Lorenzo Giaretto | Searcharxiv

arXiv2026

On elliptic systems with $k$-wise interactions in the strong competition regime: uniform H\"older 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 $3 \leq k \leq d$, 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-$\beta$ H\"older bounds up to an explicit threshold exponent depending only on the dimension of the space and on the order $k$ of the interaction. As $\beta \to +\infty$, we show that minimizers converge strongly in $H^1$ and in H\"older 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 H\"older regularity and we derive some basic extremality conditions.

math.AP↗

arXiv2025

On least energy solutions for a nonlinear Schrödinger system with $K$-wise interaction

In this paper we establish existence and properties of minimal energy solutions for the weakly coupled system $$ \begin{cases} -Δu_i + λ_i u_i = μ_i|u_i|^{Kq-2}u_i + β|u_i|^{q-2}u_i\prod_{j\neq i}|u_j|^q & \text{in }\mathbb{R}^d, \qquad u_i \in H^1(\mathbb{R}^d), \end{cases}\qquad i=1,\dots, K, $$ characterized by $K$-wise interaction (namely the interaction term involves the product of all the components). We consider both attractive ($β>0$) and repulsive cases ($β<0$), and we give sufficient conditions on $β$ in order to have least energy fully non-trivial solutions, if necessary under a radial constraint. We also study the asymptotic behavior of least energy fully non-trivial radial solutions in the limit of strong competition $β\to -\infty$, showing partial segregation phenomena which differ substantially from those arising in pairwise interaction models.

math.AP↗