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We are concerned with determining values of

In [

there exist

Using the bifurcation theory of Rabinowitz [

Let (

The results of Theorem

However, no results on the existence of nodal solutions, even positive solutions, have been established for one-dimensional

If

The purpose of this paper is to study the bifurcation behavior of one-dimensional

there exists

there exists

The main tool is the global bifurcation techniques in [

The rest of this paper is arranged as follows. In Section

Let

Let

We start by considering the following auxiliary problem:

Since the bifurcation points of

We define the operator

The following spectrum result plays a fundamental role in our study.

Let (

the set of all eigenvalues of the problem (

for

the eigenfunction corresponding to

Using the Gronwall inequality, we can easily show that all zeros of eigenfunction corresponding to eigenvalue

It is very known that

For

We divide the proof into two cases.

If

If

This together with Lemma

We first show that the principle eigenvalue function

The eigenvalue function

We only show that

Let

To do this, let

On applying the Dominated Convergence Theorem, we find that

Relation (

Thus, to prove (

Let us fix

Thus,

Similarly, we can also obtain that

We note that (

On the other hand, since

Now, letting

Finally, combining (

This and the variational characterization of

Using Remark

For fixed

Let

(i) Let

(ii) Let

We will only prove the case

It is easy to show that

For the existence of bifurcation branches for (

Let

Define the Nemytskii operators

Then, it is clear that

Notice that (

Assume that (

Finally, we give a key lemma that will be used in Section

Let

Let

Let

After taking a subsequence if necessary, we may assume that

Let

Let (

Then, (

We only prove the case of

Considering the results of Lemma

It is clear that any solution of (

In this case, it follows that

Let

We divide the equation

By the continuity and compactness of

We claim that

Suppose on the contrary that

By the openness of

On the contrary, we suppose that

Since

Let

However, this contradicts (

Assume that

If

If there exists

Applying the same method used in Step 1 of Case 1, after taking a subsequence and relabeling if necessary, it follows that

Thus,

This paper was supported by the NSFC (nos. 11061030, 11361047, and 11201378), SRFDP (no. 20126203110004), and Gansu Provincial National Science Foundation of China (no. 1208RJZA258).