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

Abstract

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