On the Regularizing Effect of Some Absorption and Singular Lower Order Terms in Classical Dirichlet Problems with L1Data

We are interested in existence and regularity results concerning the solution to the following problem $$\left\{ \begin{array}{l} - \Delta u + {u^8} = \frac{{f\left( x \right)}}{{{u^\gamma }}}in\,\Omega \\ u > 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,in\,\Omega \\ u = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,on\,\partial \Omega \\ \end{array} \right.\, $${−Δu+u8=f(x)uγinΩu>0inΩu=0on∂Ω where Ω is an open and bounded subset of ℝN, 0 < γ ≤ 1, s ≥ 1 and fis a nonnegative function that belongs to some Lebesgue space.