Boundary blow-up solutions to degenerate elliptic equations with non-monotone inhomogeneous terms

Abstract

Given a non-negative, and non-trivial continuous real-valued function h on Ω̄×[0,∞) such that h(x,0)=0 for all x∈Ω, we study the boundary value problem ∫Δ ∞u=h(x,u)in Ωu=∞on ∂Ω, where Ω⊆ RN,N<2 is a bounded domain and Δ ∞ is the ∞-Laplacian, a degenerate elliptic operator. In this paper, we investigate conditions on the inhomogeneous term h(x,t) under which Problem (BVP) admits a solution or fails to admit a solution in C(Ω). Some notable features of this work are that h(x,t) is not required to have any special structure, and no monotonicity condition is imposed on h(x,t). Furthermore, h(x,t) may be allowed to vanish in either of the variables. © 2011 Elsevier Ltd. All rights reserved.