Non-existence criteria for solutions of the Lane–Emden–Fowler system

By means of a simple proof we show that the system − Δ u i = p i ( | x | ) f i ( u i + 1 ) and − Δ u d = p d ( | x | ) f d ( u 1 ) for i = 1 , d − 1 ¯ on R N , where N > 2 , f i i = 1 , d ¯ : ( 0 , ∞ ) → ( 0 , ∞ ) are continuous functions bounded in a neighborhood at infinity such that lim s i ↘ 0 f i i = 1 , d ¯ ( s i ) = + ∞  and p i i = 1 , d ¯  are positive radial functions which are continuous on R N , has no positive radial solutions that decay to zero at infinity provided ∫ 0 ∞ r ∑ i = 1 d p i ( r ) d r = ∞ , with r : = | x | . Moreover, a non-existence result for the case N = 2 is obtained.