Solutions for singular quasilinear Schrödinger equations with one parameter

$i\frac{\partial \psi}{\partial t}=- \Delta \psi+\psi + \bar{\omega} (|\psi |^2)\psi- \lambda \rho(|\psi|^2)\psi-\kappa\Delta \rho(|\psi|^2)\rho'(|\psi|^2)\psi, x \in \Omega,$

where $\bar{\omega} (\tau^2) \tau \rightarrow +\infty$ as $\tau \rightarrow 0$ and, $\lambda>0$ is a parameter and $\Omega$ is a ball in $\mathcal{R}^N$. This problem is studied in connection with the following quasilinear eigenvalue problem

$-\Delta \Psi-\kappa\Delta \rho(|\Psi|^2)\rho'(|\Psi|^2)\Psi =\lambda_1 \rho(|\Psi|^2)\Psi, x \in \Omega,$

Indeed, we establish the existence of solutions for the above Schrödinger equation when $\lambda$ belongs to a certain neighborhood of the first eigenvalue $\lambda_1$ of the above eigenvalue problem. The main feature of this paper is that the nonlinearity $ \bar{\omega} ( |\psi |^2)\psi$ is unbounded around the origin and also the presence of the second order nonlinear term. Our analysis shows the importance of the role played by the parameter $\lambda$ combined with the nonlinear nonhomogeneous term $-\Delta \rho(|\psi|^2)\rho'(|\psi|^2)\psi$. The proofs are based on various techniques related to variational methods and implicit function theorem.