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

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 {Δu=b(x)f(u),x∈Ω,u(x)=+∞,x∈∂Ω, where ΩΩ is a smooth bounded domain in RNRN. The weight function b(x)b(x) is a non-negative continuous function in the domain, which vanishes on the boundary of the underlying domain ΩΩ at different rates according to the point of the boundary. f(u)f(u) is locally Lipschitz continuous satisfying the Keller–Osserman condition and f(u)/uf(u)/u is increasing on (0,∞)(0,∞). It is worth emphasizing that we obtain the main results for a large class of nonlinear terms ff, which is regularly varying at infinity with index p∈Rp∈R (that is for all ξ>0ξ>0, limu→∞f(ξu)/f(u)=ξplimu→∞f(ξu)/f(u)=ξp), instead of the restriction: f(u)∼Hupf(u)∼Hup for sufficiently large uu and some positive constants H>0H>0, p>1p>1 as in the series of papers of J. L. Gómez.