Hölder bounds and regularity of emerging free boundaries for strongly competing Schrödinger equations with nontrivial grouping

Abstract

We study regularity issues for systems of elliptic equations of the type \[ -Δu_i=f_{i,β}(x)-β\sum_{j\neq i} a_{ij} u_i |u_i|^{p-1}|u_j|^{p+1} \] set in domains $\Omega \subset {\mathbb{R}}^N$, for $N\ge 1$. The paper is devoted to the derivation of ${\mathcal{C}}^{0,\alpha }$ estimates that are uniform in the competition parameter $\beta >0$, as well as to the regularity of the limiting free-boundary problem obtained for $\beta \to +\infty$. The main novelty of the problem under consideration resides in the non-trivial grouping of the densities: in particular, we assume that the interaction parameters $a_{ij}$ are only non-negative, and thus may vanish for specific couples $(i,j)$. As a main consequence, in the limit $\beta \to +\infty$, densities do not segregate pairwise in general, but are grouped in classes which, in turn, form a mutually disjoint partition. Moreover, with respect to the literature, we consider more general forcing terms, sign-changing solutions, and an arbitrary $p>0$. In addition, we present a regularity theory of the emerging free-boundary, defined by the interface among different segregated groups. These equations are very common in the study of Bose-Einstein condensates and are of key importance for the analysis of optimal partition problems related to high order eigenvalues.

arXiv commentFinal version, Nonlinear Analysis: Theory, Methods & Applications, 138 (2016), 388--427. Special Volume in honor of Juan Luis Vázquez for his 70th birthday