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

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.

$-\Delta_p u = k_1(|x| )f(u,v),$ for $|x| > 1$ and $x \in \mathbb R^N, $

$-\Delta_p v = k_2(|x|)g(u,v),$ for $|x| > 1 $ and $x \in \mathbb R^N, $

$u(x) = v(x) =0,$ for $|x| =1, $

$u(x), v(x) \rightarrow 0 $ as $|x| \rightarrow +\infty,$

where $1 < p < N $ and $\Delta_p u=$ div $(|\nabla u|^{p-2}\nabla u )$ is the p-Laplacian operator. We consider nonlinearities that are either superlinear or sublinear.

*quasilinear*Schrödinger equations in dimension $N\ge 3$: we investigate the case of

*unbounded*potentials which can not be covered in a standard function space setting. This leads us to use a nonstandard penalization technique, in a

*borderline*Orlicz space framework, which enable us to build up a one parameter family of classical solutions which have finite energy and exhibit, as the parameter goes to zero, a concentrating behavior around some point which is localized.

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