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Research Article Dead Core Problems for Singular Equations with φ-Laplacian Ravi P. Agarwal, Donal O’Regan, and Svatoslav Stanˇek Received 27 May 2007; Accepted 6 September 2007 Recommended by Ivan Kiguradze

The paper discusses the existence of positive solutions, dead core solutions, and pseudo dead core solutions of the singular problem (φ(u )) + f (t,u ) = λg(t,u,u ), u (0) = 0, βu (T) + αu(T) = A. Here λ is a positive parameter, β ≥ 0, α, A > 0, f may be singular at t = 0 and g is singular at u = 0. Copyright © 2007 Ravi P. Agarwal et al. 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. 1. Introduction Throughout the paper L1 [a,b] denotes the set of integrable functions on [a,b], L1loc (a,b] the set of functions x : (a,b] → R which are integrable on [a − ε,b] for arbitrary small ε > 0, AC[a,b] is the set of absolutely continuous function on [a,b], and ACloc (a,b] is the set of functions x : (a,b] → R which are absolutely continuous on [a − ε,b] for arbitrary small ε > 0. Let T be a positive number. If G ⊂ R j ( j = 1,2) then Car([0, T] × G) stands for the set of functions h : [0,T] × G → R satisfying the local Carath´eodory conditions on [0, T] × G, that is, (j) for each z ∈ G, the function h(·,z) : [0,T] → R is measurable; (jj) for a.e. t ∈ [0,T], the function h(t, ·) : G → R is continuous; (jjj) for each compact set M ⊂ G, there exists δM ∈ L1 [0,T] such that |h(t,z)| ≤ δM (t) for a.e. t ∈ [0,T] and all z ∈ M. We will write h ∈ Car((0,T] × G) if h ∈ Car([a,T] × G) for each a ∈ (0,T]. We consider the singular boundary value problem  

φ u (t)

u (0) = 0,











+ f t,u (t) = λg t,u(t),u (t) , βu (T) + αu(T) = A,

λ > 0,

(1.1)

β ≥ 0, α,A > 0,

(1.2)

2

Boundary Value Problems

depending on the positive parameter λ. Here φ ∈ C 0 [0, ∞), f ∈ Car((0,T] × [0, ∞)) is nonnegative, f (t,0) = 0 for a.e. t ∈ [0,T], g ∈ Car([0,T] × D) is positive, where D = (0,A/α] × [0, ∞) and g is singular at the value 0 of its first space variable. We say that g is singular at the value 0 of its first space variable provided lim g(t,x, y) = ∞ for a.e. t ∈ [0,T] and each y ∈ [0, ∞).

x→0+

(1.3)

A function u ∈ C 1 [0,T] is called a positive solution of problem (1.1), (1.2) if u > 0 on [0,T], φ(u ) ∈ ACloc (0,T], u satisfies (1.2), and (1.1) holds for a.e. t ∈ [0,T]. We say that u ∈ C 1 [0,T] satisfying (1.2) is a dead core solution of problem (1.1), (1.2) if there exists t0 ∈ (0,T) such that u = 0 on [0,t0 ], u > 0 on (t0 ,T], φ(u ) ∈ AC[t0 ,T] and (1.1) holds for a.e. [t0 ,T]. The interval [0, t0 ] is called the dead core of u. If u(0) = 0, u > 0 on (0,T], φ(u ) ∈ ACloc (0,T], u satisfies (1.2) and (1.1) a.e. on [0,T], then u is called a pseudo dead core solution of problem (1.1), (1.2). The aim of this paper is to discuss the existence of positive solutions, dead core solutions, and pseudo dead core solutions to problem (1.1), (1.2). Alt