Nonlocal Kirchhoff Superlinear Equations with Indefinite Nonlinearity and Lack of Compactness

Abstract

AbstractWe study the following Kirchhoff equation: (K)$$ - \left({1 + b\int_{{{\mathbb R}^3}} |\nabla u{|^2}dx} \right)\Delta u + V(x)u = f(x, u), \quad x \in {{\mathbb R}^3}. $$A 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 the mountain pass theorem.