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

TL;DR

This paper proves the continuity of solutions to a class of anisotropic elliptic equations with $L^1$ data, extending known results for the isotropic $N$-Laplacian to more general anisotropic cases.

Contribution

It establishes continuity of solutions for anisotropic $N$-Laplacian equations with $L^1$ right-hand side under specific conditions, generalizing classical results.

Findings

Solutions are continuous under the given conditions.

Conditions include a specific relation among the exponents $p_i$ and a limit condition on $f$.

Results recover known isotropic cases when all $p_i$ are equal to $N$.

Abstract

We establish the continuity of bounded solutions to the anisotropic elliptic equation $$-\sum\limits_{i=1}^N\Big(|u_{x_i}|^{p_i-2} u_{x_i}\Big)_{x_i}=f(x),\quad x\in \Omega,\quad f(x)\in L^1(\Omega)$$ 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 $$\lim\limits_{\rho\rightarrow 0}\,\sup\limits_{x\in \Omega}\int\limits^{\rho}_0\Big(\int\limits_{B_r(x)}|f(y)|\,dy\Big)^{\frac{1}{N-1}}\frac{dr}{r}=0.$$ In the standard case $p_1=...=p_N=N$, these conditions recover the known results for the $N$-Laplacian.