| Summary: | We show the existence and nonexistence of entire positive solutions for semilinear elliptic system with gradient term ?u+|?u|=p(|x|)f(u,v)?u+|?u|=p(|x|)f(u,v), ?v+|?v|=q(|x|)g(u,v)?v+|?v|=q(|x|)g(u,v) on RNRN, N?3N?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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