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 $\mathrm{\Delta }u=u^{q(x)}$ in a smooth bounded domain $\Omega \subset {\mathbb{R}}^n$. 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 $\partial \Omega$ is compatible with the occurrence of dead cores, i.e., nonempty interior regions where solutions vanish.