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) ≡ ∫s0 f(t) dt satisfies ∫1(ψ(s)) − 1/2 ds < ∞. 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| → ∞.


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Reviewed at MR1728197

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Journal of Mathematical Analysis and Applications