Entropy solutions of the Dirichlet problem for a class of non-linear elliptic fourth-order equations with right-hand sides in $L^1$

In this paper we introduce and study the notion of an entropy solution of the Dirichlet problem for a class of non-linear elliptic fourth-order equations whose right-hand sides admit arbitrary growth with respect to the variable corresponding to the unknown function and belong to the space $L^1$ for each fixed value of this variable. We prove the existence and uniqueness of an entropy solution. We establish the existence of so-called $H$-solutions and $W$-solutions of the problem and prove that the entropy solutions belong to certain Sobolev spaces.