A Necessary and Sufficient Condition for Existence of Large Solutions to Semilinear Elliptic Equations

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We consider the semilinear equation Δu=p(x)f(u) on a domain Ω⊆Rn, n≥3, where f is a nonnegative, nondecreasing continuous function which vanishes at the origin, and p is a nonnegative continuous function with the property that any zero of p is contained in a bounded domain in Ω such that p is positive on its boundary. For Ω bounded, we show that a nonnegative solution u satisfying u(x)→∞ as x→∂Ω exists if and only if the function ψ(s)≡∫s0f(t)dt satisfies ∫∞1(ψ(s))−1/2ds<∞. For Ω unbounded (including Ω=Rn), we show that a similar result holds where u(x)→∞ as |x|→∞ within Ω and u(x)→∞ as x→∂Ω if p(x) decays to zero rapidly as |x|→∞.