Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient

Giovany M. Figueiredo; Tarcyana S. Figueiredo-Sousa; Cristian Morales-Rodrigo; Antonio Suárez

doi:10.3934/dcdsb.2018311

Article Contents

2019, Volume 24, Issue 8: 3689-3711. Doi: 10.3934/dcdsb.2018311

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Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient

1.
Dpto. de Matemática, Campus Universitário Darcy Ribeiro, Universidade de Brasília, 70910-900, Brasília - DF, Brazil
2.
Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Fac. de Matemáticas, Univ. de Sevilla, Sevilla, C/. Tarfia s/n, 41012, Spain

Received: February 28, 2018

Revised: June 30, 2018

Early access: October 2018

Published: July 2019

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Abstract

In this paper we study a stationary problem arising from population dynamics with a local and nonlocal variable diffusion coefficient. We show the existence of an unbounded continuum of positive solutions that bifurcates from the trivial solution. The global structure of this continuum depends on the value of the nonlocal diffusion at infinity and the relative position of the refuge of the species and of the sets where it diffuses locally and not locally, respectively.

Keywords:

Bifurcation method,
variable local and nonlocal diffusion coefficient.

Mathematics Subject Classification: Primary: 35B09, 35B32; Secondary: 35J60, 35P30.

Citation:

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Figure 1. Bifurcation diagrams when $\lambda _0<\lambda _\infty<\infty$ and $\lambda _\infty<\lambda _0<\infty$, respectively

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Figure 2. Bifurcation diagrams when $\lambda _\infty = 0$ and $\lambda_\infty = \infty$, respectively. For example, this last diagram appears when $b\geq b_0>0$

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