Global bifurcation and constant sign solutions of discrete boundary value problem involving p -Laplacian

We study the unilateral global bifurcation result for the one-dimensional discrete p -Laplacian problem where (cid:2) u ( t ) = u ( t + 1) – u ( t ) is a forward diﬀerence operator, ϕ p ( s ) = | s | p –2 s (1 < p < + ∞ ) is a one-dimensional p -Laplacian operator. λ is a positive real parameter, a : [1, T + 1] Z → [0,+ ∞ ) and a ( t 0 ) > 0 for some t 0 ∈ [1, T + 1] Z , g : [1, T + 1] Z × R 2 → R satisﬁes the Carathéodory condition in the ﬁrst two variables. We show that ( λ 1 ,0) is a bifurcation point of the above problem, and there are two distinct unbounded continua C + and C – , consisting of the bifurcation branch C from ( λ 1 ,0), where λ 1 is the principal eigenvalue of the eigenvalue problem corresponding to the above problem. Let T > 1 be an integer, Z denote the integer set for m , n ∈ Z with m < n , [ , Z := m , } . the the above details about the existence of constant sign solutions for the following problem: where f ∈ C ( R , R ) with sf ( s ) > 0 for s = 0.

There is no report on the global structure of solution sets by the bifurcation theory for discrete p-Laplacian problem (1.1). Although there are a great amount of papers researching the bifurcation phenomenon of p-Laplacian problem, but those results are not unilateral, and their conclusions are all obtained in the differential case. As we know, the proofs are based on the local properties of solutions of (1.1) bifurcating from (λ 1 , 0) (see Lemma 3.4). Although the proof of the above result follows the same steps as for the semilinear case from [7], his methods cannot be applied directly to the quasilinear discrete problem. In addition, the main reason is that the spectrum of the discrete p-Laplacian eigenvalue problem is unknown.
In 2002, Anane [1] discussed the spectra of the following differential p-Laplacian problem: λ is the spectral parameter. They obtained the following result.
The first eigenvalue λ 1 is of special importance. However, the method in [1] has no effect on the spectral study of the discrete p-Laplacian problem, whether the first eigenvalue of the discrete p-Laplacian problem is simple or not, and the properties of the corresponding eigenfunction are both unknown.
Of course, the next natural question is: does the unilateral bifurcation version exist for quasilinear difference problem (1.1)? Furthermore, what is the existence of a positive solution or a negative one for a nonlinear p-Laplacian difference problem (1.1)? In this paper, we give a positive answer for those questions.
This paper is organized as follows. In Sect. 2, we show the existence of the principal eigenvalue and the sign of the corresponding eigenfunctions for the one-dimensional discrete p-Laplacian eigenvalue problem, which will be of interest for us. In Sect. 3, we establish the unilateral global bifurcation theory for (1.1). In Sect. 4, as an application, we prove that there exist constant sign solutions for problem (4.1) (see Sect. 4) according to the different behavior of nonlinear term f at 0 and ∞.

The existence of the principal eigenvalue
In this section, we consider the existence of the principal eigenvalue and the sign of the corresponding eigenfunction for the discrete p-Laplacian eigenvalue problem Lemma 2.2 λ 1 (a) is the first eigenvalue of (2.1), then the first eigenvalue λ 1 (a) is the minimum of the Rayleigh quotient, that is, Furthermore, λ 1 (a) < λ(a), where λ(a) is some other eigenvalue of (2.1).
Proof Combining the equation of (2.1) with Lemma 2.1, the conclusion is clearly established.
Applying a similar method to prove [4, Proposition 1.10] with obvious changes, we can obtain the following theorem.

Unilateral global bifurcation results for (1.1)
In this section, we establish the unilateral global bifurcation theory for (1.1).
We consider the following auxiliary problem: where h : [1, T + 1] Z → R. It can be easily seen that problem (3.1) is equivalently written as where G p : R → E maps bounded sets of R into relative compacts of E. We define the operator T where λ 2 (a) is the second eigenvalue of problem (2.1).
Proof Using the similar method of [15, Lemma 2.8], we can get the conclusion of this theorem.

Define Nemytskii operators
Then it is clear that H is a continuous operator which maps bounded sets of R × E into the bounded sets of R. Obviously, (1.1) can be equivalently written as It is easy to see that F : R × E → E is completely continuous and F(λ, 0) = 0, ∀λ ∈ R.
For convenience, we abbreviate λ 1 (a) as λ 1 . Our first main result for (1.1) is the following theorem.
Theorem 3.1 For p > 1, λ 1 is a bifurcation point of (1.1) and the associated bifurcation branch C in R × E whose closure contains (λ 1 , 0), then either (i) C is unbounded in R × E, or (ii) C contains a pair (λ, 0), where λ is an eigenvalue of (2.1) and λ = λ 1 .
then g is nondecreasing with respect to u and uniformly for a.e. t ∈ [1, T + 1] Z and λ on bounded sets. It is easy to see that {h n } is a bounded sequence in R, thus we can assume that v n → v 0 and v 0 = 1 as n → ∞, This implies that λ is an eigenvalue of (2.1), this is a contradiction. From the homotopic invariance of the degree (refer to the proof method of Theorem 2.10 in [15]) and Lemma 3.1, we conclude that It is easy to see that (3.6) and (3.7) contradict (3.2). Thus (λ 1 , 0) is a bifurcation point of (1.1). By the global bifurcation theory [18], we can get the existence of a global branch of solutions of (1.1) emanating from (λ 1 , 0).
With the help of the definition of λ 1 , we see that λ k > λ 1 . According to the Sturm comparison theorem, if λ > λ 1 , then the eigenfunction u corresponding to λ must change sign in [0, (3.8), and the compactness of G p , we can get v m → v 0 and v 0 = 1.
Hence, v 0 ∈ S k , where S k denotes the set of functions in E which must change sign in [0, T + 2] Z . Therefore, when m is sufficiently large, v m ∈ S k , combining the definition of S k with Lemma 2.3, it can be seen that this is a contradiction.
If there exists t 0 ∈ [0, T + 1] Z such that one of the following cases holds: Proof (i) By virtue of the equation of (1.1), we have Combining u(t 0 ) = 0 with the assumption of g, we obtain we can get u(t 0 + 2) = 0. Further, u(t 0 -1) = u(t 0 + 1) = 0, step by step, it follows that u ≡ 0.
(ii) Using the same method, we conclude that Hence, u ≡ 0. Proof In view of the conclusion of Theorem 3.1 and Lemma 3.2, we only need to prove Since u ∈ ∂S, by Lemma 3.3, we obtain u ≡ 0. Let ω n = u n u n , then ω n satisfies ω n = G p -λ n a(t)ϕ p ω n (t) -g(t, u n (t), λ n ) u n p-1 . (3.9) Combining (3.5), (3.9) with the compactness of G p , we obtain ω n → ω 0 and ω 0 = 1. Obviously, there is Hence there exists λ = λ k (k = 1). Furthermore, . For convenience, let us introduce a few notations. Given any λ ∈ R and 0 < s < +∞, we consider an open neighborhood of (λ 1 , 0) in E defined by By the Hahn-Banach theorem [26], there exists a linear functional l ∈ E * , where E * denotes the dual space of E such that l(φ 1 ) = 1 and E 0 = u ∈ E | l(u) = 0 .

Moreover, for each
there are s ∈ R and y ∈ E 0 (unique) such that u = sφ 1 + y and |s| > η u . Let δ > 0 be the constant from Lemma 3.4. For 0 < ε ≤ δ, we define D v λ 1 ,ε to be the com- containing (λ 1 , 0), and C v to be the closure of 0<ε≤δ C v λ 1 ,ε . Obviously, C v is connected. By Lemma 3.4, the definition of C v is independent of the choice of η and C = C + ∪ C -.
Similar to Dancer's result [9, Theorem 2], we can obtain the following unilateral global bifurcation result. Remark 3.2 Although the proof of the above result is similar to that of the semilinear case in [9], the method cannot be used directly in this paper. In fact, the proof of Lemma 3 in [9] strictly depends on the linear characteristics of L. Therefore, we use Lemma 3.4 and functional analysis to prove the above result.
Then the definition of implies A similar result holds with λ 1 + σ replaced by λ 1σ , By our hypothesis, for λ ∈ [λ 1σ , λ 1 + σ ], the homotopy (λ, ·) is admissible on B δ 1 . The homotopy invariance of the degree ensures that Subtracting (3.10) from (3.11) and using (3.12), we have The proof of this lemma is complete.

Lemma 3.6
If 0 < ε < δ, zero is an isolated solution of (λ 1 , u) = 0 and Tλ 1 ,ε is bounded in E, then Proof The proof of Lemma 2 in Dancer [9] is also valid for the quasilinear case, so we omit the proof.
Lemma 3.7 Lemma 3.6 holds without the assumption that zero is an isolated solution of (λ 1 , u) = 0.
We divide the proof into two cases.
For any 0 < ε < δ, we assume that Tλ 1 ,ε is bounded in E. Define T n to be the component of S n \ (B ε (λ 1 , 0) ∩ K + η ) containing (λ 1 , 0). It is easy to see that the limit of T n is Tλ 1 ,ε , so T n is bounded in E. Suppose that Lemma 3.7 is false, then The definition of Tλ 1 ,ε implies Since Tλ 1 ,ε is bounded, we can find a constant R > 0 such that Tλ 1 ,ε ⊂ B R (λ 1 , 0). By these facts and the classical topological result from Whyburn [21], we obtain that where k 1 , k 2 are disjoint compact subsets of K and Tλ 1 ,ε ⊂ k 1 , Applying Lemma 3.6 to n , we can see that By the connectedness of T n , there exists (λ n , u n ) ∈ ∂U ∩ T n . We assume that there are u n u * in E and λ n → λ * in R. Letting n → +∞ on the both of n (λ n , u n ) = 0 and using the compact and continuous properties of G p , we can show that n (λ * , u * ) = 0. It is easy to see that this contradicts the definition of U. We have completed the proof.
Proof of Theorem 3.3 Define Tλ 1 to be the closure of 0<ε≤δ Tλ 1 ,ε , then Tλ 1 ⊆ C -. Suppose that Cis bounded. Then, by Lemma 3.7, for any 0 < ε ≤ δ, we obtain It follows that Furthermore, for every open set U in E, which satisfies (λ 1 , 0) ∈ U and U ⊆ B δ (λ 1 , 0), (3.16) implies 0). It is easy to know that E is a compact metric space under the induced topology of E and T is a closed subset of E.
Combining Theorem 3.2 and Theorem 3.3, we can obtain the following unilateral global bifurcation result.
Proof We can find a bounded neighborhood O of (λ 1 , 0) such that Without loss of generality, we may suppose that (3.18) holds. Next, we only need to prove that both C + and Care unbounded. Suppose on the contrary that C + is bounded, the case for Cis similar. By Theorem 3.3, we know that in view of (3.18), there exists (λ * , u * ) ∈ C + ∩ Csuch that (λ * , u * ) = (λ 1 , 0) and u * ∈ S + ∩ S -.
This contradicts the definitions of S + and S -.

Constant sign solutions for nonlinear discrete p-Laplacian problem
In this section, we use Theorem 3.4 to prove the existence of constant sign solutions for the discrete p-Laplacian problem where a : [1, T + 1] Z → [0, +∞) and a(t 0 ) > 0 for some t 0 ∈ [1, T + 1] Z , f ∈ C(R, R) with sf (s) > 0 for s = 0. We will discuss the existence of constant sign solutions according to the different behavior of nonlinear term f at 0 and ∞. Denote Definition 4.1 (see [17]) Let X be a Banach space, {C n | n = 1, 2, 3, . . .} be a family of subsets of X. Then the superior D of C n is defined by Lemma 4.2 (see [19]) Let X be a Banach space, C n is a component of X, assume that: (i) There exist z n ∈ C n (n = 1, 2, . . .) and z * ∈ X such that z n → z * ; (ii) lim n→∞ r n = ∞, where r n = sup{ x : x ∈ C n }; (iii) For every R > 0, ( ∞ n=1 C n ) ∩ R is a relative compact set of X, where R = {x ∈ X : x ≤ R}. Then D := lim sup n→∞ C n contains an unbounded component C such that z * ∈ C.
The main results of this section are the following.  Applying Theorem 3.4 to problem (4.1), it can be seen that there exist two different unbounded connected components C + and C -, which bifurcate from ( λ 1 f 0 , 0), and Next, we prove that C v is unbounded in the direction of the λ axis. Assume on the contrary that Then there exists a sequence {(μ k , y k )} ⊂ C v such that lim k→∞ y k = ∞, |μ k | ≤ C 0 for some positive constant C 0 independent of k. This implies that  Set v k (t) = y k (t) Choosing a subsequence and relabeling if necessary, it follows that there exists (μ * , v * ) Moreover, from (4.2), (4.3), and f ∞ = 0, it follows that Hence, v * (t) ≡ 0 for t ∈ [0, T + 2] Z . This contradicts (4.4). Therefore, we show that lim k→∞ μ k = 0. Suppose on the contrary that choosing a subsequence and relabeling if necessary, μ k ≥ M 2 for some constant M 2 > 0. By (4.2), we obtain Consequently, we have Hence, we have from (4.5) that y k must change its sign on [0, T + 2] Z for k large enough. However, this is impossible. Thus, lim k→∞ μ k = 0. So, we prove that C v joins ( λ 1 f 0 , 0) with (0, ∞).
By simple calculation, we know f 0 = f ∞ , f ∞ = f 0 . By applying Theorem 4.1, we can get the conclusions of this theorem under the inversion ω → ω ω 2 = u.
Proof Using the conclusion of Theorem 4.2 and the method of Theorem 4.3, we can easily prove the conclusion of this theorem.