Abstract
In this paper, we study singular φ -Laplacian nonlocal boundary value problems with a nonlinearity which does not satisfy the L 1 -Carathéodory condition. The existence, nonexistence and/or multiplicity results of positive solutions are established under two different asymptotic behaviors of the nonlinearity at ∞.
Highlights
We introduce a solution operator related to boundary value problem (BVP) (1) and (2)
U is a positive solution to BVP (1) and (2) if and only if T (λ, u) = u for some (λ, u) ∈ (0, ∞) × K
{(0, 0)} and (ii ) for any (λ, u) ∈ C \ {(0, 0)}, u is a positive solution to BVP (1) and (2) with λ > 0
Summary
Where w ∈ C ([0, 1], (0, ∞)), φ : R → R is an odd increasing homeomorphism, λ ∈ [0, ∞) is a parameter, h ∈ C ((0, 1), (0, ∞)) and f ∈ C ([0, 1] × (0, ∞), R). The condition ( A1) on the odd increasing homeomorphism φ was first introduced by Wang in [1] where the existence, nonexistence and/or multiplicity of positive solutions to quasilinear elliptic equations were studied. Karakostas ([2,3]) introduced a sup-multiplicative-like function as an odd increasing homeomorphism φ satisfies the following condition. When φ(s) = s, w ≡ 1 and λ = 1, Webb and Infante [16] considered problem (1) with various nonlocal boundary conditions involving a Stieltjes integral with a signed measure and gave several sufficient conditions on the nonlinearity f = f (t, u) for the existence and multiplicity of positive solutions via fixed point index theory. For an odd increasing homeomorphism φ satisfying ( A1), Kim and Jeong [4] studied various existence results for positive solutions to BVP (1) and (2) with λ = 1.
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