Nonlocal Kirchhoff Superlinear Equations with Indefinite Nonlinearity and Lack of Compactness

Abstract

Abstract We study the following Kirchhoff equation: (K) −1+b∫ℝ3|∇u|2dxΔu+V(x)u=f(x,u),x∈ℝ3.$$ - \left({1 + b\int_{{{\mathbb R}^3}} |\nabla u{|^2}dx} \right)\Delta u + V(x)u = f(x, u), \quad x \in {{\mathbb R}^3}. $$ A feature of this paper is that the nonlinearity f$f$ and the potential V$V$ are indefinite, hence sign-changing. Under some appropriate assumptions on V$V$ and f$f$ , we prove the existence of two different solutions of the equation via the Ekeland variational principle and the mountain pass theorem.