A necessary and sufficient condition for the existence of large solutions to sublinear elliptic systems

We prove that the elliptic system Δ u = p ( | x | ) v α , Δ v = q ( | x | ) u β on R n ( n ⩾ 3 ) where 0 < α ⩽ 1 , 0 < β ⩽ 1 , and p and q are nonnegative continuous functions has a nonnegative entire radial solution satisfying lim | x | → ∞ u ( x ) = lim | x | → ∞ v ( x ) = ∞ 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 . This extends previous existence results for both the single equation and systems.