Quasilinear Lane-Emden equations with absorption and measure data

Abstract

We study the existence of solutions to the equation $-\Gd_pu+g(x,u)=\mu$ when $g(x,.)$ is a nondecreasing function and $\gm$ a measure. We characterize the good measures, i.e. the ones for which the problem as a renormalized solution. We study particularly the cases where $g(x,u)=\abs x^{\beta}\abs u^{q-1}u$ and $g(x,u)=\abs x^{\tau}\rm{sgn}(u)(e^{\tau\abs u^\lambda}-1)$. The results state that a measure is good if it is absolutely continuous with respect to an appropriate Lorentz-Bessel capacities.