Existence of Large Solutions to Semilinear Elliptic Equations with Multiple Terms

We consider the semilinear elliptic equation 4u = p(x)uα + q(x)uβ on a domain Ω ⊆ Rn, n ≥ 3, where p and q are nonnegative continuous functions with the property that each of their zeroes is contained in a bounded domain Ωp or Ωq, respectively in Ω such that p is positive on the boundary of Ωp and q is positive on the boundary of Ωq. For Ω bounded, we show that there exists a nonnegative solution u such that u(x) −→ ∞ as x −→ ∂Ω if 0 < α ≤ β,β > 1, and that such a solution does not exist if 0 < α ≤ β ≤ 1. For Ω = Rn, we establish conditions on p and q to guarantee the existence of a nonnegative solution u satisfying u(x) −→∞ as |x| −→∞ for 0 < α ≤ β,β > 1, and for 0 < α ≤ β ≤ 1. For Ω = Rn and 0 < α ≤ β < 1, we also establish conditions on p and q for the existence and nonexistence of a solution u where u is bounded on Rn.