## Differential and Integral Equations

- Differential Integral Equations
- Volume 22, Number 5/6 (2009), 495-518.

### 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.

#### Article information

**Source**

Differential Integral Equations, Volume 22, Number 5/6 (2009), 495-518.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019603

**Mathematical Reviews number (MathSciNet)**

MR2501681

**Zentralblatt MATH identifier**

1240.35209

**Subjects**

Primary: 35J92: Quasilinear elliptic equations with p-Laplacian

Secondary: 35B65: Smoothness and regularity of solutions 35D30: Weak solutions 35J25: Boundary value problems for second-order elliptic equations 35J70: Degenerate elliptic equations 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

#### Citation

Ioku, Norisuke. Brezis-Merle type inequality for a weak solution to the $N$-Laplace equation in Lorentz-Zygmund spaces. Differential Integral Equations 22 (2009), no. 5/6, 495--518. https://projecteuclid.org/euclid.die/1356019603