Entire large solutions for semilinear elliptic equations

We analyze the semilinear elliptic equation Δu=ρ(x)f(u), u>0 in RD (D⩾3), with a particular emphasis put on the qualitative study of entire large solutions, that is, solutions u such that . Assuming that f satisfies the Keller–Osserman growth assumption and that ρ decays at infinity in a suitable sense, we prove the existence of entire large solutions. We then discuss the more delicate questions of asymptotic behavior at infinity, uniqueness and symmetry of solutions.