Uniqueness in Wloc1,p(x)(Ω) 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 W l o c 1 , p ( x ) ( Ω ) -solutions and its continuity up to portions of the boundary for the elliptic equation − Δ p ( 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 ⊂ R N , 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 W l o c 1 , 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.