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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.03381 |
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| _version_ | 1866912362388258816 |
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| author | Savchenko, Mariia Skrypnik, Igor Yevgenieva, Yevgeniia |
| author_facet | Savchenko, Mariia Skrypnik, Igor Yevgenieva, Yevgeniia |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03381 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the continuity of solutions to the anisotropic $N$-Laplacian with $L^1$ lower order term Savchenko, Mariia Skrypnik, Igor Yevgenieva, Yevgeniia Analysis of PDEs 35B65, 35D30, 35J92 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. |
| title | On the continuity of solutions to the anisotropic $N$-Laplacian with $L^1$ lower order term |
| topic | Analysis of PDEs 35B65, 35D30, 35J92 |
| url | https://arxiv.org/abs/2505.03381 |