On weak solutions of boundary value problems for some general differential equations

We study general settings of the Dirichlet problem, the Neumann problem, and other boundary value problems for equations and systems of the form $\mathcal{L}^+ A\mathcal{L}u=f$ with general (matrix, generally speaking) differential operation $\mathcal{L}$ and some linear or non-linear operator $A$ acting in $L^k_2(\Omega)$-spaces. For these boundary value problems, results on well-posedness, existence and uniqueness of a weak solution are obtained. As an operator $A$, we consider Nemytskii and integral operators. The case of operators involving lower-order derivatives is also studied.