Large positive solutions to an elliptic system of competitive type with nonhomogeneous terms

Abstract

In this paper, we study the elliptic system of competitive type with nonhomogeneous terms $ \Delta u = u^pv^q+h_1(x) $, $ \Delta v = u^rv^s+h_2(x) $ in $ \Omega $ with two types of boundary conditions: (Ⅰ) $ u = v = +\infty $ and (SF) $ u = +\infty $, $ v = f $ on $ \partial\Omega $, where $ f > 0 $, $ (p-1)(s-1)-qr > 0 $, and $ \Omega \subset \mathbb{R^N} $ is a smooth bounded domain. The nonhomogeneous terms $ h_1(x) $ and $ h_2(x) $ may be unbounded near the boundary and may change sign in $ \Omega $. First, for a single semilinear elliptic equation with a singular weight and nonhomogeneous term, boundary asymptotic behaviour of large positive solutions is established. Using this asymptotic behaviour, we show existence of large positive solutions for this elliptic system with the boundary condition (SF), existence of maximal solution, boundary asymptotic behaviour and uniqueness of large positive solutions for this elliptic system with (Ⅰ).

Keywords

• elliptic system
• large solutions
• nonhomogeneous term
• existence
• boundary asymptotic behaviour
• uniqueness