is the unique point where a bifurcation from the trivial solution occurs. Moreover, by Lemma 3.4, the curve bifurcates to the right. We denote this local curve by

and continue

to the right as long as it is possible. Meanwhile, by Lemma 3.6, there is no positive solution of (1.1) which has the maximum value

on

. So, if

, then

. From (1.1), we have

where
. Obviously, there exists a constant
such that
if
is bounded. Hence,
cannot blow up.

On the other hand, Lemma 3.7 and the implicit function theorem ensure that
cannot stop at a finite point
.

From the above discussion, we see that
can be extended continuously to infinity and
. Meanwhile, the maximum values of all positive solutions of (1.1) are less than
.

Now, we consider positive solutions of (1.1), for which the maximum value on
is greater than
.

Let us return to consider (3.6) as the bifurcation problem from infinity. Note that (3.6) is also the same as to (1.1). Since
by Theorem
and Corollary
in [18], there exists a subcontinuum
of positive solutions of (3.6) which meets
. Take
as an interval such that
and
as a neighborhood of
whose projection on
lies in
and whose projection on
is bounded away from
. Then, there exists a neighborhood
such that any positive solution
of (1.1) satisfies
for
and some
and
at
, where
denotes the normalized eigenvector of (3.2) corresponding to
. So,

Hence,
is a continuous curve, and we denote it by
. It tends to the right from Lemma 3.4(ii). From Lemma 3.7 and the implicit function theorem,
can be continued to a maximal interval of definition over the
axis. We claim that
*∖*
cannot blow up if
is bounded. In fact, suppose that there exists a positive solutions sequence
of (1.1) and
such that
as
. Then, by Lemma 3.5(ii),
. This is a contradiction. On the other hand, the implicit function theorem implies that
cannot stop at a finite point
. Thus,
and
if
.

Finally, we show that both curves

and

are the only two positive solutions curves of (1.1). On the contrary, suppose that

is a positive solution of (1.1) with

. Without loss of generality, assume that

. Note that

is nondegenerate, so we can extend it to form a curve. We denote this curve by

and the corresponding maximal interval of definition by

. Since all positive solutions of (1.1) are nondegenerate, according to the implicit function theorem, we must have that

It follows that
from Lemma 3.5(ii). But all solutions near
can be parameterized by
for
and some
; thus,
. This contradicts that
.

Similarly, we can show that every positive solution of (1.1), the maximum value on

of which is less than

lies on

.

- (ii)
The result (ii) is a corollary of (i).

Next, we will give directly other theorems. Their proofs are similar to that of Theorem 3.8. So, we omit them.

Theorem 3.9.

Let
and (H2)–(H4) hold. Then, the following are considered.

(i) All positive solutions of (1.1) lie on a single continuous curve
. And
bifurcates from
to the right to a unique degenerate positive solution
of (1.1), then it tends to the left to
.

(ii) Equation (1.1) has no positive solution for
, and has exactly one positive solution for
, and has exactly two positive solutions for
(see Figure 2).

Remark 3.10.

In fact, if we reverse the inequalities in (H1), (H1
), (H2), we will obtain corresponding results similar to Theorems 3.8 and 3.9.

Also using the method in this paper, we can obtain the exact numbers of positive solutions for the Dirichlet problem

where
is a parameter. We assume that

(H)
with
, and
for all
.

Definition 3.11
.

where
is the first eigenvalue of the corresponding linear problem of (3.19).

Theorem 3.12.

Let (H1
), (H2), (H3), and (H4
) hold. Then, the following are considered.

(i) All positive solutions of (3.19) lie on a single continuous curve
. And
bifurcates from
to the right to a unique degenerate positive solution
of (3.19), then it tends to the left to
.

(ii) Equation (1.1) has no positive solution for
but has exactly one positive solution for
and has exactly two positive solutions for
.

Theorem 3.13.

Let (H1), (H2), (H3), (H4
) hold. Then

(i) All positive solutions of (3.19) lie on two continuous curves
and
without intersection.
bifurcates from
to infinity and
;
bifurcates from
to infinity and
. There is no degenerate positive solution on these curves. For any
,
, and for any
,
.

(ii) Equation (3.19) has no positive solution for
, and has exactly one positive solution for
, and has exactly two positive solutions for
.

Remark 3.14.

Theorems 3.12 and 3.13 extend the main result Theorem
in [10], where
for
.