Multiplicity of positive solutions to semilinear elliptic boundary value problems

Kenichiro Umezu

doi:10.1155/s1085337599000147

Abstract

We study semilinear elliptic boundary value problems of one parameter dependence where the number of positive solutions is discussed. Our main purpose is to characterize the critical value given by the infimum of such parameters for which positive solutions exist. Our approach is based on super‐ and sub‐solutions, and relies on the topological degree theory on the positive cones of ordered Banach spaces. A concrete example is also presented.

Highlights

Let D be a bounded domain of Euclidean space RN, N ≥ 2, with smooth boundary ∂D
We study the following semilinear elliptic boundary value problem: Lu := − ∆ + c(x) u = λf (u) in D, Bu := a(x) ∂u + ∂n
We prove here that problem (1.1) has at least two distinct positive solutions in the open interval given by (1.17)

Summary

KENICHIRO UMEZU

We study semilinear elliptic boundary value problems of one parameter dependence where the number of positive solutions is discussed. Our main purpose is to characterize the critical value given by the infimum of such parameters for which positive solutions exist. Our approach is based on super- and sub-solutions, and relies on the topological degree theory on the positive cones of ordered Banach spaces.

Introduction

Here we assert that

Then we obtain