In this paper, we obtain the existence of positive solution for a system of quasilinear Schrödinger equations with concave nonlinearities which is related to several applications in Hydrodynamics, Heidelberg Ferromagnetism and Magnus Theory, Condensed Matter Theory, Dissipative Quantum Mechanics and nanotubes and fullerene-related structures. The quasilinear Schrödinger problem is studied by considering a suitable change of variables which transforms the original problem in to a semilinear one. By means of the several properties of the change of variables, constructions of suitable sub-supersolutions, monotonic iteration arguments and the Galerkin method, we obtain the existence of solution for the semilinear problem. The paper is divided in two parts. In the first one, we use the method of sub-supersolutions to obtain a solution for the problem. In the second part, we use the Galerkin method and a comparison argument to obtain a solution for the system considered. An important feature is that the sub-supersolution approach is rare in the literature for the type of problem considered here and the Galerkin method was not used to consider quasilinear Schrödinger equations.

在本文中，我们获得了具有凹非线性的拟线性Schrödinger方程组的正解的存在，该方程组与流体力学，海德堡铁磁性和马格努斯理论，凝聚态理论，耗散量子力学以及与纳米管和富勒烯有关的结构中的几种应用有关。通过考虑变量的适当变化来研究拟线性Schrödinger问题，该变量将原始问题转换为半线性问题。通过变量变化的几种性质，合适的子超解的构造，单调迭代参数和Galerkin方法，我们获得了半线性问题解的存在性。本文分为两部分。在第一个中，我们使用子超解法来获得问题的解决方案。在第二部分中，我们使用Galerkin方法和比较参数来获取所考虑系统的解决方案。一个重要的特征是，对于此处考虑的问题类型，子超解法在文献中很少见，而Galerkin方法并未用于考虑拟线性Schrödinger方程。