Existence of weak solutions for quasilinear Schrödinger equations with a parameter

In this paper, we study the following quasilinear Schrödinger equation of the form \begin{equation*} -\Delta_{p}u+V(x)|u|^{p-2}u-\left[\Delta_{p}(1+u^{2})^{\alpha/2}\right]\frac{\alpha u}{2(1+u^{2})^{(2-\alpha)/2}}=k(u),\qquad x\in \mathbb{R}^{N}, \end{equation*} where $p$-Laplace operator $\Delta_{p}u={\rm div}(|\nabla u|^{p-2}\nabla u)\ (1