Bounded solutions for quasilinear modified Schrödinger equations

In this paper we establish a new existence result for a quasilinear elliptic problem stated in $R^N$ which generalizes the modified Schrödinger equation. Here, we suppose that the principal part $A:R^N imesR ightarrowR$ is a $mathcal{C}^{1}$--Caratheodory function and the given nonlinear term satisfies a subcritical growth and the Ambrosetti--Rabinowitz condition. Since the coefficient of the principal part depends also on the solution itself, we study the interaction of two different norms in a suitable Banach space so to obtain a ``good'' variational approach. Thus, by means of approximation arguments on bounded sets we can state the existence of a nontrivial weak bounded solution.