Large solutions of a class of degenerate equations associated with infinity Laplacian

Abstract

Abstract In this article, we investigate the boundary blow-up problem Δ ∞ h u = f ( x , u ) , in Ω , u = ∞ , on ∂ Ω , \left\{\begin{array}{ll}{\Delta }_{\infty }^{h}u=f\left(x,u),& {\rm{in}}\hspace{0.33em}\Omega ,\\ u=\infty ,& {\rm{on}}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Δ ∞ h u = ∣ D u ∣ h − 3 ⟨ D 2 u D u , D u ⟩ {\Delta }_{\infty }^{h}u=| Du\hspace{-0.25em}{| }^{h-3}\langle {D}^{2}uDu,Du\rangle is the highly degenerate operator related to infinity Laplacian which comes from the absolutely minimizing Lipschitz extension and has a close relationship with the random game named tug-of-war. When the function f 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 ) f\left(x,u)=b\left(x)g\left(u) , we establish the asymptotic estimate of the blow-up solution near the boundary under some regular conditions 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 existence of the boundary blow-up viscosity solution.