Existence and uniqueness results for solutions of nonlinear equations with right hand side in ${\displaystyle L^1}$

We prove an existence and uniqueness theorem for the elliptic Dirichlet problem for the equation div a(x,∇u) = f in a planar domain Ω. Here ${\displaystyle f\in L^1(\Omega )}$ and the solution belongs to the so-called grand Sobolev space ${\displaystyle W_0^{1,2}\}(\Omega )}$. This is the proper space when the right hand side is assumed to be only ${\displaystyle L^1}$-integrable. In particular, we obtain the exponential integrability of the solution, which in the linear case was previously proved by Brezis-Merle and Chanillo-Li.