An extreme variation phenomenon for some nonlinear elliptic problems with boundary blow-up
Florica C. Cîrstea · Comptes Rendus Mathématique · 2004
Let Ω be a smooth bounded domain in R N ( N ⩾ 2 ) and Γ ∞ be a non-empty open and closed subset of ∂ Ω . Denote by B either the Dirichlet or the mixed boundary operator on Γ B : = ∂ Ω ∖ Γ ∞ when Γ ∞ ≠ ∂ Ω . We consider the nonlinear elliptic problem Δ u + a u = b ( x ) f ( u ) in Ω , subject to B u = 0 on Γ B when Γ B ≠ ∅ , where a is a real number, b is a continuous non-negative function on Ω ¯ , while f ⩾ 0 is continuous on [ 0 , ∞ ) such that f ( u ) / u is increasing on ( 0 , ∞ ) . Assuming that f varies rapidly at infinity with index ∞ (i.e., lim u → ∞ f ( λ u ) / f ( u ) = λ ∞ for all λ > 0 ), we establish the uniqueness of the positive solution satisfying u = ∞ on Γ ∞ and describe its blow-up rate via the extreme value theory.