On the Boundedness of Generalized Solutions of Higher-Order Nonlinear Elliptic Equations with Data from an Orlicz–Zygmund Class

In the present paper, a $2m$th-order quasilinear divergence equation is considered under the condition that its coefficients satisfy the Carathéodory condition and the standard conditions of growth and coercivity in the Sobolev space $W^{m,p}(\Omega)$, $\Omega\subset \mathbb{R}^{n}$, $p>1$. It is proved that an arbitrary generalized (in the sense of distributions) solution $u\in W^{m,p}_{0}(\Omega)$ of this equation is bounded if $m\ge2$, $n=mp$, and the right-hand side of this equation belongs to the Orlicz–Zygmund space $L(\log L)^{n-1}(\Omega)$.