| Contents |
| We give a series of results on the existence, uniqueness and summability of so called proper entropy solutions of the homogeneous Dirichlet problem for the equation
$$
\sum_{|\alpha|=1,2}\,(-1)^{|\alpha|}D^\alpha A_\alpha(x,\nabla_2u)=F(x,u)\,\,\,
\text{in}\,\,\,\Omega
$$
where $\Omega$ is a bounded open set of $\mathbb R^n$ ($n>2$), $F:\Omega\times\mathbb R\to\mathbb R$ and $\nabla_2u=\{D^\alpha u:|\alpha|=1,2\}$. It is assumed that the functions $A_\alpha$ satisfy some growth and monotonicity conditions and the following strengthened coercivity condition: for a. e. $x\in\Omega$ and for every $\xi=\{\xi_\alpha\in\mathbb R:
|\alpha|=1,2\}$,
$$
\sum_{|\alpha|=1,2}\,A_\alpha(x,\xi)\xi_\alpha \geq
c\bigg(\sum_{|\alpha|=1}|\xi_\alpha|^q + \sum_{|\alpha|=2}|\xi_\alpha|^p\bigg) - g(x)
$$
where $p\in (1,n/2)$, $q\in (2p,n)$, $c>0$ and $g\in L^1(\Omega)$, $g\geq 0$. For the existence of the solutions, in particular, we require $F(\cdot,s)\in L^1(\Omega)$ for every $s\in \mathbb R$. In the case where $F(x,s)=f(x)$ and $f[\ln(1+|f|)]^\sigma\in L^1(\Omega)$ or $f[\ln\ln(e+|f|)]^\sigma\in L^1(\Omega)$ with an arbitrary $\sigma>0$, we describe dependence of summubility of the solutions on $\sigma$. |
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