Uniqueness in [formula omitted] and continuity up to portions of the boundary of positive solutions for a strongly-singular elliptic problem

Abstract

In this paper, we consider issues about existence and uniqueness of Wloc1,p(x)(Ω)-solutions and its continuity up to portions of the boundary for the elliptic equation −Δp(x)u=c(x)d(x)−β(x)u−α(x) under a general sense of zero-boundary condition for a smooth bounded domain Ω⊂RN, where α(x) and β(x) may change their signals in multiple sub-regions of this domain Ω or on its boundary. We also prove a Comparison Principle for Wloc1,p(x)(Ω)-sub and super solutions for a related problem. In addition, we present a kind of “compatibility condition” involving the trio (c,α,β) to obtain solution still with zero-boundary in the sense of the trace on portions of the boundary. Some of our results are new even for the classical Laplacian operator.