We study interior regularity issues for systems of elliptic equations of the type -Δui=fi,β(x)-β∑j≠iaijui|ui|p-1|uj|p+1 set in domains Ω ⊂ℝN, for N≥1. The paper is devoted to the derivation of C0,α estimates that are uniform in the competition parameter β>0, as well as to the regularity of the limiting free-boundary problem obtained for β→+∞. 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 aij are only non-negative, and thus may vanish for specific couples (i,j). As a main consequence, in the limit β→+∞, 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.
|Titolo:||Hölder bounds and regularity of emerging free boundaries for strongly competing Schrödinger equations with nontrivial grouping Dedicated to Prof. Juan Luis Vázquez with deep admiration and respect|
|Autori interni:||SOAVE, NICOLA|
|Data di pubblicazione:||2016|
|Appare nelle tipologie:||01.1 Articolo in Rivista|