Multiplicity of Positive Solutions to Nonlocal Boundary Value Problems with Strong Singularity

Chan-Gyun Kim

doi:10.3390/axioms11010007

Abstract

In this paper, we consider generalized Laplacian problems with nonlocal boundary conditions and a singular weight, which may not be integrable. The existence of two positive solutions to the given problem for parameter λ belonging to some open interval is shown. Our approach is based on the fixed point index theory.

Highlights

IntroductionSatisfying the L1 -Carathéodory condition, several existence results for positive solutions to φ-Laplacian boundary value problems involving linear bounded operators in the boundary conditions
Consider the following singular φ-Laplacian problem:(q(t) φ(u0 (t)))0 + λh(t) f (u(t)) = 0, t ∈ (0, 1), u (0) =Citation: Kim, C.-G
We show the existence of two positive solutions to nonlocal boundary value problems (1)–(2) for λ belonging to some open interval in the case when either f 0 = f ∞ = ∞ or f 0 = f ∞ = 0

Summary

Introduction

Satisfying the L1 -Carathéodory condition, several existence results for positive solutions to φ-Laplacian boundary value problems involving linear bounded operators in the boundary conditions. Yang [8], by using the Avery– Peterson fixed point theorem, obtained the existence of at least three positive solutions to the p-Laplacian equation with integral boundary conditions. Goodrich [9] studied perturbed Volterra integral operator equations and, as an application, established the existence of at least one positive solution to the p-Laplacian differential equation with nonlocal boundary conditions. We show the existence of two positive solutions to nonlocal boundary value problems (1)–(2) for λ belonging to some open interval in the case when either f 0 = f ∞ = ∞ or f 0 = f ∞ = 0.

Preliminaries

Proof of Theorem 1

Conclusions