Positive solutions of an integro-differential equation in all space with singular nonlinear term

Giuseppe Maria Coclite; Mario Michele Coclite

doi:10.3934/dcds.2008.22.885

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2008, Volume 22, Issue 4: 885-907. Doi: 10.3934/dcds.2008.22.885

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Positive solutions of an integro-differential equation in all space with singular nonlinear term

Giuseppe Maria Coclite^1, and
Mario Michele Coclite^1,

1.
Dipartimento di Matematica, Università di Bari, via Orabona 4, 70125 Bari

Received: December 31, 2006

Revised: October 31, 2007

Published: September 2008

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Abstract

We prove the existence of a positive solution in $W_{loc}^{2,q}$ for a semilinear elliptic integro-differential problem in $\mathbb{R}^N.$ The integral operator of the equation depends on a nonlinear function that is singular in the origin. Moreover, we prove that the averages of the solution and its gradient on the balls $\{x\in\mathbb{R}^N; |x| \le R\}, R>0,$ vanish as $R\to \infty.$

Keywords:

Integro-differential equations in all space,
singular nonlinearity,
existence of positive solutions,
asymptotic behavior.

Mathematics Subject Classification: Primary: 45G05; 45K05; 45M05; Secondary: 45M20.

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