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November  2009, 8(6): 2037-2053. doi: 10.3934/cpaa.2009.8.2037

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, Argentina

Received  November 2008 Revised  April 2009 Published  August 2009

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, 45M2.
Citation: J. García-Melián, Julio D. Rossi. A logistic equation with refuge and nonlocal diffusion. Communications on Pure & Applied Analysis, 2009, 8 (6) : 2037-2053. doi: 10.3934/cpaa.2009.8.2037
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