The existence and nonexistence of entire positive solutions of semilinear elliptic systems with gradient term

We show the existence and nonexistence of entire positive solutions for semilinear elliptic system with gradient term Δ u + | ∇ u | = p ( | x | ) f ( u , v ) , Δ v + | ∇ v | = q ( | x | ) g ( u , v ) on R N , N ⩾ 3 , provided that nonlinearities f and g are positive and continuous, the potentials p and q are continuous, c-positive and satisfy appropriate growth conditions at infinity. We find that entire large positive solutions fail to exist if f and g are sublinear and p and q have fast decay at infinity, while if f and g satisfy some growth conditions at infinity, and p, q are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded.