The Existence And Multiplicity Of Positive Solutions To The One-Dimensional P-Laplacian With Singular Sturm-Liouville Boundary Value Conditions

The existence and multiplicity of positive solutions to the p-Laplac ian together with Dirichlet, Sturm-Liouville or nonlinear boundary value conditions on finite intervals or on infinite intervals have been investigated extensively during the past years. For the special case p= 2, see [1-4] and the references therein. Ge and Ren[1] considered the following Sturm-Liouville boundary value problem authors converted solutions of the problem into the fixed points of a nonlinear integral operator using the corresponding Green's function. On the basis of the application of the fixed point theory in cones, some results for the existence and multiplicity of positive solutions of the problem (1) were obtained. Further, Lian and Ge[2] considered the problem above on the half-line, i.e.,They established sufficient conditions to guarantee the existence of at least one positive solution, a unique positive solution and multiple positive solutions for the problem (2) by using the compact theorem on infinite intervals and Krasnoselskii fixed point theorem. When f depends on the first-order derivative, the existence results are avail-able. For example, in [3] Sun et al. studied the following nonlinear singular problemwhere f was a nonnegative continuous function which may be singular at t= 0. They obtained the existence results of positive solutions under some certain conditions. We can see that some good results have been achieved for p=2.For p> 1, there are also some results, see [5-11]. In [5], Wang Junyu considered the equation subject to one of the following three pairs of boundary value condi-tions: where f was a nonnegative, lower semi-continuous function defined on [0,+00), a(t) was a nonnegative measurable function, B0(v) and B1(v) were both nondecreasing, continuous, odd functions and at least one of them satisfied the condition that there existed b＞0 such that 0≤Bj(v)≤bv,v≥0,j= 0 or 1. Through analyzing the problem, the author obtained that a positive solution of (4)-(5) was concave, so the minimum value of the solution on some closed interval could be controled by its supremum norm. An available intergral operator was defined in a cone according to the equation and the boundary value condition considered. He first showed the existence of at least one pos-itive solution for (4)-(5) assuming that f was continuous on [0,00) by applying the fixed point theorem, then obtained the existence results for the problem where f was discontinuous by approximation. Later, Wang Junyu and Kong Lingbin investigated the multiplicity of posi-tive solutions of the problem (4)-(5) in [6]. In [7], Bai et al. studied the equation (4) where f depended on x(t),χ'(t) explicitly. Further, Lian, Pang and Ge considered the problem on infinite intervals in [8], i.e.,whereχ'(∞)= (?)x'(t). Using the compact theorem on infinite intervals and Avery-Peterson fixed point theorem, they demonstrated that the problem (6) has at least three positive solutions under the certain conditions.In this paper, we consider the one-dimensional p-Laplacian and subject to one of the following three pairs of boundary value condi-tions:whereφp(s)=│s│p-2s,p＞1,λ(t), f(t,x,y) and h(t,x,y) are all non-negative continuous functions, moreover f may be singular at t=0,φ(t) is a nonnegative measurable function defined on (0,+∞),μ> 0 is a parameter,α,β,γ,δ> 0.The main purpose of this paper is to investigate the existence of positive solutions to the problem (7), (9) and the multiplicity of positive solutions to the problem (8), (9). Notice the fact that the equations we consider are quasilinear, for which the theory based on Green's function can not be applied. In addition, due to the appear-ance of A, solutions of the problems (7), (9) and (8), (9) may not be concave, and so, some efficient methods based on convexity (see for example [5]) could not be available here. In order to overcome these difficulties, a special Banach space and special cones are introduced so that we can establish the existence and multiplicity results using Kras-noselskii fixed point theorem and Avery-Peterson fixed point theorem.