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We study the following Kirchhoff equation $$- \left(1 + b \int_{\mathbb{R}^3}|\nabla u|^2 dx \right) \Delta u + V(x) u = f(x,u), \ x \in \mathbb{R}^3.$$ Aspecial 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 equationvia the Ekeland variational principle and Mountain Pass Theorem.

Nonlocal Kirchhoff superlinear equations with indefinite nonlinearity and lack of compactness