Mulltiplicity Of Positive Solutions For A Class Of Fourth-order Quasilinear Singular Differential Equations

In this paper, we consider the boundary value problems for a class of fourth-order quasilinear singular differential equations. The fourth-order differential equations arise from many fields, such as hydrodynamics, material mechanics, image processing. During the last two decades, more and more mathematicians pay attention to the fourth order differential equations.In the beginning, they discussed the following two point boundary value problems for the fourth-order ordinary differential equationThe above equation is often referred to the beam equation. It describes the deflection of a beam under a certain force. Aftabizadeh [1] and Gupta [2] discussed the case when f was either continuous or a Caratheodory function and studied the existence and uniqueness of the positive solutions for the boundary value problems. In [3], O'Regan considered the case when f was singular at t = 0 and t = 1, and proved a sufficient condition for the existence of positive solutions by using the topological transformation theorem. This result was then generalized in [4] and [5], in which the equation was changed to the following formwhere a(t) was singular at t = 0 and t = 1. The authors supposed that f was sublinear or superlinear, and obtained the sufficient and necessary condition for the existence of the positive solutions of the above problem. In [6], Q. Yao, Z. Bai discussed the singular eigenvalue problemThey also permitted the singularity of a(t), and supposed lim(f(t)/t)and lim(f(t)/t) were finity. They obtained the existence of positivesolutions and the multiplicity of positive solutions, when A is in a range. In recent years, in [7], M. Q. Feng and W. G. Ge generalized new results on the above fourth order differential equations. Under some conditions, the authors obtained the existence of the solution, and obtained the relation between the existence of solutions and the parameterλ.On the other hand, the boundary value problem for one dimensional p-Laplace equation is studied abroad. In [8], Wong Fu-Hsiang considered the boundary value problem for the following p-Laplace equationwhere m≥2 and f : [0,1]×(0,∞)→(0,∞) satisfying f(t,·) was locally Lipschitz continuous on (0,∞), and (f(t,u))/u^p-1 was strictly decreasing in u. By using the shooting method, the author obtained a sufficient and condition for the existence of the positive solutions. In [9], Liu Bing used fixed-point theory to establish the existence of the positive solution for the singular three-point boundary problem, which is as followswhere p≥2,0≤β< 1,0 <η< 1,f∈C([0, +∞), [0, +∞)), a∈C((0,1), [0, +∞)), and a(t) was allowed to have a singularity at the endpoints of (0,1). Recently, in [10], C. H. Jin and J. X. Yin obtained the existence of positive solutions for the following boundary value problem of one-dimensional p-Laplace equation with delaywhere p > 1,λ> 0 , and (?) is the correlative prescribed positive delay time.In recent years, some researchers combined the methods about fourth order differential equations and one-dimensional p-Laplace equations to study the fourth-order quasilinear differential equa-tions. In [11], Drabek and Otani considered the local existence and uniqueness of the solution for the fourth-order initial value problemwhereλ> 0,p = q> 1. In [12], this result was then generalized by Benedikt for p, q > 1 arbitrary. Later, in [13] he discussed the initial value problem for the following 2n-order quasilinear differential equationwhereλ∈R, n∈N, p, q > 1. The authors showed that there was a global solution for p≥q, while the solution can blow up for p > q. On the other hand, there was at most one solution for p < q. and for p > q they gave an example of nonuniqueness.In this paper, we study the following fourth-order quasilinear singular differential equationwhere p > 1,λ> 0 . We discuss two classes of boundary conditions for the above equation, respectively, i.e.andAs we know, at present, some researchers mostly study the boundary or initial value problem for the fourth-order linear differential equation, and the initial value problem for the fourth-order quasilinear differential equation. However, only a small number of papers have discussed the boundary value for the fourth-order quasi-linear singular differential equations. Most people use the fixed theorem to obtain the existence of positive solution. In this paper, we use the fixed point index theory and the upper and lower solutions method to obtain the existence of positive solution for the problem (1)-(2) and the problem (1), (3), respectively. And we obtain the relation between the multiplicity of positive solutions and the parameterλ.Firstly, we study the problem (1)-(2). For the case p≠2, our problem is quasilinear, and have no corresponding Green's function. The skills mentioned above in the works might be unavailable. Bea-cause of the singularity of g(t), it makes difficulties when we want to obtain the priori estimate of the positive solution. In order to overcome all these difficulties, we transform the problem (1)-(2) into a system which is equivalent to the problem (1)-(2). By using the first boundary value condition (2), we construct the integral operator, use the fixed point index theory, and obtain the existence of positive solution and multiplicity of positive solutions. Meanwhile, we show that there exists a thresholdλ* < +∞, i.e. there exists at least two positive solutions for 0 <λ<λ*, while there is no positive solution forλ>λ*.Secondly, we study the problem (1), (3). The singularity of g take the similar difficulties, when we want to obtain the priori estimate of the positive solution for our problem. We transform the problem (1) and (3) into a system which is equivalent to the problem (1), (3). We use the Green function twice to construct the integral operator which is related to the problem (1), (3). At last. we obtain the relation between the multiplicity of positive solutions for the problem (1), (3) and the parameterλ.