Multiplicity of Positive Solutions for a Fourth-Order Quasilinear Singular Differential Equation ∗

This paper is concerned with the multiplicity of positive solutions of boundary value problem for the fourth-order quasilinear singular differential equation (|u ′′ | p−2 u ′′ ) ′′ = �g(t)f(u), 0 1, � > 0. We apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of positive solutions. We have found a threshold � ∗ � ∗ , then the problem has no positive solution. In particular, there exist at least two positive solutions for 0 < � < � ∗ .


Introduction
In this paper, we consider the multiplicity of positive solutions for the following fourthorder quasilinear singular differential equation (|u ′′ | p−2 u ′′ ) ′′ = λg(t)f (u), 0 < t < 1, (1.1) subject to the boundary value conditions where p > 1 and λ is a positive parameter.
In the past few years, some fourth-order nonlinear equations have been proposed for image (signal) processing (i.e., edge detection, image denoising, etc.).And a number of authors hoped that these methods might perform better than some second-order equations [15][16][17][18][19]. Indeed, there are good reasons to consider fourth-order nonlinear equations.First, fourth-order differential damps oscillations at high frequencies (e.g., noise) much faster than second-order differential.Second, there is the possibility of having schemes that include effects of curvature, i.e., the second derivatives of the image (signal), in the dynamics.The equation (1.1) can be regarded as the analogue of the Euler-Lagrange equations from the variation problem in [15].Similarly, the solution of the equation (1.1) can be regarded as the steady-state case of the fourth-order anisotropic diffusion equation in [16][17][18][19].
Equations of the form (1.1), especially the special case p = 2, have been the subject of intensive study during the last thirty years, see for example [1,2,5,[9][10][11][12][13][14].In particular, in a recent paper [5], under some structure conditions which permit some singularities for g(t), the authors discussed the special case p = 2 subject to the boundary value conditions (1.2), and revealed the relation between the existence of positive solutions and the parameter λ.While for the case p > 1 with g(t) ≡ 1 and f (s) being of power type, in [6], P. Dráek, M. Ôani considered the corresponding initial value problem, and obtain the local existence and uniqueness of solutions, see also [7] for some extension of the results.
The main purpose of this paper is to investigate the existence, nonexistence and multiplicity of positive solutions of the problem (1.1), (1.2).Different from the known works, the equation we consider is quasilinear, which might have degeneracy or singularities.In fact, if p > 2, then the equation is degenerate at the points where u ′′ = 0; while if 1 < p < 2, then the equation has singularity at the points where u ′′ = 0.It should be noticed that there is no Green's function compared with the special case p = 2.
The main result of this paper is the following Theorem 1.1 Let (H1) and (H2) be satisfied.If then there exists a threshold 0 < λ * < +∞ such that the problem (1.1), (1.2) has no positive solution for λ > λ * , has at least one positive solution for λ = λ * , and has at least two positive solutions for 0 < λ < λ * .
Our method can also be applied to the discussion of the equation subject to another boundary value condition The corresponding extension of the result is as follows.then there exists a threshold 0 < λ * < +∞ such that the problem (1.1), (1.4) has no positive solution for λ > λ * , has at least one positive solution for λ = λ * , and has at least two positive solutions for 0 < λ < λ * .
This paper is organized as follows.As preliminaries, we state some necessary lemmas in Section 2. Subsequently, in Section 3, we apply the fixed point index theory and the upper and lower solutions method to the proof of Theorem 1.1.In the last section, we are concerned with the proof of Theorem 1.2.Owing to the similarity with the proof of Theorem 1.1, we only give a sketch and omit the details of the proof of Theorem 1.2.

Preliminaries
In this section, we first present necessary definitions and introduce some auxiliary lemmas, including those from the fixed point index theory and the theory based on the upper and lower solutions method.Subsequently, we construct some key integral operators, which is closely related to the problem (1.1), (1.2).As an important preparation to the proof of our main result, we devote the remaining part of this section to the proof of the complete continuity and monotonicity of the operator.We also give the a priori estimates on the positive solutions of the problem (1.1), (1.2).
The following two lemmas from the fixed point index theory and the theory based on upper and lower solutions method will be used to obtain the multiplicity of positive solutions, see [4].
Lemma 2.1 Let D be a cone in a real Banach space X, Ω be a bounded open subset of X with θ ∈ Ω, and EJQTDE, 2010 No. 27, p. 3 Lemma 2.2 Suppose X is a partially ordered Banach space, D is a normal cone of X, and A : D → X is an increasing and completely continuous operator.If there exist x 0 , y 0 ∈ D such that x 0 ≤ y 0 , x 0 , y 0 ⊂ D, and x 0 , y 0 are a lower solution and an upper solution of the equation x − Ax = 0, respectively, then the above equation has a minimal solution x * and a maximal solution y * in the ordered interval x 0 , y 0 , and x * ≤ y * .
The following technical lemma, see [8], will be used to characterize some properties of a related operator.
We also need the following technical lemma on the property of the function f , see [3].
Lemma 2.4 Suppose that f : [0, +∞) → (0, +∞) is continuous.For s > 0 and M > 0, there exists s > s and h 0 > 0 such that We are now in the position to make preparation to prove our main result.We use the notation ϕ p (s) = |s| p−2 s, for p > 1.Let q = p p−1 .Then ϕ q is the inverse function of ϕ p .Let E = C[0, 1] with the norm u = max 0≤t≤1 |u(t)| and It is clear that P is a normal cone of E. Define a set S={λ > 0; such that the problem (1.1), (1.2) has at least one positive solution }.
To show the existence, it is necessary to construct an appropriate operator and solve the corresponding operator equation.For this purpose, we first notice that u(t) is a solution of the problem (1.1), (1.2), if and only if u(t) is a solution of the following problem Because of (2.1) and (2.3), v(t) can be expressed by where k(t, s) is the Green function of the equation (2.1) with the boundary value (2.3), and Then u(t) can be expressed by (2.5) It is easy to prove that the Green function k(t, s) has the following properties.
Lemma 2.5 For all t, s ∈ [0, 1], we have The proof is complete.Next, we consider the following approximate problem Define a cone K ⊂ P as follows And we define an integral operator T h λ on K by (2.8) Now, we discuss the properties of the operator T h λ .First, by the definition of T h λ and the assumption (H1), we have EJQTDE, 2010 No. 27, p. 5 Lemma 2.7 Let (H1) and (H2) be satisfied.Then the operator T h λ defined by (2.8) is non-decreasing, that is, the inequality where " ≤ " is the partial order defined on K.
The complete continuity of T h λ can be obtained by the following lemma.
Proof.Firstly, we testify the complete continuity of . Moreover, because of the continuity of ϕ q (s), (1.3) and (H2), by the Lebesgue dominated convergence theorem, we have Therefore, we see that T h λ is continuous.And the compactness of the operator T h λ is easily obtained from the Arzela-Ascoli theorem.
Next, we testify that T h λ K ⊂ K.For each u ∈ K, it is easy to check that (T h λ u) ′′ (t) ≤ 0 for 0 < t < 1 and (T h λ u)(t) ≥ 0 for 0 ≤ t ≤ 1.Then according to Lemma 2.3, the following inequality holds By the definition of T h λ , namely (2.8), and a direct computation, we obtain the following lemma.

Proofs of the Main Result
In this section, we give the proof of the main result, that is, Theorem 1.1.The proof will be divided into two parts.Firstly, by the upper and lower solutions method, we investigate the basic existence of positive solutions of the problem (1.1), (1.2).Exactly, we will determine the threshold λ * of the parameter λ, such that the problem is solvable if and only if 0 < λ ≤ λ * .Furthermore, by the fixed point index theory, we establish the multiplicity of positive solutions for 0 < λ < λ * .We first present and prove the basic existence result of positive solutions of the problem (1.1), (1.2).Proof.Let S be the set defined in the previous section, namely S = {λ > 0; such that the problem (1.1), (1.2) has at least one positive solution}.
We first show that S = ∅.Let β(t) be a solution of the boundary value problem Then, by (2.5) we have k(τ, s)g(s)ds dτ.
Next, we show that if λ 1 ∈ S, then (0, λ 1 ) ⊂ S. In fact, if u λ 1 be a positive solution of the problem (1.1), (1.2), then, by Lemma 2.9, we have Therefore, for any λ ∈ (0, λ 1 ), by (2.8), we have which implies that u λ 1 is an upper solution of T 0 λ .Taking this into account, noticing the fact that the function α(t) ≡ 0 is a lower solution of T 0 λ , and using Lemma 2.2, Lemma 2.7 and Lemma 2.8, we see that the problem (1.1), (1.2) has a positive solution, and therefore λ ∈ S, which implies that (0, λ 1 ) ⊂ S.
EJQTDE, 2010 No. 27, p. 8 Now, we conclude that sup S < +∞.If this were not true, then we would have N ⊂ S, where N denotes the set of natural numbers.Therefore, for any n ∈ N, by Lemma 2.8, there exists u n ∈ K satisfying If u n ≥ 1, then we have k(s, s)g(s)δds dτ Letting n → +∞ in (3.2) and (3.3), we get a contradiction.Therefore, sup S < +∞.
We are now in a position to determine the threshold λ * .We conclude that It remains to show that λ * ∈ S. Let {λ n } be an increasing sequence in [ λ * 2 , λ * ) with λ n → λ * (n → +∞), and let u n be the corresponding solutions of the problem (1.1), (1.2) with λ replaced by λ n .By Lemma 2.10, there exists R( λ * 2 ) > 0 such that Therefore, {u n } is an equicontinuous and uniformly bounded subset of C[0, 1].By the Arzela-Ascoli theorem, {u n } has a convergent subsequence, and denoted also by itself, with u n → u * as n → +∞.Since u n = T 0 λn u n , due to the continuity of f (s), we see that 2 )], which together with the continuity of ϕ q (s) and (H2), by the Lebesgue dominated convergence theorem, implies that u * = T 0 λ * u * .Hence, by Lemma 2.9, u * is a positive solution of the problem (1.1), (1.2) with λ replaced by λ * .Summing up, we have completed the proof of the proposition.
We can now establish the multiplicity of positive solutions by the fixed point index theory.
Proof.Just as mentioned above, the arguments are based on fixed point index theory.Exactly, we apply Lemma 2.1 to calculate the indexes of the corresponding operator in two different domains, and then complete the proof by the index theory.
Let K be the set defined in the previous section, namely To calculate the index of the operator T 0 λ on some subset of K, we need to check the validity of the conditions in Lemma 2.1.By Lemma 3.1, there exists h 0 ∈ (0, 1), such that for any h ∈ (0, h 0 ) and λ ∈ (0, λ * ), the problem (2.6), (2.7) admits at least one positive solution.Let v λ (t) be a positive solution of the problem (2.6), (2.7), and Ω = {u ∈ K; u(t) < v λ (t), t ∈ [0, 1]}.It is clear that the set Ω ⊂ K is nonempty, bounded and open.If u ∈ ∂Ω, then there exists t 0 ∈ [0, 1], such that u(t 0 ) = v λ (t 0 ).Therefore, for any µ ≥ 1, h ∈ (0, h 0 ) and u ∈ ∂Ω, we have EJQTDE, 2010 No. 27, p. 10 and by Lemma 2.9, we have Hence, for any µ ≥ 1, we see that T 0 λ u = µu, u ∈ ∂Ω.Therefore, by Lemma 2.1, Now, we calculate the index of the operator T 0 λ on another relevant subset of K.For this purpose, we check the conditions of Lemma 2.1.Firstly, we check the validity of the condition (1) of the second part of Lemma 2.1.In fact, for any u ∈ K, we have where B R = {u ∈ K; u < R}.Hence the condition (1) of the second part of Lemma 2.1 is fulfilled.It remains to check the validity of the condition (2) of the second part of Lemma 2.1.In fact, if this condition were not satisfied, then there would exist a function Then, we have T 0 λ u 1 ≤ u 1 , which conflicts with (3.7).Therefore by Lemma 2.1, we have Finally, from (3.8), by the additivity of the fixed point index, we can now complete the proof of the proposition.In fact, we have Consequently, from (3.4) and (3.9), by the properties of the fixed point index, there are a fixed point of T 0 λ in Ω and a fixed point of T 0 λ in B R \Ω, respectively.Therefore by Lemma 2.9, the problem (1.1), (1.2) has at least two positive solutions.The proof is complete.
Combining Proposition 3.1 with Proposition 3.2, we have completed the proof of Theorem 1.1.

The Extension
Just as mentioned in the introduction, our approach for treating the problem (1.1), (1.2) can also be applied to the problem (1.1), (1.4).Since several details are quite similar, we only give a sketch for the proof of Theorem 1.2.
Similar to the proof of the Theorem 1.1, it is necessary to construct an appropriate operator and solve the corresponding operator equation.For this purpose, we first notice that u(t) is a solution of the problem (1.1), (1.4), if and only if u(t) is a solution of the following problem where ϕ q is the inverse function of ϕ p .Because of (4.1) and (4.3), v(t) can be expressed by where k(t, s) is the Green function of the equation (4.1) with the boundary value (4.3), and From (4.2), (4.4), and (4.5), u(t) can be expressed by Next, we consider the following approximate problem Now, we discuss the properties of the operator T h λ .From (H1), we can easily obtain the monotonicity of the operator T h λ .While the complete continuity of T h λ can be obtained by the following lemma.Lemma 4.2 Let (H1), (H2) and (1.5) be satisfied.Then the operator T h λ is completely continuous, and T h λ K ⊂ K.
Proof.Firstly, we testify the complete continuity of T h λ .The proof is parallel to that of the front part of Lemma 2.8, and so, we omit the details.Now, we testify T h λ K ⊂ K.For each u ∈ K, it is easy to check that (T 0 λ ) ′′ (t) ≤ 0 for 0 < t < 1 and (T 0 λ )(t) ≥ 0 for 0 < t < 1.Then according to Lemma 2.3, the following inequality holds which implies that (T 0 λ u)(t) ∈ K.For any u ∈ K, by (4.8), we have On the other hand, by (4.9), we have min which implies that T h λ K ⊂ K.The proof is complete.Similar to Lemma 2.10, we obtain the following a priori estimates on positive solutions of the problem (1.1), (1.4).Lemma 4.3 Let (H1), (H2) and (1.5) be satisfied.Suppose that λ ∈ S and S 1 = (λ, +∞) ∩ S ≡ ∅, where S={λ > 0; such that the problem (1.1), (1.4) has at least one positive solution }.Then there exists R(λ) > 0, such that u λ ′ ≤ R(λ), where λ ′ ∈ S 1 , and u λ ′ ∈ K is a positive solution of the problem (1.1), (1.4) with λ replaced by λ ′ .
The Proof of Theorem 1.2.The proof is parallel to that of Theorem 1.1, and so, we omit the details.

k
(s, s)g(s)δds dτ > 1. (3.6) Therefore, for any R > R > 0 and B R ⊂ K, by (3.5) and (3.6) we have Lemma 4.1 Let (H1) and (H2) be satisfied.Then the problem(1.1),(1.4)hasa positive solution u if and only if u is a fixed point of T 0 λ .And the problem (4.6), (4.7) has a positive solution u if and only if u is a fixed point of T h λ .