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

By means of a simple proof we show that the system −Δui=pi(|x|)fi(ui+1)−Δui=pi(|x|)fi(ui+1) and −Δud=pd(|x|)fd(u1)−Δud=pd(|x|)fd(u1) for i=1,d−1¯ on RN, where N>2N>2, fii=1,d¯:(0,∞)→(0,∞) are continuous functions bounded in a neighborhood at infinity such that limsi↘0fii=1,d¯(si)=+∞ and pii=1,d¯ are positive radial functions which are continuous on RN, has no positive radial solutions that decay to zero at infinity provided ∫0∞r∑i=1dpi(r)dr=∞, with r:=|x|r:=|x|. Moreover, a non-existence result for the case N=2N=2 is obtained.