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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| Published in | AIMS mathematics Vol. 6; no. 4; pp. 3821 - 3837 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
AIMS Press
01.01.2021
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| Subjects | |
| Online Access | Get full text |
| ISSN | 2473-6988 2473-6988 |
| DOI | 10.3934/math.2021227 |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Lei, Jun Suo, Hongmin |
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| Title | Multiple solutions of Kirchhoff type equations involving Neumann conditions and critical growth |
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