Abstract
This paper is concerned with the multiplicity of positive solutions of boundary value problem for the fourth-order quasilinear singular differential equation (|u ′′ | p−2 u ′′ ) ′′ = �g(t)f(u), 0 1, � > 0. We apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of positive solutions. We have found a threshold � ∗ � ∗ , then the problem has no positive solution. In particular, there exist at least two positive solutions for 0 < � < � ∗ .
Highlights
We consider the multiplicity of positive solutions for the following fourthorder quasilinear singular differential equation
In the past few years, some fourth-order nonlinear equations have been proposed for image processing
A number of authors hoped that these methods might perform better than some second-order equations [15,16,17,18,19]
Summary
We consider the multiplicity of positive solutions for the following fourthorder quasilinear singular differential equation (|u′′|p−2u′′)′′ = λg(t)f (u), 0 < t < 1,. The solution of the equation (1.1) can be regarded as the steady-state case of the fourth-order anisotropic diffusion equation in [16,17,18,19]. In a recent paper [5], under some structure conditions which permit some singularities for g(t), the authors discussed the special case p = 2 subject to the boundary value conditions (1.2), and revealed the relation between the existence of positive solutions and the parameter λ. We mainly discuss the boundary value problem for the fourth-order quasilinear differential equation (1.1), namely, the problem (1.1), (1.2). Owing to the similarity with the proof of Theorem 1.1, we only give a sketch and omit the details of the proof of Theorem 1.2
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