An elliptic cross-diffusion system describing two-species models on a bounded domain with different natural conditions

This paper is concerned with an elliptic cross-diffusion system describing two-species models on a bounded domain Ω, where Ω consists of a finite number of subdomains ΩiΩi (i=1,…,mi=1,…,m) separated by interfaces ΓjΓj (j=1,…,m−1j=1,…,m−1) and the natural conditions of the subdomains ΩiΩi are different. This system is strongly coupled and the coefficients of the equations are allowed to be discontinuous on interfaces ΓjΓj. The main goal is to show the existence of nonnegative solutions for the system by Schauder's fixed point theorem. Furthermore, as applications, the existence of positive solutions for some Lotka–Volterra models with cross-diffusion, self-diffusion and discontinuous coefficients are also investigated.