Nodal Solutions of the p-Laplacian with Sign-Changing Weight

and Applied Analysis 3 (iii) the eigenfunction corresponding to μ k (p) has exactly k − 1 simple zeros in (0, 1). Remark 3. Using the Gronwall inequality, we can easily show that all zeros of eigenfunction corresponding to eigenvalue μ ] k (p) are simple. It is very known that T2 λ is completely continuous in C 1 [0, 1].Thus, the Leray-Schauder degree dLS(I−T 2 λ , B r (0), 0) is well-defined for arbitrary r-ball B r (0) and λ ̸ = μ k , k ∈ Z and ] ∈ {+, −}. Lemma 4. For r > 0, we have dLS (I − T 2 λ , B r (0) , 0) = {{ {{ { 1, if λ ∈ (μ − 1 (2) , μ + 1 (2)) , (−1) k , if λ ∈ (μ + k (2) , μ + k+1 (2)) , k ∈ N, (−1) k , if λ ∈ (μ − k+1 (2) , μ − k (2)) , k ∈ N. (17) Proof. We divide the proof into two cases. Case 1 . λ ≥ 0. SinceT λ is compact and linear, by [20,Theorem 8.10] and Lemma 2 (ii) with p = 2,

The results of Theorem 1 have been extended to the case that the weight function changes its sign by Ma and Han [4].Bifurcation methods have been applied to study the existence of nodal solutions of nonlinear two-point, multipoint, and periodic boundary value problems; see [5][6][7][8][9] and the references therein.The results they obtained extend some well-known theorems of the existence of positive solutions for the related problems [10].
However, no results on the existence of nodal solutions, even positive solutions, have been established for one-dimensional -Laplacian equation with sign-changing weight ().It is the purpose of this paper to establish a similar result to Theorem 1 for one-dimensional -Laplacian equation with sign-changing weight.Problem with signchanging weight arises from the selection-migration model in population genetics.In this model, () changes sign corresponding to the fact that an allele  1 holds an advantage over a rival allele  2 at the same points and is at a disadvantage at others; the parameter  corresponds to the reciprocal of diffusion; for details see [11].
If () ≡ 1, Del Pino et al. [12] established the global bifurcation theory for one-dimensional -Laplacian eigenvalue problem.Peral [13] got the global bifurcation theory for -Laplacian eigenvalue problem on the unite ball.In [14], Del Pino and Manásevich obtained the global bifurcation from the principal eigenvalue for -Laplacian eigenvalue problem on the general domain.If () ≥ 0 and is singular at  = 0 or  = 1, Lee and Sim [15] also established the bifurcation theory for one-dimensional -Laplacian eigenvalue problem.However, if () changes sign, there are a few papers dealing with the -Laplacian eigenvalue problem via bifurcation techniques.In [16], Drábek and Huang established the global bifurcation from the principal eigenvalue for -Laplacian eigenvalue problem in R  .
The purpose of this paper is to study the bifurcation behavior of one-dimensional -Laplacian eigenvalue problem as follows: under the condition ( 1 ) and ( 3 ) there exists  0 ∈ (0, ∞) such that where   () = || −2  with 1 <  < +∞; ( 4 ) there exists  ∞ ∈ (0, +∞) such that Moreover, based on our global bifurcation theorem, we will prove the existence of nodal solutions for the corresponding nonlinear problem with a parameter (see Theorem 11).The main tool is the global bifurcation techniques in [17].The rest of this paper is arranged as follows.In Section 2, we establish the global bifurcation theory for one-dimensional -Laplacian eigenvalue problem with signchanging weight.In Section 3, we state and prove the main results of this paper.

Some Preliminaries
Let  be the Banach space  1 0 [0, 1] with the norm Let  =  1 (0, 1) with its usual normal ‖ ⋅ ‖  1 .We start by considering the following auxiliary problem: for a given ℎ ∈  1 (0, 1).By a solution of problem (9), we understand a function  ∈  with   (  ) absolutely continuous which satisfies (9).Problem ( 9) is equivalently written to where  :  → R is a continuous function satisfying It is known that   :  →  is continuous and maps equiintegrable sets of  into relatively compacts of .One may refer to Lee and Sim [15] for details.

Remark 3. Using the Gronwall inequality, we can easily show that all zeros of eigenfunction corresponding to eigenvalue 𝜇 ]
() are simple.
Case 1 . ≥ 0. Since  2  is compact and linear, by [20, Theorem 8.10] and Lemma 2 (ii) with  = 2, where () is the sum of algebraic multiplicity of the eigenvalues  of ( 13) satisfying ), then there are no such  at all; then If  ∈ ( +  (2),  + +1 (2)) for some  ∈ N, then This together with Lemma 2 (ii) implies the following: Case 2 . < 0. In this case, we consider a new sign-changing eigenvalue problem as follows where Thus, we may use the result obtained in Case 1 to deduce the desired result.
Proof.We will only prove the case  >  + 1 () since the proof for the other cases is similar.We also only give the proof for the case  > 2. Proof for the case 1 <  < 2 is similar.Assume that  +  () <  <  + +1 () for some  ∈ N. Since the eigenvalues depend continuously on , there exists a continuous function  : For the existence of bifurcation branches for ( 12), we will make use of the following global bifurcation theorem results.Lemma 8 (see [17]).Let  be a Banach space.Let  : R× →  be completely continuous such that (, 0) = 0 for all  ∈ R. Suppose that there exist constants ,  ∈ R, with  < , such that (, 0) and (, 0) are not bifurcation points for the equation Furthermore, assume that where   (0) = { ∈  : ‖‖ < } is an isolating neighborhood of the trivial solution for both constants  and .Let and let C be the component of S containing [, ] × {0}.Then, either Define the Nemytskii operators  : R ×  →  by  (, ) () := − ()  ( ()) .
Then, it is clear that  is continuous operator which sends bounded sets of R ×  into an equi-integrable sets of  and problem ( 12) can be equivalently written as is completely continuous in R ×  →  and (, 0) = 0, for all  ∈ R.
Finally, we give a key lemma that will be used in Section 3. Let (55) Let   ∈  be a solution of the equation Then, the number of zeros of   |  goes to infinity as  → +∞.
Proof.After taking a subsequence if necessary, we may assume that as  → +∞.It is easy to check that the distance between any two consecutive zeros of any nontrivial solution of the equation goes to zero as  → +∞.Using this with [21, Lemma 2.5], it follows the desired results.

Main Results and Its Proof
Let  ±  be the th positive or negative eigenvalue of (13).By applying Lemma 9, we will establish the main results as follows.
Step 2. We show that there exists a constant  such that   ∈ (0, ] for  ∈ N large enough.