BOUNDARY BLOW-UP SOLUTIONS TO SEMILINEAR ELLIPTIC EQUATIONS WITH NONLINEAR GRADIENT TERMS

In this article we study the blow-up rate of solutions near the boundary for the semilinear elliptic problem Delta u +/- vertical bar del u vertical bar(q) = b(x)f(u), x is an element of Omega, u(x) = infinity, x is an element of partial derivative Omega, where Q is a smooth bounded domain in R-N, and b(x) is a nonnegative weight function which may be bounded or singular on the boundary, and f is a regularly varying function at infinity. The results in this article emphasize the central role played by the nonlinear gradient term vertical bar del u vertical bar(q) and the singular weight b(x). Our main tools are the Karamata regular variation theory and the method of explosive upper and lower solutions.