Semilinear cooperative elliptic systems on Rn

• Published in: Rendiconti di matematica e delle sue applicazioni (1981) Vol. 15; no. 1; pp. 89 - 108
• Main Authors: J. FLECKINGER-PELLÉ, H. SERAG
• Format: Journal Article
• Language: English
• Published: Sapienza Università Editrice 01-01-1995
• Subjects: cooperative elliptic systems; unbounded domains; weighted sobolev spaces
• EISSN: 2532-3350
• ISSN: 1120-7183

We study here the following semilinear cooperative elliptic system defined on IRn , n > 2 : (1 – a) −∆u = aρ(x)u + bρ(x)v + f(x, u, v) x ∈ IRn , (1 – b) −∆v = cρ(x)u + dρ(x)v + g(x, u, v) x ∈ IRn , (1 – c) u −→ 0 , v −→ 0 as |x| −→ +∞. Here a, b, c, d are constants such that b, c > 0 ; ρ, f and g are given functions; ρ is nonnegative and tends to 0 at ∞. We first establish necessary and sufficient conditions on the coefficients for having a Maximum Principle for the linear System. Then we show that these conditions ensure existence of solutions for the linear System and for the semilinear System when f and g satisfy some ”sublinear” condition. Under some additional assumption we also derive uniqueness of the solutions. Finally we show that our results can be extended to N × N systems, N > 2.