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) Delta u = b(x)f(u). x is an element of Omega, u vertical bar(partial derivative Omega) = +infinity, where Omega is a bounded domain with smooth boundary in R-N, and b is an element of C-alpha((Omega) over bar) which is positive in Omega and may be vanishing on the boundary and rapidly varying near the boundary. Further, when f(s) = s(p) +/- f(1)(s) for s sufficiently large, where p > 1 and f(1) is normalized regularly varying at infinity with index p(1) is an element of (0, p), we show the influence of the geometry of Omega on the boundary behavior for solutions to the problem. We also give the existence and uniqueness of solutions. (C) 2010 Elsevier Inc. All rights reserved.