Blow-up rates of large solutions for elliptic equations

In this paper, we mainly study the boundary behavior of solutions to boundary blow-up elliptic problems for more general nonlinearities f (which may be rapidly varying at infinity) Δu=b(x)f(u), x∈Ω, u|∂Ω=+∞, where Ω is a bounded domain with smooth boundary in RN, and which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary. Further, when f(s)=sp±f1(s) for s sufficiently large, where p>1 and f1 is normalized regularly varying at infinity with index p1∈(0,p), we show the influence of the geometry of Ω on the boundary behavior for solutions to the problem. We also give the existence and uniqueness of solutions.