Conditions for the Limit Summability of Solutions of Nonlinear Elliptic Equations with Degenerate Coercivity and $L^1$-Data

We study entropy and weak solutions of the Dirichlet problem for a class of second-order nonlinear elliptic equations with degenerate coercivity and right-hand side~$f$ in $L^{1}(\Omega)$, where $\Omega$ is a bounded open set in ${\mathbb R}^n$ ($n\geqslant 2$). The growth condition on the coefficients of the equations admits any their growth with respect to the unknown function itself. Estimates for the distribution function of an entropy solution and its gradient are obtained using a function~$\tilde{f}\colon[0,+\infty)\to{\mathbb R}$ generated by the function~$f$. Applying these estimates, we establish integral conditions on the function~$\tilde{f}$ which guarantee the belonging of entropy solutions and their gradients to limit Lebesgue spaces. As a consequence, we obtain conditions for the belonging of entropy solutions to a limit Sobolev space $W^{1,r}_{0}(\Omega)$ and, as a particular case, to the space $W^{1,1}_{0}(\Omega)$. In addition, we establish conditions for the existence of weak solutions of the considered problem belonging to the space $W^{1,r}_{0}(\Omega)$. The obtained results generalize the known ones for equations whose coefficients satisfy the usual coercivity condition.