Optimal uniform bounds for competing variational elliptic systems with variable coefficients

Abstract

Let $\Omega \subset \mathbb{R}^N$ be an open set. In this work we consider solutions of the following gradient elliptic system \[ -\text{div}(A(x)\nabla u_{i,\beta}) = f_i(x,u_{i,\beta}) + a(x)\beta |u_{i, \beta}|^{\gamma -1}u_{i, \beta} \mathop{\sum_{j=1}^l}_{j\neq i} |u_{j, \beta}|^{\gamma + 1}, \] for $i=1,\ldots, l$. We work in the competitive case, namely $\beta<0$. Under suitable assumptions on $A$, $a$, $f_i$ and on the exponent $\gamma$, we prove that uniform $L^\infty$-bounds on families of positive solutions $\{u_\beta\}_{\beta<0}=\{(u_{1,\beta},\ldots, u_{l,\beta})\}_{\beta<0}$ imply uniform Lipschitz bounds (which are optimal). One of the main points in the proof are suitable generalizations of Almgren's and Alt-Caffarelli-Friedman's monotonicity formulas for solutions of such systems. Our work generalizes previous results, where the case $A(x)=Id$ (i.e. the operator is the Laplacian) was treated.