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.


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.)