Boundary Blow-up Problems Involved In Infinity Laplacian Operator

In this paper,we study the existence,comparison principle,uniqueness and asymptotic estimate of the boundary blow-up solution for infinity Laplacian problem.In Chapter 1,we introduce the research background and significance of the infinity Laplacian operator and the boundary blow-up solutions,review the research results of the blow-up solutions of the infinity Laplacian problem,and summarize the main work and innovation points of this paper.In Chapter 2,we introduce the definition of the viscosity solution of the infinity Laplacian equation and the definition and some properties of regularly varying functions.In Chapter 3,we study the boundary blow-up problem related to the infinity Laplacian where Δ∞hu=|Du|h-3(D2uDu,Du>is the highly degenerate and h-homogeneous operator.When q>h>1,we establish the existence of the boundary blow-up viscosity solution.Moreover,when the domain satisfies some regular condition,we establish the asymptotic estimate of the blow-up solution near the boundary.By the asymptotic estimate and the comparison principle,we obtain the uniqueness result of the large solution.We also give the non-existence of the large solution for the case q ≤ h.In Chapter 4,we investigate the boundary blow-up problem related to the infinity Laplacian When the function f satisfies the Keller-Osserman type condition,we establish the existence of the boundary blow-up viscosity solution.Moreover,for the separable case f(x,u)=b(x)g(u),we establish the asymptotic estimate of the blow-up solution near the boundary under some regular condition of the domain.Based on the asymptotic estimate and comparison principle,we obtain the uniqueness of the large viscosity solution.During this procedure,we also study the non-existence of the large solution.For the separable case,we show that the Keller-Osserman type condition is sufficient and necessary for the non-existence of the boundary blow-up viscosity solution.In this process we also consider a broader class of equationsWhen A(p)satisfies certain structural conditions,we ues the perturbation method to establish the comparison principle and stability of the viscosity solution.