Entire large solutions to semilinear elliptic systems

We consider the problem of existence of positive solutions to the elliptic system Δ u = p ( | x | ) v α , Δ v = q ( | x | ) u β on R n ( n ⩾ 3 ) which satisfies lim | x | → ∞ u ( x ) = lim | x | → ∞ v ( x ) = ∞ . The parameters α and β are positive, and the nonnegative functions p and q are continuous and min { p ( r ) , q ( r ) } does not have compact support. We show that if α β ⩽ 1 , then such a solution exists if and only if the functions p and q satisfy ∫ 0 ∞ t p ( t ) ( t 2 − n ∫ 0 t s n − 3 Q ( s ) d s ) α d t = ∞ , ∫ 0 ∞ t q ( t ) ( t 2 − n ∫ 0 t s n − 3 P ( s ) d s ) β d t = ∞ with P ( r ) = ∫ 0 r τ p ( τ ) d τ and Q ( r ) = ∫ 0 r τ q ( τ ) d τ . For α β > 1 , we show that a solution exists if either of the above conditions fails to hold; i.e., one of the integrals is finite. These extend all known results for the given system.