A system of superlinear elliptic equations in a cylinder

Abstract The article is concerned with the existence of positive solutions a semi-linear elliptic system defined in cylinder $$\Omega =\Omega '\times (0,a)\subset {\mathbb {R}}^n$$ Ω = ′ × ( 0 , a ) ⊂ R n , where '\subset {R}}^{n-1}$$ - 1 bounded and smooth domain. couples superlinear equation whole $$ another (or linear) at bottom \{0\}$$ { } . Possible applications for such systems are interacting substances (gas fluid bottom) or competing species cylindrical habitat (insects air plants on ground). We provide priori $$L^\infty L ∞ bounds all when nonlinear terms satisfy certain growth conditions. It interesting that due to structure our restrictions weaker than those pioneering result by Brezis–Turner single equation. Using topological arguments, we prove these particular systems.