Topological Methods in Nonlinear Analysis

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 $V$.

Article information

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

Dates
First available in Project Euclid: 25 February 2019

https://projecteuclid.org/euclid.tmna/1551063640

Digital Object Identifier
doi:10.12775/TMNA.2018.044

Mathematical Reviews number (MathSciNet)
MR3939153

Zentralblatt MATH identifier
07068334

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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