Liouville theorems for mixed local and nonlocal indefinite equations
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918471502135296 |
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| author | Wang, Pengyan Wu, Leyun |
| author_facet | Wang, Pengyan Wu, Leyun |
| contents | We investigate the qualitative properties of positive solutions to mixed local-nonlocal equations with indefinite nonlinearities, emphasizing the interaction between classical and fractional Laplacians. We first establish maximum principles and prove strict monotonicity along the $x_1$-direction for mixed elliptic operators. By combining a mollified first eigenfunction with a suitable sub-solution, we derive nonexistence results for the mixed operator $ (-Δ)^s - Δ$ via a contradiction argument. These results are further extended to the parabolic setting, incorporating both the Marchaud-type fractional time derivative and the classical first-order derivative, revealing new qualitative features under dual nonlocality.
A key aspect of our approach is a careful adaptation of the method of moving planes to the mixed local-nonlocal context. By addressing the distinct scaling behaviors of local and nonlocal terms, the method yields monotonicity and Liouville-type results without standard decay assumptions, and provides a framework potentially applicable to a broader class of mixed elliptic and parabolic problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17944 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Liouville theorems for mixed local and nonlocal indefinite equations Wang, Pengyan Wu, Leyun Analysis of PDEs We investigate the qualitative properties of positive solutions to mixed local-nonlocal equations with indefinite nonlinearities, emphasizing the interaction between classical and fractional Laplacians. We first establish maximum principles and prove strict monotonicity along the $x_1$-direction for mixed elliptic operators. By combining a mollified first eigenfunction with a suitable sub-solution, we derive nonexistence results for the mixed operator $ (-Δ)^s - Δ$ via a contradiction argument. These results are further extended to the parabolic setting, incorporating both the Marchaud-type fractional time derivative and the classical first-order derivative, revealing new qualitative features under dual nonlocality. A key aspect of our approach is a careful adaptation of the method of moving planes to the mixed local-nonlocal context. By addressing the distinct scaling behaviors of local and nonlocal terms, the method yields monotonicity and Liouville-type results without standard decay assumptions, and provides a framework potentially applicable to a broader class of mixed elliptic and parabolic problems. |
| title | Liouville theorems for mixed local and nonlocal indefinite equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.17944 |