Entire positive solution to the system of quasilinear elliptic equations

Abstract

The quasilinear elliptic system div ( | ∇ u | p - 2 ∇ u ) + f 1 ( x ) u α 1 + g 1 ( x ) u - β 1 + h 1 ( x ) u γ 1 Q ( v ) = 0 , div ( | ∇ v | q - 2 ∇ v ) + f 2 ( x ) v α 2 + g 2 ( x ) v - β 2 + h 2 ( x ) v γ 2 Q ( u ) = 0 with p , q > 1 , 0 < α 1 , β 1 , γ 1 < p - 1 , 0 < α 2 , β 2 , γ 2 < q - 1 is considered in R N . Under suitable hypotheses on functions f i , g i , h i ( i = 1 , 2 ) and Q, it is shown that this system possesses an entire positive solution ( u , v ) . These extend recent work of Qiu and Yao in [Lingyun Qiu, Miaoxin Yao, Entire positive solution to the system of nonlinear elliptic equations, Boundary Value Problems, 2006 (2006), Article ID 32492, pp. 1–11], which considered the semilinear elliptic system.