Nonlocal Kirchhoff superlinear equations with indefinite nonlinearity and lack of compactness

Abstract

We study the following Kirchhoff equation $$-\left(1+b{\int }_{{\mathbb{R}}^3}|\nabla u{|}^{2}dx\right)\hat{I}”u+V(x)u=f(x,u),x\in {\mathbb{R}}^3.$$ 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.