On elliptic systems with $k$-wise interactions in the strong competition regime: uniform Hölder bounds and properties of the limiting configurations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918383476277248 |
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| author | Giaretto, Lorenzo |
| author_facet | Giaretto, Lorenzo |
| contents | In this paper we investigate a class of variational reaction-diffusion systems with strong competition driven by beyond-pairwise interactions. The model involves $d$ nonnegative components interacting through $k$-wise terms, with $3 \leq k \leq d$, and includes symmetric interaction coefficients accounting for multi-component effects as well as suitable nonlinear terms. We focus on minimal energy solutions, proving uniform-in-$β$ Hölder bounds up to an explicit threshold exponent depending only on the dimension of the space and on the order $k$ of the interaction. As $β\to +\infty$, we show that minimizers converge strongly in $H^1$ and in Hölder spaces to a partially segregated configuration, characterized as minimizer of a natural variational problem under a $k$-segregation constraint. Finally, we prove that every minimizer of the limit problem enjoys the Hölder regularity and we derive some basic extremality conditions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_10949 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On elliptic systems with $k$-wise interactions in the strong competition regime: uniform Hölder bounds and properties of the limiting configurations Giaretto, Lorenzo Analysis of PDEs 35R35, 35B25 (Primary) 35J61, 35J47 (Secondary) In this paper we investigate a class of variational reaction-diffusion systems with strong competition driven by beyond-pairwise interactions. The model involves $d$ nonnegative components interacting through $k$-wise terms, with $3 \leq k \leq d$, and includes symmetric interaction coefficients accounting for multi-component effects as well as suitable nonlinear terms. We focus on minimal energy solutions, proving uniform-in-$β$ Hölder bounds up to an explicit threshold exponent depending only on the dimension of the space and on the order $k$ of the interaction. As $β\to +\infty$, we show that minimizers converge strongly in $H^1$ and in Hölder spaces to a partially segregated configuration, characterized as minimizer of a natural variational problem under a $k$-segregation constraint. Finally, we prove that every minimizer of the limit problem enjoys the Hölder regularity and we derive some basic extremality conditions. |
| title | On elliptic systems with $k$-wise interactions in the strong competition regime: uniform Hölder bounds and properties of the limiting configurations |
| topic | Analysis of PDEs 35R35, 35B25 (Primary) 35J61, 35J47 (Secondary) |
| url | https://arxiv.org/abs/2603.10949 |