Bounded solutions for quasilinear modified Schrödinger equations

Abstract

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 $V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$ suitable measurable positive function, which generalizes the modified Schrödinger equation. Here, we suppose that $A:{\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is a ${\mathcal {C}}^{1}$-Caratheodory function such that $A_t(x,t) = \frac{\partial A}{\partial t} (x,t)$ and a given Carathéodory function $g:{\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ 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.