During my work, I came to the question of existence of weak solutions to the following elliptic equation
$$\triangle u + \partial_{1} u + \partial_2(f(x_1,x_2,u)) = 0 \hbox{ in } \Omega$$ with homogeneous Neumann and inhomogeneous Dirichlet boundary conditions on the boundary of bounded open set $\Omega\subset{\bf R}^2$ with smooth boundary. Function $f$ is in ${\rm L}^\infty(\Omega) \cap {\rm L}^2(\Omega)$.
I have two questions:
1) could you suggest me some references (articles or books) where I might find existence results for the above problem?
2) could you also suggest me some references on maximum principle results for such problems (namely, if the function is bounded on the Dirichlet part of the boundary, is it bounded (not necessary with the same bound) on the whole domain?)
I have tried searching in Gilbarg-Trudinger monograph, but they usually consider only Dirichlet problem and/or their function $f$ is smooth. Some other articles I've stumbled upon usually deal with special forms of function $f$ (some polynomial of $u$) but they never consider it to be dependent on spatial variables as well.
I am not an expert in nonlinear elliptic equations, so I might be missing some obvious well-known references.
Thank You.