Brezis-Merle type inequality for a weak solution to the $N$-Laplace equation in Lorentz-Zygmund spaces

Abstract

We consider a regularity estimate for a solution of the homogeneous Dirichlet problem for $N$-Laplace equations in a bounded domain $\Omega\subset{\mathbb R}^N$ with external force $f\in L^1(\Omega)$. Introducing the generalized Lorentz-Zygmund space, we show the multiple exponential integrability of the Brezis-Merle type for an entropy solution of the Dirichlet problem of the $N$-Laplace equation. We also discuss conditions on $f$ that guarantee the solutions are bounded.