Nonlocal Kirchhoff superlinear equations with indefinite nonlinearity and lack of compactness

TL;DR

This paper investigates the existence of solutions for a nonlocal Kirchhoff equation with indefinite, sign-changing potential and nonlinearity, employing variational methods to overcome the lack of compactness in an unbounded domain.

Contribution

It establishes the existence of two solutions for the Kirchhoff equation with indefinite potential and nonlinearity using variational techniques.

Findings

Proved existence of two solutions under certain conditions.

Applied Ekeland variational principle and Mountain Pass Theorem.

Addressed challenges due to indefinite and sign-changing terms.

Abstract

We study the following Kirchhoff equation $$- \left(1 + b \int_{\mathbb{R}^3} |\nabla u|^2 dx \right) \Delta u + V(x) u = f(x,u), \ x \in \mathbb{R}^3.$$ A special feature of this paper is that the nonlinearity $f$ and the potential $V$ are indefinite, hence sign-changing. Under some appropriate assumptions on $V$ and $f$, we prove the existence of two different solutions of the equation via the Ekeland variational principle and Mountain Pass Theorem.