BOUNDARY BLOW-UP SOLUTIONS TO EQUATIONS INVOLVING THE INFINITY LAPLACIAN

Abstract

Abstract In this paper, we study the boundary blow-up problem related to the infinity Laplacian $$ \begin{align*}\begin{cases} \Delta_{\infty}^h u=u^q &\mathrm{in}\; \Omega, \\ u=\infty &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$ where $\Delta _{\infty }^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator associated with the infinity Laplacian arising from the stochastic game named Tug-of-War. 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. As an application of the asymptotic estimate and the comparison principle, we obtain the uniqueness result of the large solution. We also give the nonexistence of the large solution for the case $q \leq h.$