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

摘要

In this paper we study the asymptotic behavior of boundary blow-up solutions to the equation?_∞u = b(x)f(u)in ?, where A_∞is the∞-Laplacian, the nonlinearity f is a positive, increasing function in (0, ∞), and the weighted function b e C(?)is positive in?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.