Multiplicity and concentration for Kirchhoff type equations around topologically critical points in potential

Abstract

We consider the multiplicity and concentration of solutions for the Kirchhoff Type Equation \[ -\varepsilon^2 M\bigg( \varepsilon^{2-N}\int_{\mathbb{R}^N} |\nabla v|^2dx \bigg) \Delta v+V(x)v=f(v), \quad \mathrm{in }\ \mathbb{R}^N. \] Under suitable conditions on functions $M$, $V$ and $f$, we obtain the existence of positive solutions concentrating around the local maximum points of $V$, which gives an affirmative answer to the problem raised in [21]. Moreover, we also obtain multiplicity of solutions which are affected by the topology of critical points set of potential $V$.