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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/2511.04447 |
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| _version_ | 1866915602639093760 |
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| author | Malanchini, Paolo Bisci, Giovanni Molica Secchi, Simone |
| author_facet | Malanchini, Paolo Bisci, Giovanni Molica Secchi, Simone |
| contents | We study the regularizing effect arising from the interaction between the coefficient \(a\) of the zero order term and the datum \(f\) in the problem $$ \left\lbrace \begin{array}{ll}
-\mathcal{L}u + a(x) g(u) = f(x) \quad &\mbox{in} \;\; Ω,
u = 0 \quad &\mbox{on} \;\; \partialΩ, \end{array} \right. $$ where $Ω\subseteq\mathbb{R}^N$ is a bounded domain and $\mathcal{L}$ is an $X$-elliptic operator introduced by Lanconelli and Kogoj. If $f \in L^1(Ω)$, we prove that the \(Q\)-condition introduced by Arcoya and Boccardo is sufficient to ensure the existence and boundedness of solutions in the framework of $X$-elliptic operators as well. Finally, we prove the existence of a bounded solution for linear problems under a more general condition between $f$ and $a$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04447 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularizing effect of the interplay between coefficients in linear and semilinear $X$-elliptic equations Malanchini, Paolo Bisci, Giovanni Molica Secchi, Simone Analysis of PDEs We study the regularizing effect arising from the interaction between the coefficient \(a\) of the zero order term and the datum \(f\) in the problem $$ \left\lbrace \begin{array}{ll} -\mathcal{L}u + a(x) g(u) = f(x) \quad &\mbox{in} \;\; Ω, u = 0 \quad &\mbox{on} \;\; \partialΩ, \end{array} \right. $$ where $Ω\subseteq\mathbb{R}^N$ is a bounded domain and $\mathcal{L}$ is an $X$-elliptic operator introduced by Lanconelli and Kogoj. If $f \in L^1(Ω)$, we prove that the \(Q\)-condition introduced by Arcoya and Boccardo is sufficient to ensure the existence and boundedness of solutions in the framework of $X$-elliptic operators as well. Finally, we prove the existence of a bounded solution for linear problems under a more general condition between $f$ and $a$. |
| title | Regularizing effect of the interplay between coefficients in linear and semilinear $X$-elliptic equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.04447 |