Bounded solutions for quasilinear modified Schrödinger equations

In this paper we establish a new existence result for the quasilinear elliptic problem $$\begin{aligned} -\mathrm{div}(A(x,u)|\nabla u|^{p-2}\nabla u) +\frac{1}{p} A_t(x,u)|\nabla u|^p + V(x)|u|^{p-2} u = g(x,u)\quad \text{ in } {\mathbb {R}}^N, \end{aligned}$$ with $$N\ge 2$$ , $$p>1$$ and $$V:{\mathbb {R}}^N\rightarrow {R}}$$ suitable measurable positive function, which generalizes modified Schrödinger equation. Here, suppose that $$A:{\mathbb {R}}^N\times {R}}\rightarrow is $${\mathcal {C}}^{1}$$ -Caratheodory function such $$A_t(x,t) \frac{\partial A}{\partial t} (x,t)$$ given Carathéodory $$g:{\mathbb has subcritical growth satisfies Ambrosetti–Rabinowitz condition. Since coefficient of principal part depends also on solution itself, study interaction two different norms Banach space so to obtain “good” variational approach. Thus, by means approximation arguments bounded sets can state nontrivial weak solution.