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Positive solutions for some competitive elliptic systems

1Department of Mathematics, Rabigh College of Sciences and Arts, King Abdulaziz University, P.O. Box 344, Rabigh, 21911, Kingdom of Saudi Arabia

© 2014 Mathematical Institute, Slovak Academy of Sciences. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

Citation Information: Mathematica Slovaca. Volume 64, Issue 1, Pages 61–72, ISSN (Online) 1337-2211, DOI: 10.2478/s12175-013-0187-1, March 2014

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Using some potential theory tools and the Schauder fixed point theorem, we prove the existence of positive bounded continuous solutions with a precise global behavior for the semilinear elliptic system Δu = p(x)u α ν r in domains D of ℝn, n ≥ 3, with compact boundary (bounded or unbounded) subject to some Dirichlet conditions, where α ≥ 1, β ≥ 1, r ≥ 0, s ≥ 0 and the potentials p, q are nonnegative and belong to the Kato class K(D).

MSC: Primary 31B35, 35B09, 35B50, 35J08, 35J57

Keywords: positive solutions; Green function; Kato class; elliptic systems; Maximum principle

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