Bounded solutions for quasilinear modified Schrödinger equations

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 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 $A_t(x,t)=\frac{\partial A}{\partial t}(x,t)$ 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.

Preprint

References in corpus (1)

• Existence of radial bounded solutions for some quasilinear elliptic equations in R^N

Cited by in corpus (1)

• Multiple solutions for coupled gradient-type quasilinear elliptic systems with supercritical growth