Entire large solutions to semilinear elliptic systems

We consider the problem of existence of positive solutions to the elliptic system Δu=p(|x|)vαΔu=p(|x|)vα, Δv=q(|x|)uβΔv=q(|x|)uβ on RnRn (n⩾3n⩾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)}min{p(r),q(r)} does not have compact support. We show that if αβ⩽1αβ⩽1, then such a solution exists if and only if the functions p and q satisfy∫0∞tp(t)(t2−n∫0tsn−3Q(s)ds)αdt=∞,∫0∞tq(t)(t2−n∫0tsn−3P(s)ds)βdt=∞ with P(r)=∫0rτp(τ)dτ and Q(r)=∫0rτq(τ)dτ. For αβ>1αβ>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.