In the last years much progress has been achieved in KAM theory concerning bifurcation of quasi-periodic solutions of Hamiltonian or reversible partial differential equations. We provide an overview of the state of the art in this field.

In this paper, a weak type $(1,1)$ estimate is established for the higher order commutator introduced by Christ and Journé which is defined by

$$ T[a_1,\cdots,a_l]f(x)=p.v. \int_{R^d} K(x-y)\Big(\prod_{i=1}^lm_{x,y}a_i\Big)\cdot f(y)dy, $$

where $K$ is the standard Calderόn-Zygmund convolution kernel on $\mathbb{R}^d (d\geq2)$ and $m_{x,y}a_i=\int_0^1a_i(sx+(1-s)y)ds$.

For the Schrödinger system

$$\left\{\begin{array}{ll}-\Delta u_j +\lambda_j u_j =\sum\limits_{i=1}^k \beta_{ij} u_i^2 u_j\quad \mbox{in}\ \ \ \ \mathbb R^N,\\ u_j(x)\to0\quad\text{ as }\ \ |x|\to\infty,  j=1,\cdots,k,\end{array}\right.$$

where $k\geq 2$ and $N=2, 3$, we prove that for any $\lambda_j>0$ and $\beta_{jj}>0$ and any positive integers $p_j$, $j=1,2,\cdots,k$, there exists $b>0$ such that if $\beta_{ij}=\beta_{ji}\leq b$ for all $i\neq j$ then there exists a radial solution $(u_1,u_2,\cdots,u_k)$ with $u_j$ having exactly $p_j-1$ zeros. Moreover, there exists a positive constant $C_0$ such that if $\beta_{ij}=\beta_{ji}\leq b\ (i\neq j)$ then any solution obtained satisfies

$$\sum_{i,j=1}^k|\beta_{ij}|\int_{\mathbb R^N}u_i^2u_j^2\leq C_0.$$

Therefore, the solutions exhibit a trend of phase separations as $\beta_{ij}\to-\infty$ for $i\neq j.$

In this paper, we prove a form of the strong comparison principle for a class of fully nonlinear subelliptic operators of the form $\nabla^2_{H,s}\psi + L(\cdot,\psi,\nabla_{H}\psi)$ on the Heisenberg group, which include the CR invariant operators.