This paper is concerned with the following quasilinear Schrödinger system in the entire space $\mathbb R^{N}$($N\geq3$): $$ \begin{cases} -\Delta u+A(x)u-\frac{1}{2} \triangle(u^{2})u=\frac{2\alpha}{\alpha+\beta} |u|^{\alpha-2}u|v|^{\beta},\\ -\Delta v+Bv-\frac{1}{2}\triangle(v^{2}) v=\frac{2\beta}{\alpha+\beta}|u|^{\alpha} |v|^{\beta-2}v. \end{cases} $$ By establishing a suitable constraint set and studying related minimization problem, we prove the existence of ground state solution for $\alpha,\beta > 1$, $2 < \alpha+\beta < \frac{4N}{N-2}$. Our results can be looked on as a generalization to results by Guo and Tang (Ground state solutions for quasilinear Schrödinger systems, J. Math. Anal. Appl. 389 (2012) 322).

We consider the sharp interface limit of the Allen-Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field method under the vanishing on the boundary and some uniform boundedness property of the discrepancy measure. For this purpose, we extend the usual Brakke flow under these boundary conditions by the first variations for varifolds on the boundary.