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

M. V. Voitovich

Matematičeskie zametki, Tome 99 (2016) no. 6, pp. 855-866

Résumé

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 $L(\log L)^{n-1}(\Omega)$.

Détail

Keywords: quasilinear divergence equation, generalized solution, Sobolev space, Orlicz–Zygmund space.