*Accepted Paper*

**Inserted:** 29 jul 2015

**Journal:** Archive Ration. Mech. Anal.

**Year:** 2015

**Doi:** 10.1007/s00205-015-0867-9

**Links:**
link preliminary version

**Abstract:**

For a class of systems of semi-linear elliptic equations, including \[ -\Delta u_i=f_i(x,u_i) - \beta u_i\sum_{j\neq i}a_{ij}u_j^p,\qquad i=1,\dots,k, \] 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 \emph{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.