Multiple solutions of Kirchhoff type equations involving Neumann conditions and critical growth

In this paper, we consider a Neumann problem of Kirchhoff type equation \begin{equation*} \begin{cases} \displaystyle-\left(a+b\int_{\Omega}|\nabla u|^2dx\right)\Delta u+u= Q(x)|u|^4u+\lambda P(x)|u|^{q-2}u, &\rm \mathrm{in}\ \ \Omega, \\ \displaystyle\frac{\partial u}{\partial v}=0, &\rm \m...

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Bibliographic Details
Published inAIMS mathematics Vol. 6; no. 4; pp. 3821 - 3837
Main Authors Lei, Jun, Suo, Hongmin
Format Journal Article
LanguageEnglish
Published AIMS Press 01.01.2021
Subjects
critical growth
kirchhoff type equation
neumann problem
nontrivial solution
variation methods
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ISSN2473-6988
2473-6988
DOI10.3934/math.2021227

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Summary:In this paper, we consider a Neumann problem of Kirchhoff type equation \begin{equation*} \begin{cases} \displaystyle-\left(a+b\int_{\Omega}|\nabla u|^2dx\right)\Delta u+u= Q(x)|u|^4u+\lambda P(x)|u|^{q-2}u, &\rm \mathrm{in}\ \ \Omega, \\ \displaystyle\frac{\partial u}{\partial v}=0, &\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} where $\Omega$ $\subset$ $\mathbb{R}^3$ is a bounded domain with a smooth boundary, $a,b>0$, $1<q<2$, $\lambda>0$ is a real parameter, $Q(x)$ and $P(x)$ satisfy some suitable assumptions. By using the variational method and the concentration compactness principle, we obtain the existence and multiplicity of nontrivial solutions.
ISSN:2473-6988
2473-6988
DOI:10.3934/math.2021227