Bounded solutions for quasilinear modified Schr\"odinger equations

TL;DR

This paper proves the existence of bounded solutions for a class of quasilinear modified Schrödinger equations with solution-dependent coefficients, using a novel variational approach and approximation techniques.

Contribution

It introduces a new existence result for quasilinear elliptic problems with solution-dependent coefficients, extending the modified Schrödinger equation framework.

Findings

Existence of nontrivial bounded solutions established.

Use of a new variational approach involving interaction of two norms.

Approximation methods on bounded sets facilitate the proof.

Abstract

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