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Existence of a Solution for a Non-Local Problem in ℝN via Bifurcation Theory

Part of: Elliptic equations and systems Mathematical biology in general

Published online by Cambridge University Press:  21 May 2018

Claudianor O. Alves ,
Romildo N. de Lima  and
Marco A. S. Souto
Claudianor O. Alves
Affiliation:
Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, CEP: 58429-900, Campina Grande – PB, Brazil (romildo@mat.ufcg.edu.br; coalves@dme.ufcg.edu.br; marco@dme.ufcg.edu.br)
Romildo N. de Lima*
Affiliation:
Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, CEP: 58429-900, Campina Grande – PB, Brazil (romildo@mat.ufcg.edu.br; coalves@dme.ufcg.edu.br; marco@dme.ufcg.edu.br)
Marco A. S. Souto
Affiliation:
Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, CEP: 58429-900, Campina Grande – PB, Brazil (romildo@mat.ufcg.edu.br; coalves@dme.ufcg.edu.br; marco@dme.ufcg.edu.br)
*
*Corresponding author.
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Abstract

In this paper, we study the existence of a solution for the following class of non-local problems: P

$$\eqalign{\big\{&-\Delta u=\left(\lambda f(x)-\int_{{open R}^N}K(x,y)\vert u(y)\vert ^{\gamma}\hbox{d}y\right)u\quad \mbox{in } \R^{N}, \cr &\lim_{\vert x\vert \to +\infty}u(x)=0,\quad u \gt 0 \quad \text{in } {open R}^{N},}$$
where N ≥ 3, λ > 0, γ ∈ [1, 2), f : ℝ → ℝ is a positive continuous function and K : ℝN × ℝN → ℝ is a non-negative function. The functions f and K satisfy some conditions that permit us to use bifurcation theory to prove the existence of a solution for (P).

Keywords

non-local logistic equationsa priori boundspositive solutions

MSC classification

Primary: 35J15: Second-order elliptic equations
Secondary: 35J60: Nonlinear elliptic equations 92B05: General biology and biomathematics
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 3 , August 2018 , pp. 825 - 845
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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