A system of superlinear elliptic equations in a cylinder

Abstract

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