Bounded solutions for quasilinear modified Schrödinger equations

In this paper we establish a new existence result for the quasilinear elliptic problem $$-\mathrm{d}\mathrm{i}\mathrm{v}(A(x,u)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)+\frac{1}{p}{A}_t(x,u)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p+V(x)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u=g(x,u)in{\mathbb{R}}^N,$$ with $N\ge 2$, $p>1$ and $V:{\mathbb{R}}^N\to \mathbb{R}$ suitable measurable positive function, which generalizes the modified Schrödinger equation. Here, we suppose that $A:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ is a ${\mathcal{C}}^1$-Carathéodory function such that $A_t(x,t)=\frac{\mathrm{\partial }A}{\mathrm{\partial }t}(x,t)$ and a given Carathéodory function $g:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ 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.