On boundary blow-up solutions to equations involving the ∞∞-Laplacian

Given a non-negative, continuous function hh on View the MathML sourceΩ¯×R such that h(x,0)=0h(x,0)=0 for all x∈Ωx∈Ω, h(x,t)>0h(x,t)>0 in Ω×(0,∞)Ω×(0,∞), and h(x,t)h(x,t) non-decreasing in tt for each x∈Ωx∈Ω, we study the boundary value problem View the MathML source{Δ∞u=h(x,u)in Ωu=∞on ∂Ω Turn MathJax on where View the MathML sourceΩ⊆RN,N≥2 is a bounded domain and Δ∞Δ∞ is the ∞∞-Laplacian, a degenerate elliptic operator. We provide sufficient conditions on hh under which the above problem admits a solution, or fails to admit a solution. A necessary and sufficient condition on ff is given for a solution to exist in the special case when h(x,t)=b(x)f(t)h(x,t)=b(x)f(t). In the latter case an asymptotic boundary behavior of solutions will be studied. As an application a sufficient condition on ff will be given to ensure the uniqueness of solutions in case bb is a constant.