In this paper, our aim is to study thefollowing critical Choquard equation onthe Heisenberg group:\begin{equation*} \begin{cases} \displaystyle{-\Delta_H u }={\mu}|u|^{q-2}u+\int_{\Omega}\frac{|u(\eta)|^{Q_{\lambda}^{\ast}}}{|\eta^{-1}\xi|^{\lambda}}d\eta|u|^{Q_{\lambda}^{\ast}-2}u&\mbox{in }\ \Omega, \\ u=0 &\mbox{on }\ \partial\Omega,\end{cases}\end{equation*}where$\Omega\subset \mathbb{H}^N$is a smooth bounded domain,$\Delta_H$ is the Kohn-Laplacianon the Heisenberg group$\mathbb{H}^N$,$1 < q < 2$ or$2 < q < Q_\lambda^\ast$,$\mu > 0$,$0 < \lambda < Q=2N+2$, and$Q_{\lambda}^{\ast}=\frac{2Q-\lambda}{Q-2}$is the critical exponent.Using the concentration compactnessprinciple and the critical point theory,we prove that the aboveproblem has the least two positivesolutions for$1 < q < 2$ in the caseof low perturbations (small values of$\mu$), andhas a nontrivial solution for$2 < q < Q_\lambda^\ast$ in the case ofhigh perturbations (large values of$\mu$). Moreover, for$1 < q < 2$,we also show that there is a positiveground state solution, and for$2 < q < Q_\lambda^\ast$, there are at least$n$pairs of nontrivial weak solutions.

In this paper, we consider the Maxwell-Dirac system in 3 dimension under zero magnetic field. We prove the global well-posedness and modified scattering for small solutions in the weighted Sobolev class. Imposing the Lorenz gauge condition, (and taking the Dirac projection operator), it becomes a system of Dirac equations with Hartree type nonlinearity with a long-range potential as $|x|^{-1} $. We perform the weighted energy estimates. In this procedure, we have to deal with various resonance functions that stem from the Dirac projections. We use the spacetime resonance argument of Germain-Masmoudi-Shatah ([14, 15, 16]),as well as the spinorial null-structure. On the way, we recognize a long-range interaction which is responsible for a logarithmic phase correction in the modified scattering statement. This result was obtained by Cloos in his dissertation [9],via a different technique (see Remark 1.2).

This paper deals with the local null controllability of a coupled nonlinear parabolic system which is the Kuramoto-Sivashinsky-Korteweg-de Vries equation coupled with the heat equation through first order derivatives.More precisely, we prove the local null controllability of the system with a single localized interior control acting on either of the equations of the coupled system.Additionally, we also investigate the same issue with a single periodic boundary control acting through zeroth order derivatives of either of the components.We employ the well-known moment method to study the controllability of the corresponding linearized system around the origin with a suitable control cost$Ce^{\frac{C}{T}}$, for some$C > 0$ as$T\to 0^+ $.Further, the local null controllability of the original nonlinear system is shown by combining the source term method introduced in[50] and a Banach fixed point argument.