Optimal uniform bounds for competing variational elliptic systems with variable coefficients

Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$ be an open set. In this work we consider solutions of the following gradient elliptic system $$-\text{div}(A(x)\mathrm{\nabla }{u}_{i,\beta })=f_i(x,u_{i,\beta })+a(x)\beta \mathrm{\mid }{u}_{i,\beta }{\mathrm{\mid }}^{\gamma -1}{u}_{i,\beta }{}_{j\mathrm{\ne }i}\mathrm{\mid }{u}_{j,\beta }{\mathrm{\mid }}^{\gamma +1},$$ for $i=1,\dots ,l$. 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 $L^{\mathrm{\infty }}$-bounds on families of positive solutions $\{u_{\beta }{\}}_{\beta <0}=\{(u_{1,\beta },\dots ,u_{l,\beta }){\}}_{\beta <0}$ 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.