Asymptotic estimates of boundary blow-up solutions to the infinity Laplace equations

In this paper we study the asymptotic behavior of boundary blow-up solutions to the equation Delta(infinity)u = b(x)f(u) in Omega, where Delta(infinity) is the co-Laplacian, the nonlinearity f is a positive, increasing function in (0, co), and the weighted function b EC(Omega) is positive in Omega and may vanish on the boundary. We first establish the exact boundary blow-up estimates with the first expansion when f is regularly varying at infinity with index p > 3 and the weighted function b is controlled on the boundary in some manner. Furthermore, for the case of f (s) = s(P) (1 + cg (s)), with the function g normalized regularly varying with index -q <0, we obtain the second expansion of solutions near the boundary. It is interesting that the second term in the asymptotic expansion of boundary blow-up solutions to the infinity Laplace equation is independent of the geometry of the domain, quite different from the boundary blow-up problems involving the classical Laplacian. (c) 2014 Elsevier Inc. All rights reserved.