Optimal uniqueness theorems and exact blow-up rates of large solutions

In this paper, we prove some optimal uniqueness results for large solutions of a canonical   class of semilinear equations under minimal regularity conditions on the weight function in front of the non-linearity and combine these results with the localization method introduced in [López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45] to prove that any large solution LL of Δu=a(x)upΔu=a(x)up, p>1p>1, a>0a>0, must satisfylimx→x0L(x)Fx0(dist(x,∂Ω))=I0-pp-1p+1p-1p+1p-1foreachx0∈∂Ω,whereFx0(t):=∫t∞∫0sfx01p+1-p+1p-1ds,I0:=limt↓0Fx0(t)Fx0′′(t)[Fx0′(t)]2and fx0fx0 is any smooth extension of the boundary normal section of a   at x0∈∂Ωx0∈∂Ω, i.e.,fx0(t)=a(x0-tnx0),t>0,t∼0.Subsequently, nx0nx0 stands for the outward unit normal at x0∈∂Ωx0∈∂Ω. Therefore, the theory can be extended to cover the general case when fx0∈L1p+1.