Positive solutions of second-order singular boundary value problem with a Laplace-like operator
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By use of the concavity of solution for an associate boundary value problem, existence criteria of positive solutions are given for the Dirichlet BVP (Φ(u )) + λa(t) f (t,u) = 0, 0 < t < 1, u(0) = 0 = u(1), where Φ is odd and continuous with 0 < l1 ≤ ((Φ(x) − Φ(y))/ (x − y)) ≤ l2 , a(t) ≥ 0, and f may change sign and be singular along a curve in [0,1] × R+ . 1. Introduction For the Sturm-Liouville boundary value problem (BVP)
Φ u
+ λa(t) f (t,u) = 0,
α1 u(0) − β1 u (0) = 0 = α1 u(1) + β2 u (1),
(1.1)
there has been much work done for some special cases in order to search the existence of positive solutions. For example, Erbe and Wang [3] studied the case for Φ(v) = v, Wang [8] discussed the problem with boundary conditions replaced by nonlinear ones, Sun and Ge [7] dealt with the problem for the existence of multiple positive solutions in case α1 = β2 = 0 and β1 = α2 = 1, Avery et al. [2] researched the existence of twin positive solutions for the case Φ(v) = v, α1 = β2 = 1, β1 = α2 = 0, and He and Ge [6] discussed the existence of multiple positive solutions. In all the above-mentioned articles f is supposed to be nonnegative. When Φ(v) = v, Agarwal et al. [1] as well as Ge and Ren [4] discussed the existence of positive solutions without nonnegativity condition imposed on f . As for the general BVP
p(t)Φ u
+ λp(t) f (t,u) = 0, u(0) = 0 = u(1),
0 < t < 1,
(1.2)
Hai et al. [5] studied the existence of positive solutions with f ≥ −M. When Φ is odd and Φ−1 is concave, they proved that there are λ∗ , λ > 0 such that BVP (1.2) has at least one positive solution if λ ∈ (0,λ∗ )[λ > λ] under the condition limu→∞ f (t,u)/Φ(u) = ∞ uniformly for t ∈ [0,1]. The restriction, Φ−1 being concave, excludes the case Φ(u) = |u| p−2 u, 1 < p < 2. Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:3 (2005) 289–302 DOI: 10.1155/JIA.2005.289
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Singular BVP with a Laplace-like operator
In this paper, we want to give theorems for the existence of positive solutions for the BVP
Φ u
+ λa(t) f (t,u) = 0,
0 < t < 1, λ > 0,
u(0) = 0 = u(1),
(1.3)
without the restriction f (t,u) ≥ −M for (t,u) ∈ [0,1] × R+ and without Φ−1 being concave. We suppose throughout this paper that 1−δ 1 (H1) a ∈ C((0,1), R+ ) and for a δ ∈ (0,(1/2)), 0 < δ a(t)dt ≤ 0 a(t)dt < ∞; (H2) Φ is odd, continuous with 0 < l1 ≤
Φ(x) − Φ(y) ≤ l 2 < ∞, x−y
x = y.
(1.4)
Obviously (H2) implies that Φ−1 (s) exists and
Φ−1 (x) − Φ−1 (y) 1 1 0< ≤ ≤ < ∞, l2 x−y l1
x = y.
(1.5)
2. Preliminary lemmas Lemma 2.1. Suppose (H1)-(H2) hold. Then for λM ∈ R,
Φ u
+ λa(t)M = 0,
0 < t < 1,
(2.1)
u(0) = 0 = u(1) has a unique solution wλM (t) =
t 0
Φ−1 λM c −
s 0
a(τ)dτ
ds
(2.2)
with c satisfying 1 0
Φ−1 λM c −
s 0
a(τ)dτ
ds = 0.
(2.3)
Proof. It is easy to show that u = w(t) is a solution to BVP (2.1) if and only if u(t) is expressed in (2.2) with c satisfying (2.3). Now we show that there is only one c which makes 1 (2.3) h
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