Weak Eigenfunctions for the Linearization of Extremal Elliptic Problems

g(0) > 0 and lim u→∞ g(u) u = ∞. We consider solutions u of (1.1) which are nonnegative in Ω. Typical examples are g(u) = e and g(u) = (1+u) with p > 1. Problem (1.1) and its connections with combustion theory have been extensively studied; see [CR], [G], [JL], [BCMR] and [BE]. It is known that there exists 0 < λ∗ < ∞ such that (i) for 0 ≤ λ < λ∗, there is a minimal classical solution uλ of (1.1) (see [CR]); (ii) for λ = λ∗, there exists a weak solution u∗ of (1.1) (Lemma 5 of [BCMR]); (iii) for λ > λ∗, there is no weak solution of (1.1) (Corollary 2 of [BCMR]). The notion of weak solution of (1.1) as considered in [BCMR] is the following: