Existence and concentration of solutions for a Kirchhoff type problem with potentials

Jun  Wang; Lu  Xiao

doi:10.3934/dcds.2016111

Article Contents

2016, Volume 36, Issue 12: 7137-7168. Doi: 10.3934/dcds.2016111

This issue Previous Article Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping Next Article Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials

Existence and concentration of solutions for a Kirchhoff type problem with potentials

Jun Wang^1, and
Lu Xiao^2,

1.
Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013
2.
School of management, Jiangsu University, Zhenjiang, Jiangsu 212013

Received: January 31, 2016

Revised: May 31, 2016

Published: May 2017

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Abstract

In this paper, we concern with the following semilinear Kirchhoff type equation \begin{equation*} \begin{cases} -\left(\varepsilon^{2}a+b\varepsilon\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u+V(x)u=K(x)f(u)+Q(x)|u|^{p-2}u, &x\in\mathbb{R}^{3},\\ u\in H^{1}(\mathbb{R}^{3}), u>0 &x\in\mathbb{R}^{3}, \end{cases} \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b$ are positive constants, $V, K$ and $Q$ are positive bounded functions and $p\in(4,6]$, $f$ is a continuous superlinear and subcritical nonlinearity. On the one hand, for subcritical case, i.e., $p\in(4,6)$, we prove that there are three families of semiclassical positive solutions for $\varepsilon>0$ small, one is concentrating on the set of minima of $V$, the rest of two families of solutions are concentrating on the sets of maxima of $K$ and $Q$ respectively. On the other hand, we also prove the multiplicity and concentration of positive solutions for critical case($p=6$). The novelty is that we prove some new concentration phenomena for the positive solutions.

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Mathematics Subject Classification: Primary: 35J61, 35J20, 35Q55; Secondary: 49J40.

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