Global continuum and multiple positive solutions to one-dimensional p-Laplacian boundary value problem

We show the global structure of the set of positive solutions of a discrete Dirichlet problem involving the p-Laplacian difference operator suggesting suitable conditions on the weight function and nonlinearity. We obtain existence and multiplicity of positive solutions for λ lying in various intervals in R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}$\end{document} by using the directions of a bifurcation and the Picone-type identity for discrete p-Laplacian operators.


Introduction and main result
Let T > 1 be an integer, T := [1, T] Z = {1, 2, . . . , T},T := {0, 1, . . . , T + 1}. In this paper, we are concerned with existence and multiplicity of positive solutions of the discrete boundary value problem [ϕ p ( u(x -1))] + λh(x)f (u(x)) = 0, x ∈ T, u(0) = u(T + 1) = 0, (1.1) where ϕ p (s) = |s| p-2 s, p > 1, λ > 0 is the parameter, f ∈ C(R + , R + ), f (s) > 0 for all s > 0 and h :T → R + with 0 < h * ≤ h(t) ≤ h * on T for some h * , h * ∈ (0, ∞). Existence of positive solutions for discrete boundary value problems involving the p-Laplacian difference operator has been studied by several authors, we refer to Agarwal et al. [1], Chu and Jiang [4], and [7,8,10,12] as well as the references therein. Very recently, Nastasi et al. [14,15] also obtained some existence results for discrete (p, q)-Laplacian equations. In particular, by virtue of bifurcation techniques, Bai and Chen [3] established some results of existence of positive solutions for (1.1) according to the asymptotic behavior of f at 0 and ∞. However, the sublinear and superlinear conditions imposed on the nonlinearities only deduced a relatively simple "shape of the component", and they provided no information on the existence of at least three positive solutions.
It is the purpose of this paper to show that (1.1) has three positive solutions for λ lying in various intervals in R suggesting suitable conditions on the weight function and nonlinearity by using the directions of a bifurcation and the Picone-type identity (for related results, we refer to [6,19,22]) for discrete p-Laplacian operators due to Řehák [20]. We shall make the following assumptions: In assumption (A4) below and throughout, we use the following standard notations.
Let μ 1 be the first eigenvalue of the following problem: Then the first eigenvalue μ 1 is the minimum of the Reyleigh quotient, that is, Let χ 1 be the principal eigenvalue of the eigenvalue problem wheret ∈ T satisfies T 2 ≤t ≤ T+1 2 , and let w 1 be an eigenfunction corresponding to χ 1 . Furthermore, we assume that (A4) there exist s 0 > 0 and 0 < σ < 1 such that Arguing the shape of bifurcation, we have the following main result. Remark 1.2 To obtain our main goals, we shall employ a bifurcation technique due to Sim and Tanaka [21]. They showed that one-dimensional p-Laplacian of differential equation coupled with Dirichlet boundary condition has three positive solutions. Moreover, several papers have also been devoted to elliptic equations involving the fractions of the Laplacian (see [2,13]). For other multiplicity results for related problems, we refer to [16][17][18].
The rest of the paper is organized as follows. In Sect. 2, we show the existence of bifurcation from the first eigenvalue for the corresponding problem according to the standard argument and the rightward direction of bifurcation. In Sect. 3, the change of direction of bifurcation is given. The final section is devoted to showing a priori bound of solutions for (1.1) and completing the proof of Theorem 1.1.

Rightward bifurcation
In this section, we study global bifurcation phenomena from the trivial branch with the rightward direction under suitable assumptions on h and f . We need the following preliminary lemma.
Multiplying the equation of (2.1) by -φ -1 and by a direct computation, one has According to the definition of μ 1 , we know that φ -1 is an eigenfunction of (1.2) with eigenvalue μ 1 .
We claim that φ -1 > 0. Assume that there exists t 0 ∈ T such that φ -1 (t 0 ) = 0, then It deduces that φ -1 ≡ 0 from repeating the steps as above, which is a contradiction. Consequently, φ 1 = -φ -1 < 0 is of one sign in T. (ii) Let u and v be two eigenfunctions corresponding to μ 1 , we only need to prove that there exists c ∈ R such that u = cv.
From (i), we know that u and v are of one sign, we can suppose that From the equation of (1.2), one has and c = min t∈T By similar methods, we get And accordingly, It deduces that u = cv from repeating the steps as above. Moreover, coupled with (i) and (ii), the eigenfunction corresponding to μ 1 can be chosen to be positive on T.

2)
Proof According to discrete Rolle's theorem (see [9]), there exists x 0 ∈ T such that u(x 0 ) = 0 or u(x 0 -1) u(x 0 ) < 0. Then, by a direct computation, it is easy to see that By virtue of (A2) and (A3), we get Proof Set v n := u n / u n . Then it is easy to see that v n = 1. It follows from Lemma 2.3 that v n is bounded, so there is a subsequence of v n uniformly convergent to a limit v. Furthermore, there exists a subsequence of it such that v n (0) converges to some constant c. We again denote by {v n } the subsequence. We note that v ∈ Y , v(0) = v(T + 1) = 0, and v = 1. Rewriting the equation of (1.1) with (λ, u) = (λ n , u n ), we obtain (2.5) Dividing both sides of (2.5) by u n p-1 , we get Since u n (x) → 0 for all x ∈T, we can get f (u n (t)) ϕ p (u n (t)) → f 0 for each fixed t ∈T. It follows that w n (x) converges to for each fixed x ∈T. Therefore, by recalling (2.6), one has v n (x) = x-1 s=0 ϕ -1 p w n (s) .
The fact coupled with (2.7) yields that v n (x) converges to which implies that v is a nontrivial solution of (1.2) with λ = μ 1 , and hence v ≡ φ 1 .
Proof Suppose on the contrary that there exists a sequence {(λ n , u n )} such that (λ n , u n ) ∈ C, which satisfies λ n → μ 1 /f 0 , u n → 0, and λ n ≤ μ 1 /f 0 . According to Lemma 2.4, there exists a subsequence of {u n }, for convenience denoted by {u n }, such that u n u n converges uniformly to φ 1 onT, where φ 1 (x) > 0 is the first eigenfunction of (1.2) with φ 1 = 1. Multiplying the equation of (1.1) with (λ, u) = (λ n , u n ) by u n and by a direct computation, one has and accordingly It follows from Lemma 2.4 that, after taking a subsequence and relabeling if necessary, then together with (2.8) one has That is, From condition (A2), we have This contradicts λ n < μ 1 /f 0 .

Directional turn of bifurcation
In this section, we show that the connected components grow to the left at some point under (A4) condition. In Lemma 3.3 and throughout, we use the following well-known conceptions of a generalized zero and a simple generalized zero at t ∈ T in [9]. Definition 3.1 Suppose that a function y :T → R. If y(t 0 ) = 0, then t 0 is a zero of y. If y(t 0 ) = 0 or y(t 0 )y(t 0 + 1) < 0 for some t 0 ∈ {1, . . . , T -1}, then y has a generalized zero at t 0 ∈ T. Lemma 3.2 Assume that (A1) holds. Let u be a positive solution of (1.1). Then there exists t 0 ∈ T such that u = u(t 0 ). Moreover, Proof It is an immediate consequence of the fact that u is concave down inT.
Following similar arguments as in the proof of Lemma 3.1 of Dai and Ma [5], we have Lemma 3.3 Let P k ≥ p k for k ∈ [m, n + 1] Z . Also let y(k), z(k) be solutions of the following difference equations: ϕ p y(k) + p k ϕ p y(k + 1) = 0, respectively. If y(m) = y(n + 1) = 0 but without any generalized zeros in [m + 1, n] Z , then either there exists τ ∈ [m + 1, n] Z such that τ is a generalized zero of z or P k = p k and y(k) Proof If z has a generalized zero in [m + 1, n] Z , the conclusion is done. If there is no generalized zero of z on [m, n + 1] Z , then we can assume without loss of generality that y > 0, z > 0 in [m + 1, n] Z . By the Picone-type identity [11,20], we have By a direct computation, one has y(n + 1) ϕ p (z(n + 1)) ϕ p z(n + 1) ϕ p y(n + 1)ϕ p y(n + 1) ϕ p z(n + 1) The left-hand side of (3.2) equals zero. Hence, the right-hand side of (3.2) also equals zero. Since and the equality holds if and only if y(k) = y(k)( z(k)/z(k)), we conclude that there exists a constant ν = 0 such that z(k) = νy(k) and P k = p k . Proof Let u be a positive solution of (1.1). It follows from Lemma 3.2 that We note that u is a solution of Suppose on the contrary that λ ≥ μ 1 /f 0 . Then, for x ∈ I, we have from (A4) that then [ϕ p ( (y(x -1)))] + χ 1 ϕ p (y(x)) = 0, x ∈ [b + 1, b +t -1] Z , y(b) = 0, y(b +t) = 0.
It deduces from Lemma 3.3 that u has at least one generalized zero on I. This contradicts the fact that u(x) > 0 on I.