## Positive Solutions for Fourth-Order Singular -Laplacian Differential Equations with Integral Boundary Conditions

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Research Article Positive Solutions for Fourth-Order Singular p-Laplacian Differential Equations with Integral Boundary Conditions Xingqiu Zhang1, 2 and Yujun Cui3 1

Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China 2 Department of Mathematics, Liaocheng University, Liaocheng, Shandong 252059, China 3 Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao 266510, China Correspondence should be addressed to Xingqiu Zhang, [email protected] Received 7 April 2010; Accepted 12 August 2010 Academic Editor: Claudianor O. Alves Copyright q 2010 X. Zhang and Y. Cui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By employing upper and lower solutions method together with maximal principle, we establish a necessary and suﬃcient condition for the existence of pseudo-C3 0, 1 as well as C2 0, 1 positive solutions for fourth-order singular p-Laplacian diﬀerential equations with integral boundary conditions. Our nonlinearity f may be singular at t  0, t  1, and u  0. The dual results for the other integral boundary condition are also given.

1. Introduction In this paper, we consider the existence of positive solutions for the following nonlinear fourth-order singular p-Laplacian diﬀerential equations with integral boundary conditions: 

  ϕp x t  ft, xt, xt, 0 < t < 1, 1 x0  gsxsds, x1  0, 0













ϕp x 0  ϕp x 1 

1 0

  hsϕp x s ds,

1.1

2

Boundary Value Problems

where ϕp t  |t|p−2 · t, p ≥ 2, ϕq  ϕ−1 p , 1/p 1/q  1, f ∈ CJ × R × R , R , J  0, 1, 1 R  0, ∞, R  0, ∞, I  0, 1, and g, h ∈ L1 0, 1 is nonnegative. Let σ1  0 1 − 1 1 sgsds, σ2  0 hsds. Throughout this paper, we always assume that 0 ≤ 0 gsds < 1, 1 0 < 0 hsds < 1 and nonlinear term f satisfies the following hypothesis:

H ft, u, v : J × R × R → R is continuous, nondecreasing on u and nonincreasing on v for each fixed t ∈ J, and there exists a real number b ∈ R such that, for any r ∈ J, ft, u, rv ≤ r −b ft, u, v,

∀t, u, v ∈ J × R × R ,

1.2

there exists a function ξ : 1, ∞ → R , ξl < l and ξl/l2 is integrable on 1, ∞ such that ft, lu, v ≤ ξlft, u, v,

∀t, u, v ∈ J × R × R , l ∈ 1, ∞.

1.3

Remark 1.1. Condition H is used to discuss the existence and uniqueness of smooth positive solutions in 1. i Inequality 1.2 implies that ft, u, cv ≥ c−b ft, u, v,

if c ≥ 1.

1.4

Conversely, 1.4 implies 1.2. ii Inequality 1.3 implies that   −1 ft, cu, v ≥ ξ c−1 ft, u, v,

if 0 < c < 1.

1.5

Conversely, 1.5 implies 1.3. Remark 1.2. Typical functions that satisfy condition H are those taking the form ft, u, v   −μj  ni1 ai tuλi m , where ai , bj ∈ C0, 1, 0 < λi < 1, μj > 0 i  1, 2, . . . , m; j1 bj tu j  1, 2, . . . , m. Remark 1.3. It fo