The exact boundary blow-up rate of large solutions for semilinear elliptic problems

In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem −△u=λu−a(x)up,u|∂Ω=+∞−△u=λu−a(x)up,u|∂Ω=+∞, where ΩΩ is a bounded smooth domain in RNRN. The weight function a(x)a(x) in front of the nonlinearity can vanish on the boundary of the domain ΩΩ at different rates according to the point x0x0 of the boundary. The decay rate of the weight function a(x)a(x) may not be approximated by a power function of distance near the boundary ∂Ω∂Ω. We combine the localization method of [J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45] with some previous radially symmetric results of [T. Ouyang, Z. Xie, The uniqueness of blow-up solution for radially symmetric semilinear elliptic equation, Nonlinear Anal. 64 (9) (2006) 2129–2142] to prove that any large solution u(x)u(x) must satisfy limx→x0u(x)K(bx0∗(dist(x,∂Ω)))−β=1for each x0∈∂Ω, where bx0∗(r)=∫0r∫0sbx0(t)dtds,K=[β((β+1)C0−1)]1p−1,β=1p−1,C0=limr→0(∫0rbx0(t)dt)2bx0∗(r)bx0(r) and bx0(r)bx0(r) is the boundary normal section of a(x)a(x) at x0∈∂Ωx0∈∂Ω, i.e., bx0(r)=a(x0−rnx0),r>0,r∼0, and nx0 stands for the outward unit normal vector at x0∈∂Ωx0∈∂Ω.