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

Abstract

We are concerned with determining values of 𝛾, for which there exist nodal solutions of the boundary value problem (|𝑢^′|^𝑝−2𝑢^′)^′+𝛾𝑚(𝑡)𝑓(𝑢)=0,𝑡∈(0,1), 𝑢(0)=𝑢(1)=0, where 𝑚∈𝐶[0,1] is a sign-changing function, 𝑓∶ℝ→ℝ with 𝑓(𝑠)𝑠>0. The proof of our main results is based upon global bifurcation techniques.

1. Introduction

In [1], Ma and Thompson considered determining values of 𝑟, for which there exist nodal solutions of the boundary value problem 𝑢^′′(𝑡)𝑓(𝑢)(),𝑢(0)(1)+𝑟𝑚=0,𝑡∈0,1=𝑢=0, (1) under the following assumptions:(𝐻₁)𝑓∈𝐶(ℝ,ℝ) with 𝑠𝑓(𝑠)>0 for 𝑠≠0;(𝐻₂)𝑚∶[0,1]→[0,+∞) is continuous and does not vanish identically on any subinterval of [0,1];(𝐻₃)there exist 𝑓₀, 𝑓_∞∈(0,+∞) such that 𝑓₀=lim_|𝑠|→0𝑓(𝑠)𝑠,𝑓_∞=lim_{|𝑠|→+∞}𝑓(𝑠)𝑠. (2)

Using the bifurcation theory of Rabinowitz [2, 3], they proved the following.

Theorem 1. Let (𝐻₁), (𝐻₂), and (𝐻₃) hold. Assume that, for some 𝑘∈ℕ, either 𝜆_𝑘𝑓_∞𝜆<𝑟<^𝑘𝑓₀𝜆𝑜𝑟^𝑘𝑓₀𝜆<𝑟<^𝑘𝑓_∞. (3) Then (1) has two solutions 𝑢⁺_𝑘 and 𝑢⁻_𝑘 such that 𝑢⁺_𝑘 has exactly 𝑘−1 zeros in (0,1) and is positive near 0 and 𝑢⁻_𝑘 has exactly 𝑘−1 zeros in (0,1) and is negative near 0.

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 [5–9] 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 𝑝-Laplacian equation with sign-changing weight 𝑚(𝑡). It is the purpose of this paper to establish a similar result to Theorem 1 for one-dimensional 𝑝-Laplacian equation with sign-changing weight. Problem with sign-changing weight arises from the selection-migration model in population genetics. In this model, 𝑚(𝑡) changes sign corresponding to the fact that an allele 𝐴₁ holds an advantage over a rival allele 𝐴₂ at the same points and is at a disadvantage at others; the parameter 𝑟 corresponds to the reciprocal of diffusion; for details see [11].

If 𝑚(𝑡)≡1, Del Pino et al. [12] established the global bifurcation theory for one-dimensional 𝑝-Laplacian eigenvalue problem. Peral [13] got the global bifurcation theory for 𝑝-Laplacian eigenvalue problem on the unite ball. In [14], Del Pino and Manásevich obtained the global bifurcation from the principal eigenvalue for 𝑝-Laplacian eigenvalue problem on the general domain. If 𝑚(𝑡)≥0 and is singular at 𝑡=0 or 𝑡=1, Lee and Sim [15] also established the bifurcation theory for one-dimensional 𝑝-Laplacian eigenvalue problem. However, if 𝑚(𝑡) changes sign, there are a few papers dealing with the 𝑝-Laplacian eigenvalue problem via bifurcation techniques. In [16], Drábek and Huang established the global bifurcation from the principal eigenvalue for 𝑝-Laplacian eigenvalue problem in ℝ^𝑁.

The purpose of this paper is to study the bifurcation behavior of one-dimensional 𝑝-Laplacian eigenvalue problem as follows: 𝜑_𝑝𝑢^′′(𝑡)𝑓(𝑢)(),𝑢(0)(1)+𝛾𝑚=0,𝑡∈0,1=𝑢=0, (4) under the condition (𝐻₁) and(𝐻₂)𝑚(𝑡)∈𝐶[0,1] changes sign and meas{𝑥∈[0,1]∣𝑚(𝑡)=0}=0; (5)(𝐻₃)there exists 𝑓₀∈(0,∞) such that 𝑓₀=lim_|𝑠|→0𝑓(𝑠)𝜑_𝑝(𝑠), (6)where 𝜑_𝑝(𝑠)=|𝑠|^𝑝−2𝑠 with 1<𝑝<+∞;(𝐻₄)there exists 𝑓_∞∈(0,+∞) such that 𝑓_∞=lim_{|𝑠|→+∞}𝑓(𝑠)𝜑_𝑝(𝑠). (7)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 𝑝-Laplacian eigenvalue problem with sign-changing weight. In Section 3, we state and prove the main results of this paper.

2. Some Preliminaries

Let 𝐸 be the Banach space 𝐶¹₀[0,1] with the norm ‖𝑢‖=max‖𝑢‖_∞,𝑢^′_∞. (8)

Let 𝑌=𝐿¹(0,1) with its usual normal ‖⋅‖_𝐿¹.

We start by considering the following auxiliary problem: 𝜑_𝑝𝑢^′′(),𝑢(0)(1)=ℎ,𝑡∈0,1=𝑢=0, (9) for a given ℎ∈𝐿¹(0,1). By a solution of problem (9), we understand a function 𝑢∈𝐸 with 𝜑_𝑝(𝑢^′) absolutely continuous which satisfies (9). Problem (9) is equivalently written to 𝑢(𝑡)=𝐺_𝑝(ℎ)(𝑡)∶=^𝑡₀𝜑_𝑝⁻¹𝑎(ℎ)+^𝑠₀ℎ(𝜏)𝑑𝜏𝑑𝑠, (10) where 𝑎∶𝑌→ℝ is a continuous function satisfying ¹₀𝜑_𝑝⁻¹𝑎(ℎ)+^𝑠₀ℎ(𝜏)𝑑𝜏𝑑𝑠=0. (11) It is known that 𝐺_𝑝∶𝑌→𝐸 is continuous and maps equi-integrable sets of 𝑌 into relatively compacts of 𝐸. One may refer to Lee and Sim [15] for details.

Since the bifurcation points of 𝜑_𝑝𝑢^′(𝑡)^′(𝑡)𝑓(𝑢(𝑡(),𝑢(0)(1)+𝜆𝑚))=0a.e.in0,1=𝑢=0 (12) is related to the eigenvalues of the problem 𝜑_𝑝𝑢^′(𝑡)^′+𝜆𝑚(𝑡)𝜑_𝑝(𝑢(𝑡(),𝑢(0)(1)))=0a.e.in0,1=𝑢=0. (13)

We define the operator 𝑇^𝑝_𝜆∶𝐸→𝐸 by 𝑇^𝑝_𝜆(𝑢)(𝑡)=^𝑡₀𝜑_𝑝⁻¹𝑎−𝜆𝑚𝜑_𝑝(𝑢(𝜏−))_𝑠0𝜆𝑚(𝜏)𝜑_𝑝(𝑢(𝜏))𝑑𝜏𝑑𝑠=∶𝐺_𝑝−𝜆𝑚𝜑_𝑝(𝑢)(𝑡). (14) Then 𝑇^𝑝_𝜆∶𝐸→𝐸 is completely continuous and problem (13) is equivalent to 𝑢=𝑇^𝑝_𝜆(𝑢). (15)

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

Lemma 2 (see [18, 19]). Let (𝐻₂) hold. Then(i)the set of all eigenvalues of the problem (13) is two infinite sequences of simple eigenvalues as follows: 0<𝜇⁺₁𝑝<𝜇⁺₂𝑝<⋯<𝜇⁺_𝑘𝑝<⋯,lim_𝑘→+∞𝜇⁺_𝑘𝑝=+∞,0>𝜇⁻₁𝑝>𝜇⁻₂𝑝>⋯>𝜇⁻_𝑘𝑝>⋯,lim_𝑘→+∞𝜇⁻_𝑘𝑝=−∞; (16)(ii)for 𝑘∈ℕ and 𝜈∈{+,−}, Ker(𝐼−𝑇^𝑝_{𝜇𝜈𝑘(𝑝)}) is a space of 𝐸 with dimensional 1;(iii)the eigenfunction corresponding to 𝜇^𝜈_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^𝜈_𝑘(𝑝) are simple.
It is very known that 𝑇²_𝜆 is completely continuous in 𝐶¹[0,1]. Thus, the Leray-Schauder degree 𝑑_LS(𝐼−𝑇²_𝜆,𝐵_𝑟(0),0) is well-defined for arbitrary 𝑟-ball 𝐵_𝑟(0) and 𝜆≠𝜇^𝜈_𝑘,𝑘∈ℤ and 𝜈∈{+,−}.

Lemma 4. For 𝑟>0, we have 𝑑_𝐿𝑆𝐼−𝑇²_𝜆,𝐵_𝑟(0)=𝜇,01,𝑖𝑓𝜆∈⁻₁(2),𝜇⁺₁(2,())−1^𝑘,𝑖𝑓𝜆∈𝜇⁺_𝑘(2),𝜇⁺_𝑘+1(2)(),𝑘∈ℕ,−1^𝑘,𝑖𝑓𝜆∈𝜇⁻_𝑘+1(2),𝜇⁻_𝑘(2),𝑘∈ℕ. (17)

Proof. We divide the proof into two cases.
Case  1. 𝜆≥0. Since 𝑇²_𝜆 is compact and linear, by [20, Theorem 8.10] and Lemma 2 (ii) with 𝑝=2, 𝑑_LS𝐼−𝑇²_𝜆,𝐵_𝑟(0),0=(−1)^𝑚(𝜆), (18) where 𝑚(𝜆) is the sum of algebraic multiplicity of the eigenvalues 𝜇 of (13) satisfying 𝜇⁻¹𝜆>1.
If 𝜆∈[0,𝜇⁺₁(2)), then there are no such 𝜇 at all; then 𝑑_LS𝐼−𝑇²_𝜆,𝐵_𝑟(0),0=(−1)^𝑚(𝜆)=(−1)⁰=1. (19)
If 𝜆∈(𝜇⁺_𝑘(2),𝜇⁺_𝑘+1(2)) for some 𝑘∈ℕ, then 𝜇⁺_𝑗(2)⁻¹𝜆>1,𝑗∈{1,⋯,𝑘}. (20)
This together with Lemma 2 (ii) implies the following: 𝑑_LS𝐼−𝑇²_𝜆,𝐵_𝑟(0),0=(−1)^𝑘. (21)
Case  2. 𝜆<0. In this case, we consider a new sign-changing eigenvalue problem as follows 𝑢^′′+𝜆𝑚(𝑡)(),𝑢(0)(1)𝑢=0,𝑡∈0,1=𝑢=0, (22) where 𝜆=−𝜆, 𝑚(𝑡)=−𝑚(𝑡). It is easy to check that 𝜇⁺_𝑘(2)=−𝜇⁻_𝑘(2),𝑘∈ℕ. (23) Thus, we may use the result obtained in Case 1 to deduce the desired result.

We first show that the principle eigenvalue function 𝜇^𝜈₁∶(1,+∞)→ℝ is continuous.

Proposition 5. The eigenvalue function 𝜇^𝜈₁∶(1,+∞)→ℝ is continuous.

Proof. We only show that 𝜇⁺₁∶(1,+∞)→ℝ is continuous since the case of 𝜇⁻₁ is similar. In the following proof, we will shorten 𝜇⁺₁ to 𝜇₁. From the variational characterization of 𝜇₁(𝑝), it follows that 𝜇₁𝑝=sup𝜇>0∣𝜇¹₀𝑚(𝑡)|𝑢|^𝑝≤𝑑𝑡₁₀𝑢^′𝑝𝑑𝑡,∀𝑢∈𝐶^∞_𝑐(0,1). (24)
Let {𝑝_𝑗}^∞_𝑗=1 be a sequence in (1,+∞) convergent to 𝑝>1. We will show that lim_𝑗→+∞𝜇₁𝑝_𝑗=𝜇₁𝑝. (25)
To do this, let 𝑢∈𝐶^∞_𝑐(0,1). Then, from (24), 𝜇₁𝑝_𝑗¹₀𝑚(𝑡)|𝑢|^𝑝𝑗𝑑𝑡≤¹₀𝑢^′𝑝𝑗𝑑𝑡. (26)
On applying the Dominated Convergence Theorem, we find that limsup_𝑗→+∞𝜇₁𝑝_𝑗¹₀𝑚(𝑡)|𝑢|^𝑝𝑑𝑡≤¹₀𝑢^′𝑝𝑑𝑡. (27)
Relation (27), the fact that 𝑢 is arbitrary and (24) yield limsup_𝑗→+∞𝜇₁𝑝_𝑗≤𝜇₁𝑝. (28)
Thus, to prove (25), it suffices to show that liminf_𝑗→+∞𝜇₁𝑝_𝑗≥𝜇₁𝑝. (29) Let {𝑝_𝑘}^∞_𝑘=1 be a subsequence of {𝑝_𝑗}^∞_𝑗=1 such that lim_𝑘→+∞𝜇₁(𝑝_𝑘)=liminf_𝑗→+∞𝜇₁(𝑝_𝑗).
Let us fix 𝜀₀>0 so that 𝑝−𝜀₀>1 and, for each 0<𝜀<𝜀₀, 𝑊₀^{1,𝑝−𝜀}(0,1) is compactly embedded into 𝐿^𝑝+𝜀(0,1). For 𝑘∈ℕ, let us choose 𝑢_𝑘∈𝑊^1,𝑝𝑘₀(0,1) such that ¹₀𝑢^′_𝑘^𝑝𝑘𝑑𝑡=1,₁₀𝑢^′_𝑘^𝑝𝑘𝑑𝑡=𝜇₁𝑝_𝑘¹₀𝑚(𝑡)𝑢_𝑘^𝑝𝑘𝑑𝑡. (30)(31) For 0<𝜀<𝜀₀, there exists 𝑘₀∈ℕ such that 𝑝−𝜀<𝑝_𝑘<𝑝+𝜀 for any 𝑘≥𝑘₀. Thus, for 𝑘≥𝑘₀, (30) and Hölder's inequality imply that ¹₀𝑢^′_𝑘^𝑝−𝜀𝑑𝑡≤1. (32) This shows that {𝑢_𝑘}^∞_𝑘=𝑘0 is a bounded sequence in 𝑊₀^{1,𝑝−𝜀}(0,1). Passing to a subsequence if necessary, we can assume that 𝑢_𝑘⇀𝑢 in 𝑊₀^{1,𝑝−𝜀}(0,1) and hence that 𝑢_𝑘→𝑢 in 𝐿^𝑝+𝜀(0,1). Furthermore, 𝑢∈𝐿^𝑝(0,1) and 𝑢_𝑘→𝑢 in 𝐿^𝑝𝑘(0,1) for 𝑘≥𝑘₀. It follows that ¹₀𝑢_𝑘^𝑝𝑘𝑑𝑡−¹₀|𝑢|^𝑝𝑘≤𝑑𝑡₁₀𝑝_𝑘𝑢+𝜃𝑢_𝑘^𝑝𝑘−1𝑢_𝑘≤−𝑢𝑑𝑡𝑝+𝜀¹₀𝑢+𝜃𝑢_𝑘^𝑝𝑘𝑑𝑡^{(𝑝𝑘−1)/𝑝𝑘}×¹₀𝑢_𝑘−𝑢^𝑝𝑘𝑑𝑡^1/𝑝𝑘≤𝑝+𝜀‖𝑢‖_𝑝𝑘+𝑢_𝑘𝑝𝑘^{𝑝𝑘−11}₀𝑢_𝑘−𝑢^𝑝𝑘𝑑𝑡^1/𝑝𝑘⟶0 (33) as 𝑘→+∞. It is clear that ¹₀|𝑢|^𝑝𝑘𝑑𝑡−¹₀|𝑢|^𝑝𝑑𝑡⟶0as𝑘⟶+∞. (34)
Thus, ¹₀𝑢_𝑘^𝑝𝑘𝑑𝑡⟶¹₀|𝑢|^𝑝𝑑𝑡. (35)
Similarly, we can also obtain that ¹₀𝑚⁺(𝑡)𝑢_𝑘^𝑝𝑘𝑑𝑡⟶¹₀𝑚⁺(𝑡)|𝑢|^𝑝𝑑𝑡,₁₀𝑚⁻(𝑡)𝑢_𝑘^𝑝𝑘𝑑𝑡⟶¹₀𝑚⁻(𝑡)|𝑢|^𝑝𝑑𝑡, (36) where 𝑚⁺(𝑡)=max{𝑚(𝑡),0} and 𝑚⁻(𝑡)=−min{𝑚(𝑡),0}. Therefore, ¹₀𝑚(𝑡)𝑢_𝑘^𝑝𝑘=𝑑𝑡₁₀𝑚⁺(𝑡)𝑢_𝑘^𝑝𝑘𝑑𝑡−¹₀𝑚⁻(𝑡)𝑢_𝑘^𝑝𝑘⟶𝑑𝑡₁₀𝑚⁺(𝑡)|𝑢|^𝑝𝑑𝑡−¹₀𝑚⁻(𝑡)|𝑢|^𝑝=𝑑𝑡₁₀𝑚(𝑡)|𝑢|^𝑝𝑑𝑡. (37)
We note that (30) and (31) imply that 𝜇₁𝑝_𝑘¹₀𝑚(𝑡)𝑢_𝑘^𝑝𝑘𝑑𝑡=1 (38) for all 𝑘∈ℕ. Thus, letting 𝑘 go to +∞ in (38) and using (37), we find that liminf_𝑗→+∞𝜇₁𝑝_𝑘¹₀𝑚(𝑡)|𝑢|^𝑝𝑑𝑡=1. (39)
On the other hand, since 𝑢_𝑘⇀𝑢 in 𝑊₀^{1,𝑝−𝜀}(0,1), from (32) we obtain that 𝑢^{′𝑝−𝜀}_𝑝−𝜀≤liminf_𝑘→+∞𝑢^′_𝑘^𝑝−𝜀_𝑝−𝜀≤1^𝜀/𝑝. (40)
Now, letting 𝜀→0⁺ and applying Fatou's Lemma, we find that 𝑢^′𝑝_𝑝≤1. (41) Hence, 𝑢∈𝑊^1,𝑝(0,1); here 𝑊^1,𝑝(0,1) denotes the radially symmetric subspace of 𝑊^1,𝑝(0,1). We claim that actually 𝑢∈𝑊₀^1,𝑝(0,1). Indeed, we know that 𝑢∈𝑊₀^{1,𝑝−𝜀}(0,1) for each 0<𝜀<𝜀₀. For 𝜙∈𝐶^∞_𝑐(ℝ), it is easy to see that ¹₀𝑢𝜙^′𝑑𝑡≤𝑢^′_𝑝−𝜀𝜙_{(𝑝−𝜀)′},𝑖=1,…,𝑁. (42) Then, letting 𝜀→0⁺, we obtain that ¹₀𝑢𝜙^′𝑑𝑡≤𝑢^′_𝑝𝜙_𝑝′,𝑖=1,…,𝑁, (43) where 𝑝^′=𝑝/(𝑝−1). Since 𝜙 is arbitrary, from Proposition IX-18 of [21], we find that 𝑢∈𝑊₀^1,𝑝(0,1), as desired.
Finally, combining (39) and (41), we obtain that liminf_𝑗→+∞𝜇₁𝑝_𝑘¹₀𝑚(𝑡)|𝑢|^𝑝𝑑𝑡≥¹₀𝑢^′𝑝𝑑𝑡. (44)
This and the variational characterization of 𝜇₁(𝑝) imply (29) and hence (25). This concludes the proof of the lemma.

Using Remark 3, Lemma 2, and Proposition 5, we will show that all eigenvalue functions 𝜇^±_𝑘∶(1,+∞)→ℝ, 2≤𝑘∈ℕ are continuous.

Lemma 6. For fixed 2≤𝑘∈ℕ and 𝜈∈{+,−}, 𝜇^𝜈_𝑘(𝑝) as a function of 𝑝∈(1,+∞) is continuous.

Proof. Let 𝑢^𝜈_𝑘 be an eigenfunction corresponding to 𝜇^𝜈_𝑘(𝑝). By Lemma 2 and Remark 3, we know that 𝑢 has exactly 𝑘−1 simple zeros in 𝐼; that is, there exist 𝑐_𝑘,1,…,𝑐_{𝑘,𝑘−1}∈𝐼 such that 𝑢(𝑐_𝑘,1)=⋯=𝑢(𝑐_{𝑘,𝑘−1})=0. For convenience, we set 𝑐_𝑘,0=0, 𝑐_𝑘,𝑘=1, and 𝐽_𝑖=(𝑐_{𝑘,𝑖−1},𝑐_𝑘,𝑖) for 𝑖=1,…,𝑘. Let 𝜇^𝜈₁(𝑝,𝑚/𝐽_𝑖,𝐽_𝑖) denote the first positive or negative eigenvalue of the restriction of problem (13) on 𝐽_𝑖 for 𝑖=1,…,𝑘. Lemma 3 of [18] follows that 𝜇^𝜈_𝑘(𝑝)=𝜇^𝜈₁(𝑝,𝑚/𝐽_𝑖,𝐽_𝑖) for 𝑖=1,…,𝑘. Using a similar proof to Proposition 5, we can show that 𝜇^𝜈₁(𝑝,𝑚/𝐽_𝑖,𝐽_𝑖) is continuous with respect to 𝑝 for 𝑖=1,…,𝑘. Therefore, 𝜇^𝜈_𝑘(𝑝) is also continuous with respect to 𝑝.

Lemma 7. (i) Let {𝜇⁺_𝑘(𝑝)}_𝑘∈ℕ be the sequence of positive eigenvalues of (13). Let 𝜆 be a constant with 𝜆≠𝜇⁺_𝑘(𝑝) for all 𝑘∈ℕ. Then, for arbitrary 𝑟>0, deg𝑇^𝑝_𝜆,𝐵_𝑟(0),0=(−1)^𝛽, (45) where 𝛽 is the number of eigenvalues 𝜇⁺_𝑛(𝑝) of problem (13) less than 𝜆.
(ii) Let {𝜇⁻_𝑘(𝑝)}_𝑘∈ℕ be the sequence of negative eigenvalues of (13). Consider 𝜆≠𝜇⁻_𝑘(𝑝), 𝑘∈ℕ; then deg𝑇^𝑝_𝜆,𝐵_𝑟(0),0=(−1)^𝛽,∀𝑟>0, (46) where 𝛽 is the number of eigenvalues 𝜇⁻_𝑘(𝑝) of problem (25) larger than 𝜆.

Proof. We will only prove the case 𝜆>𝜇⁺₁(𝑝) since the proof for the other cases is similar. We also only give the proof for the case 𝑝>2. Proof for the case 1<𝑝<2 is similar. Assume that 𝜇⁺_𝑘(𝑝)<𝜆<𝜇⁺_𝑘+1(𝑝) for some 𝑘∈ℕ. Since the eigenvalues depend continuously on 𝑝, there exists a continuous function 𝜒∶[2,𝑝]→ℝ and 𝑞∈[2,𝑝] such that 𝜇⁺_𝑘(𝑞)<𝜒(𝑞)<𝜇⁺_𝑘+1(𝑞) and 𝜆=𝜒(𝑝). Define Φ𝑞,𝑢=𝑢−𝐺_𝑞−𝜒𝑞𝑚(𝑡)𝜑_𝑞(𝑢). (47)
It is easy to show that Φ(𝑞,𝑢) is a compact perturbation of the identity such that, for all 𝑢≠0, by definition of 𝜒(𝑞), Φ(𝑞,𝑢)≠0, for all 𝑞∈[2,𝑝]. Hence, the invariance of the degree under homotopology and the classical result for 𝑝=2 imply deg𝑇^𝑝_𝜆,𝐵_𝑟(0),0=deg𝑇²_𝜆,𝐵_𝑟(0),0=(−1)^𝑘. (48)

For the existence of bifurcation branches for (12), we will make use of the following global bifurcation theorem results.

Lemma 8 (see [17]). Let 𝑋 be a Banach space. Let 𝐹∶ℝ×𝑋→𝑋 be completely continuous such that 𝐹(𝜆,0)=0 for all 𝜆∈ℝ. Suppose that there exist constants 𝜌, 𝜂∈ℝ, with 𝜌<𝜂, such that (𝜌,0) and (𝜂,0) are not bifurcation points for the equation 𝑢−𝐹(𝜆,𝑢)=0. (49) Furthermore, assume that deg𝐼−𝐹𝜌,⋅,𝐵_𝑟(0),0≠deg𝐼−𝐹𝜂,⋅,𝐵_𝑟(0),0, (50) where 𝐵_𝑟(0)={𝑢∈𝑋∶‖𝑢‖<𝑟} is an isolating neighborhood of the trivial solution for both constants 𝜌 and 𝜂. Let 𝒮=)∶()()∪×{0},(𝜆,𝑢𝜆,𝑢𝑖𝑠𝑎𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑜𝑓49𝑤𝑖𝑡ℎ𝑢≠0𝜌,𝜂 (51) and let 𝒞 be the component of 𝒮 containing [𝜌,𝜂]×{0}. Then, either (i)𝒞 is unbounded in ℝ×𝑋 or(ii)𝒞∩[(ℝ⧵[𝜌,𝜂])×{0}]≠∅.

Define the Nemytskii operators 𝐻∶ℝ×𝐸→𝑌 by 𝐻(𝜆,𝑢)(𝑡)∶=−𝜆𝑚(𝑡)𝑓(𝑢(𝑡)). (52)

Then, it is clear that 𝐻 is continuous operator which sends bounded sets of ℝ×𝐸 into an equi-integrable sets of 𝑌 and problem (12) can be equivalently written as 𝑢=𝐺_𝑝∘𝐻(𝜆,𝑢)∶=𝐹(𝜆,𝑢). (53)𝐹 is completely continuous in ℝ×𝐸→𝐸 and 𝐹(𝜆,0)=0, for all 𝜆∈ℝ.

Notice that (12) with 𝜆=0 has only the trivial solution. Applying this fact and Lemma 8 and the same method to prove [15, Theorem 2.1] with obvious changes, we may obtain the following.

Lemma 9. Assume that (𝐻₁), (𝐻₂), and (𝐻₃) hold. Then, for fixed 𝑝>1 and for fixed 𝜎∈{+,−}, each (𝜇^𝜈_𝑘(𝑝)/𝑓₀,0) is a bifurcation point of (12) and the associated bifurcation branch (𝒞^𝜈_𝑘)^𝜎 satisfies the following; (1)(𝒞^𝜈_𝑘)^𝜎 is unbounded in 𝐸;(2)(𝒞^𝜈_𝑘)^𝜎⊂(ℝ×Φ^𝜎_𝑘)∪{(𝜇^𝜈_𝑘(𝑝),0)}, where Φ^𝜎_𝑘 is the set of function 𝑢∈𝐶¹₀[0,1] which has exact 𝑘−1 simple zeros in (0,1), and 𝜎𝑢 is positive near 0.

Finally, we give a key lemma that will be used in Section 3. Let 𝐼⁺{[](𝑡)},𝐼∶=𝑡∈0,1∣𝑚>0₋∶={𝑡∈[0,1]∣𝑚(𝑡)<0}. (54)

Lemma 10. Let (𝐻₂) hold. Let 𝐼=[𝑎,𝑏] be such that 𝐼⊂𝐼₊ and meas𝐼>0. (55)
Let 𝑔_𝑛∶[0,1]→(0,+∞) be such that lim_𝑛→+∞𝑔_𝑛(𝑡)=+∞,𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑜𝑛𝐼. (56)
Let 𝑦_𝑛∈𝐸 be a solution of the equation 𝜑_𝑝𝑦^′_𝑛^′+𝑚(𝑡)𝑔_𝑛(𝑡)𝜑_𝑝𝑦_𝑛=0,𝑡∈(0,1). (57) Then, the number of zeros of 𝑦_𝑛|_𝐼 goes to infinity as 𝑛→+∞.

Proof. After taking a subsequence if necessary, we may assume that 𝑚(𝑡)𝑔_𝑛𝑗(𝑡)≥𝑗,𝑡∈𝐼, (58) as 𝑗→+∞. It is easy to check that the distance between any two consecutive zeros of any nontrivial solution of the equation 𝜑_𝑝𝑢^′(𝑡)^′+𝑗𝜑_𝑝(𝑢(𝑡))=0,𝑡∈𝐼, (59) goes to zero as 𝑗→+∞. Using this with [21, Lemma 2.5], it follows the desired results.

3. Main Results and Its Proof

Let 𝜇^±_𝑘 be the 𝑘th positive or negative eigenvalue of (13). By applying Lemma 9, we will establish the main results as follows.

Theorem 11. Let (𝐻₁), (𝐻₂), (𝐻₃), and (𝐻₄) hold. Assume that, for some 𝑘∈ℕ, either 𝜇𝛾∈^+𝑘𝑝𝑓_∞,𝜇⁺_𝑘𝑓₀∪𝜇⁻_𝑘𝑝𝑓₀,𝜇⁻_𝑘𝑝𝑓_∞ (60) or 𝜇𝛾∈^+𝑘𝑝𝑓₀,𝜇⁺_𝑘𝑝𝑓_∞∪𝜇⁻_𝑘𝑝𝑓_∞,𝜇⁻_𝑘𝑝𝑓₀. (61)
Then, (4) has two solutions 𝑢⁺_𝑘 and 𝑢⁻_𝑘 such that 𝑢⁺_𝑘 has exactly 𝑘−1 zeros in (0,1) and is positive near 0 and 𝑢⁻_𝑘 has exactly 𝑘−1 zeros in (0,1) and is negative near 0.

Proof. We only prove the case of 𝛾>0. The case of 𝛾<0 is similar. Consider the problem 𝜑_𝑝𝑢^′′(𝑡)𝑓(𝑢)(),𝑢(0)(1)+𝜆𝛾𝑚=0,𝑡∈0,1=0,𝑢=0. (62)
Considering the results of Lemma 9, we have that, for each integer 𝑘≥1, 𝜎∈{+,−}, there exists a continuum (𝐶⁺_𝑘)^𝜎⊆Φ^𝜎_𝑘 of solutions of (62) joining (𝜇⁺_𝑘(𝑝)/𝛾𝑓₀,0) to infinity in (0,∞)×Φ^𝜎_𝑘. Moreover, (𝐶⁺_𝑘)^𝜎⧵{(𝜇⁺_𝑘(𝑝)/𝛾𝑓₀,0)}⊂(0,∞)×Φ^𝜎_𝑘.
It is clear that any solution of (62) of the form (1,𝑢) yields a solution 𝑢 of (4). We will show that (𝐶⁺_𝑘)^𝜎 crosses the hyperplane {1}×𝐸 in ℝ×𝐸. To this end, it will be enough to show that (𝐶⁺_𝑘)^𝜎 joins (𝜇⁺_𝑘(𝑝)/𝛾𝑓₀,0) to (𝜇⁺_𝑘(𝑝)/𝛾𝑓_∞,+∞). Let (𝜂_𝑛,𝑦_𝑛)∈(𝐶⁺_𝑘)^𝜎 satisfy 𝜇_𝑛+𝑦_𝑛⟶+∞. (63) We note that 𝜂_𝑛>0 for all 𝑛∈ℕ since (0,0) is the only solution of (62) for 𝜆=0 and (𝐶⁺_𝑘)^𝜎∩({0}×𝐸)=∅.
Case  1. 𝜇⁺_𝑘(𝑝)/𝑓_∞<𝛾<𝜇⁺_𝑘(𝑝)/𝑓₀. In this case, we only need to show that 𝜇⁺_𝑘𝑝𝛾𝑓_∞,𝜇⁺_𝑘𝑝𝛾𝑓₀⊆𝜇∈ℝ∶𝜇,𝑢∈𝐶⁺_𝑘^𝜎. (64) We divide the proof into two steps.
Step  1. We show that, if there exists a constant number 𝑀>0 such that 𝜂_𝑛⊂(0,𝑀] (65) for 𝑛∈ℕ large enough, then (𝐶⁺_𝑘)^𝜎 joins (𝜇⁺_𝑘(𝑝)/𝛾𝑓₀,0) to (𝜇⁺_𝑘(𝑝)/𝛾𝑓_∞,+∞).
In this case, it follows that 𝑦_𝑛⟶+∞. (66)
Let 𝜉∈𝐶(ℝ) be such that 𝑓(𝑢)=𝑓_∞𝜑_𝑝(𝑢)+𝜉(𝑢). (67) Then, lim_{|𝑢|→+∞}𝜉(𝑢)𝜑_𝑝(𝑢)=0. (68) Let 𝜉̃(𝑢)=max_{0≤|𝑠|≤𝑢}𝜉(𝑠). (69) Then, 𝜉̃ is nondecreasing and lim_𝑢→+∞𝜉̃(𝑢)|𝑢|^𝑝−1=0. (70)
We divide the equation 𝜑_𝑝𝑦^′_𝑛^′−𝜇_𝑛𝛾𝑚(𝑡)𝑓_∞𝜑_𝑝𝑦_𝑛=𝜇_𝑛𝛾𝑚(𝑡)𝜉𝑦_𝑛 (71) by ‖𝑦_𝑛‖ and set 𝑦_𝑛=𝑦_𝑛/‖𝑦_𝑛‖. Since 𝑦_𝑛 is bounded in 𝐸, after taking a subsequence if necessary, we have 𝑦_𝑛⇀𝑦 for some 𝑦∈𝐸 and 𝑦_𝑛→𝑦 in 𝑌 with ‖𝑦‖=1. Moreover, from (70) and the fact that 𝜉̃ is nondecreasing, we have lim_𝑛→+∞𝜉𝑦_𝑛(𝑡)𝑦_𝑛^𝑝−1=0, (72) since 𝜉𝑦_𝑛(𝑡)𝑦_𝑛^𝑝−1≤𝜉̃𝑦_𝑛(𝑡)𝑦_𝑛^𝑝−1≤𝜉̃𝑦_𝑛(𝑡)_∞𝑦_𝑛^𝑝−1≤𝜉̃𝑦_𝑛(𝑡)𝑦_𝑛^𝑝−1. (73)
By the continuity and compactness of 𝐺_𝑝, it follows that 𝑦=𝐺_𝑝𝜇𝛾𝑚(𝑡)𝑓_∞𝜑_𝑝𝑦, (74) where 𝜇=lim_𝑛→+∞𝜇_𝑛, again choosing a subsequence and relabeling if necessary.
We claim that 𝑦∈𝐶⁺_𝑘^𝜎. (75)
Suppose on the contrary that 𝑦∈(𝐶⁺_𝑘)^𝜎. Since 𝑦≠0 is a solution of (74) and all zeros of 𝑦 in [0,1] are simple, it follows that 𝑦∈(𝐶⁺_ℎ)^𝜄≠(𝐶⁺_𝑘)^𝜎 for some ℎ∈ℕ and 𝜄∈{+,−}.
By the openness of 𝐸⧵(𝐶⁺_𝑘)^𝜎, we have that there exists a neighborhood 𝑈(𝑦,𝜌₀) such that 𝑈𝑦,𝜌₀⊂𝐸⧵𝐶⁺_𝑘^𝜎, (76) which contradicts the facts that 𝑦_𝑛→𝑦 in 𝐸 and 𝑦_𝑛∈(𝐶⁺_𝑘)^𝜎. Therefore, 𝑦∈𝐶^𝜈_𝑘. Moreover, by Lemma 2, 𝜇𝛾𝑓_∞=𝜇⁺_𝑘(𝑝), so that 𝜆𝜇=^𝑘𝛾𝑓_∞. (77) Therefore, (𝐶⁺_𝑘)^𝜎 joins  (𝜇⁺_𝑘(𝑝)/𝛾𝑓₀,0) to (𝜇⁺_𝑘(𝑝)/𝛾𝑓_∞,+∞).
Step  2. We show that there exists a constant 𝑀 such that 𝜇_𝑛∈(0,𝑀] for 𝑛∈ℕ large enough.
On the contrary, we suppose that lim_𝑛→+∞𝜇_𝑛=+∞. (78)
Since (𝜂_𝑛,𝑦_𝑛)∈(𝐶⁺_𝑘)^𝜎, it follows that 𝜑𝑦^′_𝑛^′+𝛾𝜂_𝑛𝑚(𝑡)𝑓𝑦_𝑛𝜑𝑦_𝑛𝜑𝑦_𝑛=0. (79)
Let 0=𝜏(0,𝑛)<𝜏(1,𝑛)<⋯<𝜏(𝑘,𝑛)=1 (80) be the zeros of 𝑦_𝑛 in [0,1]. Then, after taking a subsequence if necessary, lim_𝑛→+∞𝜏(𝑙,𝑛)∶=𝜏(𝑙,∞),𝑙∈{0,1,⋯,𝑘−1}. (81) Notice that Lemma 10 and the fact that 𝑦_𝑛 has exactly 𝑘−1 simple zeros in [0,1] yield ∪^𝑘−1_𝑙=0(𝜏(𝑙,∞),𝜏(𝑙+1,∞))∩𝐼⁺=∅, (82) which implies that meas∪^𝑘−1_𝑙=0(𝜏(𝑙,∞),𝜏(l+1,∞))∩𝐼⁻=1. (83)
However, this contradicts (𝐻₂): 0<meas𝐼⁻<1.
Case  2. 𝜇⁺_𝑘(𝑝)/𝑓₀<𝛾<𝜇⁺_𝑘(𝑝)/𝑓_∞. In this case, we have that 𝜇⁺_𝑘𝑝𝛾𝑓₀𝜇<1<^+𝑘𝑝𝛾𝑓_∞. (84)
Assume that (𝜂_𝑛,𝑦_𝑛)∈(𝐶⁺_𝑘)^𝜎 is such that lim_𝑛→+∞𝜇_𝑛+𝑦_𝑛=+∞. (85)
If 𝜂_𝑛→+∞, then we are done!
If there exists 𝑀>0, such that, for 𝑛∈ℕ sufficiently large, 𝜂_𝑛∈(0,𝑀]. (86)
Applying the same method used in Step 1 of Case 1, after taking a subsequence and relabeling if necessary, it follows that 𝜂_𝑛,𝑦_𝑛⟶𝜇⁺_𝑘𝑝𝛾𝑓_∞,+∞as𝑛⟶+∞. (87)
Thus, (𝐶⁺_𝑘)^𝜎 joins (𝜇⁺_𝑘(𝑝)/𝛾𝑓₀,0) to (𝜇⁺_𝑘(𝑝)/𝛾𝑓_∞,+∞).

Acknowledgments

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).