In this work, we establish existence of weak solutions for quasilinear Schrödinger equations where the potential is bounded from below and above by positive constants. The nonlinearity has an iteration with higher eigenvalues for the associated linear problem. Hence, the energy functional associated with our main problem admits a linking structure. The main difficulty here comes from the fact that zero is not anymore a local minimum for the energy functional. Hence, we apply a linking theorem proving existence of weak solutions for quasilinear Schrödinger equations provided that the nonlinear term is an asymptotically-superlinear function. Due to the lack of compactness for the Sobolev embeddings we need to recover some kind of compactness required in variational methods. In order to do that we apply some fine estimates together with Lions' Lemma.

在这项工作中，我们建立了拟线性Schrödinger方程弱解的存在性，其中电势从下到上被正常数所限制。对于相关的线性问题，非线性具有较高特征值的迭代。因此，与我们的主要问题相关的能量功能允许一个链接结构。这里的主要困难来自于一个事实，即零不再是能量功能的局部最小值。因此，如果非线性项是一个渐近超线性函数，我们将应用一个定理证明拟线性Schrödinger方程的弱解的存在。由于Sobolev嵌入的紧凑性不足，我们需要恢复变型方法中所需的某种紧凑性。为了做到这一点，我们与Lions's