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

Mariia Savchenko; Igor Skrypnik; Yevgeniia Yevgenieva

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arXiv · 2505.03381

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

Mariia Savchenko·Igor Skrypnik·Yevgeniia Yevgenieva

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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.

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Mariia Savchenko, Igor Skrypnik, Yevgeniia Yevgenieva. 2025-05-06. On the continuity of solutions to the anisotropic $N$-Laplacian with $L^1$ lower order term. https://arxiv.org/abs/2505.03381

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