Bounded solutions for quasilinear modified Schrödinger equations

In this paper we establish a new existence result for the quasilinear elliptic problem \[ -{\rm div}(A(x,u)|\nabla u|^{p-2}\nabla u) +\frac1p A_t(x,u)|\nabla u|^p + V(x)|u|^{p-2} u = g(x,u)\quad\mbox{ in } \R^N, \] with $N\ge 2$, $p>1$ and $V:\R^N\to\R$ suitable measurable positive function, which generalizes the modified Schrödinger equation. Here, we suppose that $A:\R^N\times\R\rightarrow\R$ is a $\mathcal{C}^{1}$--Caratheodory function such that $A_t(x,t) = \frac{\partial A}{\partial t} (x,t)$ and a given Carath\'eodory function $g:\R^N\times\R\rightarrow\R$ has a subcritical growth and satisfies 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.