Nehari-type ground state solutions for asymptotically periodic bi-harmonic Kirchhoff-type problems in $\mathbb{R}^N$

TL;DR

This paper establishes the existence of Nehari-type ground state solutions for a class of bi-harmonic Kirchhoff-type equations with asymptotically periodic potentials and nonlinearities, extending previous results in the field.

Contribution

It introduces new existence results for ground state solutions in asymptotically periodic Kirchhoff problems with sign-changing nonlinearities.

Findings

Proves existence of ground state solutions under asymptotically periodic conditions.

Extends prior results to more general nonlinearities with sign-changing properties.

Provides a framework for analyzing Kirchhoff-type problems in unbounded domains.

Abstract

We investigate the following Kirchhoff-type biharmonic equation \begin{equation}\label{pr} \left\{ \begin{array}{ll} \Delta^2 u+ \left(a+b\int_{\mathbb{R}^N}|\nabla u|^2d x\right)(-\Delta u+V(x)u)=f(x,u),\quad x\in \mathbb{R}^N,\\ u\in H^{2}(\mathbb{R}^N), \end{array} \right. \end{equation} where $a>0$, $b\geq 0$ and $V(x)$ and $f(x, u)$ are periodic or asymptotically periodic in $x$. We study the existence of Nehari-type ground state solutions of the problem just above with $f(x,u)u-4F(x,u)$ sign-changing, where $F(x,u):=\int_0^uf(x,s)d s$. We significantly extend some results from the previous literature.