Abstract
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.
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