On elliptic systems with $k$-wise interactions in the strong competition regime: uniform Hölder bounds and properties of the limiting configurations

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 $3\le k\le d$, 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-$\beta$ 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 $\beta \to +\mathrm{\infty }$, 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.