Entire solutions of quasilinear symmetric systems

Abstract

We study the following quasilinear elliptic system for all $i=1,\cdots ,m$ \begin{equation*} \label{} -div(Φ'(|\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} Φ\left(|\nabla u_i|^2\right) - Φ'\left(|\nabla u_i|^2\right) |\partial_{x_n} u_i|^2 \right] - \tilde H(u) \right) d x'\equiv C, \end{equation*} where $x=(x^{\prime },x_n)\in {\mathbb{R}}^n$ 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 $$\alpha \ge {\alpha }^{\ast }:=\underset{s>0}{\inf }\left\{\frac{2s{\Phi }^{\prime }(s)}{\Phi (s)}\right\}$$ 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.

arXiv commentTo appear in Indiana University Math Journal. 27 pages