The exact blow-up rates of large solutions for semilinear elliptic equations

Abstract

In this paper, combining the method of lower and upper solutions with the localization method, we establish the boundary blow-up rate of the large positive solutions to the singular boundary value problem { Δ u = b ( x ) f ( u ) , x ∈ Ω , u ( x ) = + ∞ , x ∈ ∂ Ω , where Ω is a smooth bounded domain in R N . The weight function b ( x ) is a non-negative continuous function in the domain, which vanishes on the boundary of the underlying domain Ω at different rates according to the point of the boundary. f ( u ) is locally Lipschitz continuous satisfying the Keller–Osserman condition and f ( u ) / u is increasing on ( 0 , ∞ ) . It is worth emphasizing that we obtain the main results for a large class of nonlinear terms f , which is regularly varying at infinity with index p ∈ R (that is for all ξ > 0 , lim u → ∞ f ( ξ u ) / f ( u ) = ξ p ), instead of the restriction: f ( u ) ∼ H u p for sufficiently large u and some positive constants H > 0 , p > 1 as in the series of papers of J. L. Gómez.