On the continuity of solutions to the anisotropic $N$-Laplacian with $L^1$ lower order term

TL;DR

The paper proves the continuity of bounded weak solutions to the anisotropic N-Laplacian with an $L^1$ lower-order term under the sharp condition $\sum_{i=1}^N 1/p_i=1$ and $\min_i p_i>1$. It adapts the Kilpeläinen–Malý iteration to anisotropic operators, employing a rescaled geometry and local energy estimates to obtain a decay of oscillation in shrinking anisotropic cubes, mediated by a Wolff-potential smallness condition on the data $f$. A sequence of levels $\{k_j\}$ and increments $\{\delta_j\}$ is built to drive the solution toward pointwise bounds, with the Kilpeläinen–Malý lemma providing the essential decay in each step. The resulting oscillation control, together with a vanishing $\theta(\rho)$ term, yields Hölder-type continuity in the domain; in the isotropic case this recovers classical $N$-Laplacian regularity results for $f\in L^1$. The work extends continuity theory to fully anisotropic elliptic equations with rough data and lower-order terms, offering a robust framework for further anisotropic regularity results.

Abstract

We establish the continuity of bounded solutions to the anisotropic elliptic equation $$-\sum\limits_{i=1}^N\Big(|u_{x_i}|^{p_i-2} u_{x_i}\Big)_{x_i}=f(x),\quad x\in Ω,\quad f(x)\in L^1(Ω)$$ under the conditions $$\min\limits_{1\leqslant i\leqslant N} p_i >1,\quad \sum\limits_{i=1}^N \frac{1}{p_i}=1$$ and $$\lim\limits_{ρ\rightarrow 0}\,\sup\limits_{x\in Ω}\int\limits^ρ_0\Big(\int\limits_{B_r(x)}|f(y)|\,dy\Big)^{\frac{1}{N-1}}\frac{dr}{r}=0.$$ In the standard case $p_1=...=p_N=N$, these conditions recover the known results for the $N$-Laplacian.

Theorems & Definitions (12)

• Definition 1.1
• Theorem 1.1
• Remark 1.1
• Lemma 2.1
• Lemma 2.2
• Proposition 3.1
• Lemma 3.1
• proof
• Claim
• proof