Topological Methods in Nonlinear Analysis

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

Yu Chen and Yanheng Ding

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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 $V$.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 53, Number 1 (2019), 183-223.

Dates
First available in Project Euclid: 25 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1551063640

Digital Object Identifier
doi:10.12775/TMNA.2018.044

Citation

Chen, Yu; Ding, Yanheng. Multiplicity and concentration for Kirchhoff type equations around topologically critical points in potential. Topol. Methods Nonlinear Anal. 53 (2019), no. 1, 183--223. doi:10.12775/TMNA.2018.044. https://projecteuclid.org/euclid.tmna/1551063640


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