Positive Solutions of Nonlinear Elliptic Equations with Nonlinear Boundary Conditions

Hua Luo

doi:10.1155/2013/175724

Abstract

This paper discusses bifurcation from interval for the elliptic eigenvalue problems with nonlinear boundary conditions and studies the behavior of the bifurcation components.

Highlights

Much effort has been devoted to the study of the nonlinear elliptic boundary value problems, in particular, to problems which arise in numerous applications, for example, in physical problems involving the steady state temperature distribution [1, 2], in problems of chemical reactions [1, 3], in the theory of stellar structures [4], and in problems of Riemannian geometry [5]
Let Ω be a bounded domain of Euclidean space RN, N ≥ 2, with smooth boundary ∂Ω
Amann, “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,” SIAM Review, vol 18, no. 4, pp. 620–709, 1976

Summary

Introduction

Much effort has been devoted to the study of the nonlinear elliptic boundary value problems, in particular, to problems which arise in numerous applications, for example, in physical problems involving the steady state temperature distribution [1, 2], in problems of chemical reactions [1, 3], in the theory of stellar structures [4], and in problems of Riemannian geometry [5]. It is worth pointing out that Umezu [9], by using a different approach based on topological degree and global bifurcation techniques [10], discusses bifurcation from infinity for (1) with a(⋅) ≡ 1 They obtained a unique bifurcation value λ∞ from infinity of (1) and there exists an unbounded, closed, and connected component in (0, ∞) × C(Ω), consisting of positive solutions of (1) and bifurcating from (λ, u) = (λ∞, ∞). They proved that all the components bifurcate into the region λ < λ∞ or λ > λ∞ under some proper conditions and f(∞) = limu → ∞(f(u)/u), g∞ = limu → ∞(g(u)/u) ∈ (0, ∞). All the components obtained by Theorem 2 bifurcate into the region λ < λ1/f∞ (resp., λ > λ1/f∞)

Bifurcation Theorem from Interval for Compact Operator

Reduction to a Compact Operator Equation

Existence of Bifurcation Values from Infinity