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 $x\in {\mathbb{R}}^m$ with $1\le m<N$, $X=(x,y)\in {\mathbb{R}}^m\times {\mathbb{R}}^{N-m}$, and $F_1,...,F_n$ are the derivatives with respect to ${\xi }^1,...,{\xi }^n$ of some $F=F(x,{\xi }^1,...,{\xi }^n)$ such that for any $i=1,...,n$ and any fixed $(x,{\xi }^1,...,{\xi }^{i-1},{\xi }^{i+1},...,{\xi }^n)\in {\mathbb{R}}^m\times {\mathbb{R}}^{n-1}$ the map ${\xi }^i\to F(x,{\xi }^1,...,{\xi }^i,...,{\xi }^n)$ belongs to $C^2(\mathbb{R})$. 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.