Journal of the
Korean Mathematical Society JKMS

In this paper, we consider the following Schr\"{o}dinger-Kirch\-hoff-type equations \[ \left[a+b\left(\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+V(x)|u|^{2})dx\right)\right][-\Delta u+V(x)u]=f(x,u), ~~\mbox{in}~ \mathbb{R}^{N}. \] Under certain assumptions on $V$ and $f$, some new criteria on the existence and multiplicity of nontrivial solutions are established by the Morse theory with local linking and the genus properties in critical point theory. Some results from the previously literature are significantly extended and complemented.

Keywords: Schr\"{o}dinger-Kirchhoff-type, Morse theory, critical groups, variational methods, genus

MSC numbers: 35J20, 35J15, 35J60