A multiplicity result for asymptotically linear Kirchhoff equations

Abstract

Abstract In this paper, we study the following Kirchhoff type equation: -\bigg{(}1+b\int_{\mathbb{R}^{N}}\lvert\nabla u|^{2}\,dx\biggr{)}\Delta u+u=a(% x)f(u)\quad\text{in }\mathbb{R}^{N},\qquad u\in H^{1}(\mathbb{R}^{N}), where {N\geq 3} , {b>0} and {f(s)} is asymptotically linear at infinity, that is, {f(s)\sim O(s)} as {s\rightarrow+\infty} . By using variational methods, we obtain the existence of a mountain pass type solution and a ground state solution under appropriate assumptions on {a(x)} .