Positive solutions of some nonlinear eigenvalue problems in exterior domains

In this paper, we give a characterization that enables us to redefine the Kato class of potential functions $K_{\infty }\left(D\right)$, on an exterior domain D with $C^{1,1}$-boundary, studied in [11]. Next, we consider a class of semilinear elliptic system of type $\Delta u=\lambda F(x,u,v)$, $\Delta v=\mu H(x,u,v)$, for some nonnegative nonlinearities F and H and positive constants λ and μ. Under adequate conditions on F and H, related to $K_{\infty }\left(D\right)$, we prove the existence of positive solutions continuous on $\bar{D}$, for $\lambda \in [0,{\lambda }_0)$ and $\mu \in [0,{\mu }_0)$ for some ${\lambda }_0,{\mu }_0\in (0,\infty ]$, whenever some conditions on the finite boundary and the infinite boundary are given.