Symmetry results for stable and monotone solutions to fibered systems of PDEs

Abstract

We study the symmetry properties for solutions of elliptic systems of the type {ll}-\dive(a_1(x,|\nabla u^1|(X))\nabla u^1(X))=F_{1}(x, u^1(X),..., u^n(X)), ... -\dive(a_n(x,|\nabla u^n|(X))\nabla u^n(X))=F_{n}(x, u^1(X),..., u^n(X)), where $x\in \R^m$ with $1\leq m< N$, $X=(x,y)\in \R^m\times \R^{N-m}$, and $F_{1},..., F_{n}$ are the derivatives with respect to $ξ^1,..., ξ^n$ of some $F=F(x,ξ^1,..., ξ^n)$ such that for any $i=1,..., n$ and any fixed $(x,ξ^1,..., ξ^{i-1},ξ^{i+1},..., ξ^n)\in \R^m\times \R^{n-1}$ the map $ξ^i\to F(x,ξ^1,...,ξ^i,..., ξ^n)$ belongs to $C^2(\R)$. We obtain a Poincaré-type formula for the solutions of the system and we use it to prove a symmetry result both for stable and for monotone solutions.