Existence of positive solutions for some nonlinear elliptic systems on the half space

Abstract

We prove some existence of positive solutions to the semilinear elliptic system $$displaylines{ Delta u =lambda p(x)g(v)cr Delta v =mu q(x)f(u) }$$ in the half space ${mathbb{R}}^n_+$, $ngeq 2$, subject to some Dirichlet conditions, where $lambda$ and $mu$ are nonnegative parameters. The functions $f, g$ are nonnegative continuous monotone on $(0,infty)$ and the potentials $p, q$ are nonnegative and satisfy some hypotheses related to the Kato class $K^infty({mathbb{R}}^n_+)$.

Keywords

• Green function
• Kato class
• elliptic systems
• positive solutions