In this paper, we consider a Neumann problem of Kirchhoff type equation
$ \begin{equation*} \begin{cases} -\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, \\ \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.
Citation: Jun Lei, Hongmin Suo. Multiple solutions of Kirchhoff type equations involving Neumann conditions and critical growth[J]. AIMS Mathematics, 2021, 6(4): 3821-3837. doi: 10.3934/math.2021227
In this paper, we consider a Neumann problem of Kirchhoff type equation
$ \begin{equation*} \begin{cases} -\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, \\ \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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Jun Lei, Hongmin Suo. Multiple solutions of Kirchhoff type equations involving Neumann conditions and critical growth[J]. AIMS Mathematics, 2021, 6(4): 3821-3837. doi: 10.3934/math.2021227
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