The Existence and Nonexistence of Entire Positive Radial Solutions of Quasilinear Elliptic Systems with Gradient Term

We study the existence and nonexistence of entire positive solutions for quasilinear elliptic systemwith gradient termdiv(|∇u|p−2∇u) +|∇u|p−1=a(|x|)f(u,v),div(|∇v|q−2∇v) +|∇v|q−1=b(|x|)g(u,v)onRN(N≥3), where nonlinearitiesfandgare positive and continuous, the potentialsaandbare continuous, c-positive and satisfy appropriate growth conditions at infinity. We have that entirelarge positive solutions fail to exist iffandgare sublinear andaandbhave fast decay at infinity,while iffandgsatisfy some growth conditions at infinity, anda,bare of slow decay or fast decayat infinity, then the system has infinitely many entire solutions, which are large or bounded.