## Brezis-Merle type inequality for a weak solution to the $$N$$-Laplace equation in Lorentz-Zygmund spaces.(English)Zbl 1240.35209

Summary: 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.

### MSC:

 35J92 Quasilinear elliptic equations with $$p$$-Laplacian 35J25 Boundary value problems for second-order elliptic equations 35J70 Degenerate elliptic equations 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)