On a nonlinear elliptic problem with singular diffusion and right hand side in \(L^1\) (Q2708189)

The authors study the following nonlinear elliptic problem: NEWLINE\[NEWLINE \begin{cases} w\nabla u-\nabla \biggl[\bigl( B_1+B_2a(u) \bigr)\nabla u\biggr]+ g(x,u)= f \quad &\text{in }\Omega\\ u=0\quad & \text{on } \partial\Omega \end{cases} \tag{1}NEWLINE\]NEWLINE where \(\Omega\subset\mathbb{R}^N\) be a bounded domain, \(w\in [L^2 (\Omega)]^N\), \(f \in L^1(\Omega)\), \(g:\Omega \times\mathbb{R}\to\mathbb{R}\) are given data, and \(\lim_{s\to s^+_0} a(s)=+\infty\), \(s_0\in\mathbb{R}\setminus \{0\}\). Under some natural conditions on the matrix diffusion coefficient, the authors introduce the concept of renormalized solution for (1) and prove existence of it. However they prove uniqueness of renormalized solutions under more restrictive assumptions.