On a nonlinear Robin problem with an absorption term on the boundary and L 1 data

We deal with existence and uniqueness of nonnegative solutions to: − Δ u = f ( x ) , in Ω , ∂ u ∂ ν + λ ( x ) u = g ( x ) u η , on ∂ Ω , \left\{\begin{array}{ll}-\Delta u=f\left(x),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ \frac{\partial u}{\partial \nu }+\lambda \left(x)u=\frac{g\left(x)}{{u}^{\eta }},\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where η ≥ 0 \eta \ge 0 and f , λ f,\lambda , and g g are the nonnegative integrable functions. The set Ω ⊂ R N ( N > 2 ) \Omega \subset {{\mathbb{R}}}^{N}\left(N\gt 2) is open and bounded with smooth boundary, and ν \nu