Archive for Rational Mechanics and Analysis

, Volume 218, Issue 2, pp 647–697 | Cite as

Uniform Bounds for Strongly Competing Systems: The Optimal Lipschitz Case

  • Nicola SoaveEmail author
  • Alessandro Zilio


For a class of systems of semi-linear elliptic equations, including
$$-\Delta u_i=f_i(x,u_i) - \beta u_i\sum_{j\neq i}a_{ij}u_j^p,\quad i=1,\dots,k,$$
for p = 2 (variational-type interaction) or p = 1 (symmetric-type interaction), we prove that uniform \({L^\infty}\) boundedness of the solutions implies uniform boundedness of their Lipschitz norm as \({\beta \to +\infty}\), that is, in the limit of strong competition. This extends known quasi-optimal regularity results and covers the optimal case for this class of problems. The proofs rest on monotonicity formulae of Alt–Caffarelli–Friedman and Almgren type in the variational setting, and on the Caffarelli–Jerison–Kenig almost monotonicity formula in the symmetric one.


Uniform Boundedness Reaction Term Uniform Bound Monotonicity Formula Lipschitz Regularity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Mathematisches InstitutJustus Liebig Universität GiessenGiessenGermany
  2. 2.Centre d’analyse et de mathématique socialesÉcole des Hautes Études en Sciences SocialesParis Cedex 13France

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