A logistic equation with refuge and nonlocal diffusion

J. García-Melián; Julio D. Rossi

doi:10.3934/cpaa.2009.8.2037

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2009, Volume 8, Issue 6: 2037-2053. Doi: 10.3934/cpaa.2009.8.2037

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A logistic equation with refuge and nonlocal diffusion

J. García-Melián^1, and
Julio D. Rossi^2,

1.
Dpto. de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271 - La Laguna
2.
Dpto. de Matemáticas, FCEyN, Universidad de Buenos Aires, 1428 – Buenos Aires

Received: November 2008

Revised: April 2009

Published: November 2009

Abstract / Introduction Related Papers Cited by

Abstract

In this work we consider the nonlocal stationary nonlinear problem $(J* u)(x) - u(x)= -\lambda u(x)+ a(x) u^p(x)$ in a domain $\Omega$, with the Dirichlet boundary condition $u(x)=0$ in $\mathbb{R}^N\setminus \Omega$ and $p>1$. The kernel $J$ involved in the convolution $(J*u) (x) = \int_{\mathbb{R}^N} J(x-y) u(y) dy$ is a smooth, compactly supported nonnegative function with unit integral, while the weight $a(x)$ is assumed to be nonnegative and is allowed to vanish in a smooth subdomain $\Omega_0$ of $\Omega$. Both when $a(x)$ is positive and when it vanishes in a subdomain, we completely discuss the issues of existence and uniqueness of positive solutions, as well as their behavior with respect to the parameter $\lambda$.

Keywords:

Nonlocal diffusion,
logistic problems.

Mathematics Subject Classification: Primary: 45C05, 45M05, 45M20.

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