Multiple solutions for a Kirchhoff-type equation with general nonlinearity

Abstract

Abstract This paper is devoted to the study of the following autonomous Kirchhoff-type equation: - M ⁢ ( ∫ ℝ N | ∇ ⁡ u | 2 ) ⁢ Δ ⁢ u = f ⁢ ( u ) , u ∈ H 1 ⁢ ( ℝ N ) , -M\biggl{(}\int_{\mathbb{R}^{N}}|\nabla{u}|^{2}\biggr{)}\Delta{u}=f(u),\quad u% \in H^{1}(\mathbb{R}^{N}), where M is a continuous non-degenerate function and N ≥ 2 {N\geq 2} . Under suitable additional conditions on M and general Berestycki–Lions-type assumptions on the nonlinearity of f, we establish several existence results of multiple solutions by variational methods, which are also naturally interpreted from a non-variational point of view.