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

Abstract

In this paper we study the asymptotic behavior of boundary blow-up solutions to the equation Δ∞u=b(x)f(u) in Ω, where Δ∞ is the ∞-Laplacian, the nonlinearity f is a positive, increasing function in (0,∞), and the weighted function b∈C(Ω¯) is positive in Ω 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)=sp(1+c˜g(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.