Large solutions for the Laplacian with a power nonlinearity given by a variable exponent

Abstract

In this paper we consider positive boundary blow-up solutions to the problem Δu=uq(x) in a smooth bounded domain Ω⊂Rn. The exponent q(x) is allowed to be a variable positive Hölder continuous function. The issues of existence, asymptotic behavior near the boundary and uniqueness of positive solutions are considered. Furthermore, since q(x) is also allowed to take values less than one, it is shown that the blow up of solutions on ∂Ω is compatible with the occurrence of dead cores, i.e., nonempty interior regions where solutions vanish.