On the Regularizing Effect of Some Absorption and Singular Lower Order Terms in Classical Dirichlet Problems with L¹ Data

Abstract

We are interested in existence and regularity results concerning the solution to the following problem

$$\left\{ \begin{array}{l} - \Delta u + {u^8} = \frac{{f\left( x \right)}}{{{u^\gamma }}}in\,\Omega \\ u > 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,in\,\Omega \\ u = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,on\,\partial \Omega \\ \end{array} \right.\, $$

where Ω is an open and bounded subset of ℝ^N, 0 < γ ≤ 1, s ≥ 1 and f is a nonnegative function that belongs to some Lebesgue space.