Study of a generalized logistic equation with nonlocal reaction term

Baoqiang Yan,Ge Gao,Jianhua Zhou

doi:10.1186/s13661-018-1066-z

Abstract

In this paper, we consider the generalized logistic equation with nonlocal reaction term \t\t\t−Δu=u(λ+b∫Ωurdx−f(u))in Ω,u>0 in Ω,u=0 on ∂Ω.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ -\\Delta u=u \\biggl(\\lambda +b \\int_{\\Omega }u^{r}\\,dx-f(u) \\biggr)\\quad \\text{in } \\Omega,\\qquad u>0 \\quad \\text{ in } \\Omega,\\qquad u=0 \\quad \\text{ on } \\partial \\Omega. $$\\end{document} Using the bifurcation and sub-supersolution method, we obtain the non-existence, existence, and uniqueness of positive solutions for different parameters on the nonlocal terms. Our works about the nonlocal elliptic problem improve the results in the previous literature.

Highlights

We present some results on the existence of an unbounded continuum of positive solutions of (1.1), the local and global behavior of the continuum, and prove the non-existence of positive solutions
4 Existence and uniqueness results first we introduce the method of sub-supersolution to some nonlocal elliptic problems
Proof By Theorem 2.1 we know the existence of an unbounded continuum C0 of positive solutions bifurcating from the trivial solution at λ = λ1

Summary

Introduction

Lemma 2.1 (See [3]) There exists a positive solution of (2.1) if and only if λ > λ1. Lemma 2.3 Assume that θλ is the unique positive solution to (2.1) for λ > λ1.

Results

Conclusion