Existence and concentration for Kirchhoff type equations around topologically critical points of the potential

Yu Chen; Yanheng Ding; Suhong Li

doi:10.3934/cpaa.2017079

Article Contents

2017, Volume 16, Issue 5: 1641-1671. Doi: 10.3934/cpaa.2017079

This issue Previous Article Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces Next Article Semilinear damped wave equation in locally uniform spaces

Existence and concentration for Kirchhoff type equations around topologically critical points of the potential

1.
Institute of Mathematics, Academy of Mathematics and Systems Science, University of Chinese Academy of Science, Chinese Academy of Science, Beijing 100190, China
2.
Department of Mathematics, Honghe University Mengzi, Yunnan 661100, China
3.
Department of Mathematics and Information Technology, Hebei Normal University of Science and Technology, Qinhuangdao, Hebei 066004, China

Received: August 31, 2016

Revised: February 28, 2017

Published: October 2017

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Abstract

We consider the existence and concentration of solutions for the following Kirchhoff Type Equations

$-\varepsilon^2 M \left( \varepsilon^{2-N} \displaystyle \int_{\mathbb{R}^N} |\nabla v|^2dx \right)Δ v+V(x)v=f(v), \mathrm{in} \ \mathbb{R}^N.$

Under suitable conditions on the continuous functions $M$, $V$ and $f$, we obtain a family of positive solutions concentrating around the local maximum or saddle points of $V$. Moreover with appropriate assumptions on $V$, we also have multiple solutions clustering respectively around three kinds of critical points of $V$.

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Mathematics Subject Classification: 35B05, 35B45.

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