Fourth-order nonlinear elliptic equations with lower order term and natural growth conditions.

We prove the existence of weak solutions of the homogeneous Dirichlet problem related toa class of nonlinear elliptic equations whose prototype is\begin{multline*}\label{modello}\sum_{|\al|=2}\Da \Big[|\DD^2 u |^{p-2}\Da u\Big] -\sum_{|\al|=1}\Da \Big[|\Du u |^{q-2}\Da u\Big]%\\%\hfill+ u \Big[ |\DD^1 u|^q + |\DD^2 u| ^p \Big] = f \end{multline*}where $\Om$ is an open bounded subset of $\R^N$ ($ N\geq 3$) with sufficiently smooth boundary, $u: \Om \rightarrow \R$ is the unknown function, $\DD^h u=\Big\{\Da u: |\al|=h \Big\}$, $|\DD^hu|=\big[\sum\limits_{|\al|=h}|\Da u|^2\big]^{\frac{1}{2}}$, for $h=1,2$, numbers $p$, $q\in [2, N[$ and$f \in L^{1}(\Om).