Some existence results for a quasilinear problem with source term in Zygmund-space

In this paper we study the existence of solution to the problem \begin{equation*} \left\{\begin{array}{l} u\in H_{0}^{1}(\Omega), \\[4pt] -\textrm{div}\,(A(x)Du)=H(x,u,Du)+f(x)+a_{0}(x)\, u\quad \text{in} \quad\mathcal{D}'(\Omega), \end{array} \right. \end{equation*} where $\Omega$ is an open bounded set of $\mathbb{R}^{2}$, $A(x)$ a coercive matrix with coefficients in $L^\infty(\Omega)$, $H(x,s,\xi)$ a Carath\'eodory function satisfying, for some $\gamma >0$, $$ -c_{0}\, A(x)\, \xi\xi\leq H(x,s,\xi)\,{\rm sign}(s)\leq \gamma\,A(x)\,\xi\xi \;\;\; {\rm a.e. }\; x\in \Omega,\;\;\;\forall s\in\mathbb{R},\;\;\; \forall\xi \in \mathbb{R}^{2}. $$ Here $f$ belongs to $L^1(\log L^1)(\Omega)$ and $a_{0} \geq 0$ to $L^{q}(\Omega )$, $q>1$. For $f$ and $a_{0}$ sufficiently small, we prove the existence of at least one solution $u$ of this problem which is such that $e^{\delta_0 |u|} -1$ belongs to $H_{0}^{1}(\Omega)$ for some $\delta_0\geq\gamma$ and satisfies an \textit{a priori} estimate.