Entire solutions of quasilinear symmetric systems

Abstract

We study the following quasilinear elliptic system for all $i=1,\cdots,m$ \begin{equation*} \label{} -div(\Phi'(|\nabla u_i|^2) \nabla u_i) = H_i(u) \quad \text{in} \ \ \mathbb{R}^n \end{equation*} where $u=(u_i)_{i=1}^m: \mathbb R^n\to \mathbb R^m$ and the nonlinearity $ H_i(u) \in C^1(\mathbb R^m)\to \mathbb R$ is a general nonlinearity. Several celebrated operators such as the prescribed mean curvature, the Laplacian and the $p$-Laplacian operators fit in the above form, for appropriate $\Phi$. We establish a Hamiltonian identity of the following form for all $x_n\in\mathbb R$ \begin{equation*}\label{} \int_{\mathbb R^{n-1}} \left(\sum_{i=1}^{m} \left[ \frac{1}{2} \Phi\left(|\nabla u_i|^2\right) - \Phi'\left(|\nabla u_i|^2\right) |\partial_{x_n} u_i|^2 \right] - \tilde H(u) \right) d x'\equiv C, \end{equation*} where $x=(x',x_n)\in\mathbb R^{n}$ and $\tilde H$ is the antiderivative of $H=(H_i)_{i=1}^m$. This can be seen as a counterpart of celebrated pointwise inequalities provided by Caffarelli, Garofalo and Segala in \cite{cgs} and by Modica in \cite{m}. For the case of system of equations, that is when $m\ge 2$, we show that as long as $$\alpha \ge \alpha^*:=\inf_{s>0}\left\{\frac{2 s \Phi'(s)}{\Phi(s)}\right\}$$ the function $I_\alpha(r):=\frac{1}{r^{n-\alpha}} \int_{B_r} \sum_{i=1}^{m} \Phi(|\nabla u_i|^2) - 2\tilde H(u)$ is monotone nondecreasing in $r$. We call this a weak monotonicity formula since for $m=1$ it is shown in \cite{cgs} that $I_\alpha(r)$ is monotone when $\alpha\ge 1$, under certain conditions on $\Phi$. We prove De Giorgi type results and Liouville theorems for $H$-monotone and stable solutions in two and three dimensions when the system is symmetric.