Normalized multi-bump solutions of nonlinear Kirchhoff equations

Zhidan Shu; Jianjun Zhang; Zhidan Shu; Jianjun Zhang

doi:10.3934/math.2024814

AIMS Mathematics

2024, Volume 9, Issue 6: 16790-16809. doi: 10.3934/math.2024814

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Normalized multi-bump solutions of nonlinear Kirchhoff equations

Zhidan Shu ,
Jianjun Zhang ^,

College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China

Received: 11 March 2024 Revised: 09 April 2024 Accepted: 16 April 2024 Published: 14 May 2024
MSC : 35A15, 35B25, 35B40

We are concerned with the existence and concentration of multi-bump solutions for the nonlinear Kirchhoff equation

$ \begin{eqnarray*} -\left ( \varepsilon ^{2}a+\varepsilon b\displaystyle {\int}_{\mathbb{R}^{3} }\left | \nabla v \right | ^{2} \mathrm {d} x \right )\Delta v+\lambda v = K(x)\left | v \right |^{2\sigma }v,\,\,\,x\in\mathbb{R}^3 \end{eqnarray*} $

with an $ L^{2} $-constraint in the $ L^{2} $-subcritical case $ \sigma\in\left(0, \, \frac{2}{3}\right) $ and the $ L^{2} $-supercritical case $ \sigma\in\left(\frac{2}{3}, \, 2 \right). $ Here $ \lambda \in \mathbb{R} $ appears as a Lagrange multiplier, $ \varepsilon $ is a small positive parameter and $ K > 0 $ possesses several local maximum points. By employing the variational gluing method and the penalization technique, we prove the existence of multi-bump solutions that are concentrated at local maximum points of $ K $ for the problem above.
- Kirchhoff equation,
- normalized solutions,
- concentration behavior
Citation: Zhidan Shu, Jianjun Zhang. Normalized multi-bump solutions of nonlinear Kirchhoff equations[J]. AIMS Mathematics, 2024, 9(6): 16790-16809. doi: 10.3934/math.2024814

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Abstract

We are concerned with the existence and concentration of multi-bump solutions for the nonlinear Kirchhoff equation

$ \begin{eqnarray*} -\left ( \varepsilon ^{2}a+\varepsilon b\displaystyle {\int}_{\mathbb{R}^{3} }\left | \nabla v \right | ^{2} \mathrm {d} x \right )\Delta v+\lambda v = K(x)\left | v \right |^{2\sigma }v,\,\,\,x\in\mathbb{R}^3 \end{eqnarray*} $

with an $ L^{2} $-constraint in the $ L^{2} $-subcritical case $ \sigma\in\left(0, \, \frac{2}{3}\right) $ and the $ L^{2} $-supercritical case $ \sigma\in\left(\frac{2}{3}, \, 2 \right). $ Here $ \lambda \in \mathbb{R} $ appears as a Lagrange multiplier, $ \varepsilon $ is a small positive parameter and $ K > 0 $ possesses several local maximum points. By employing the variational gluing method and the penalization technique, we prove the existence of multi-bump solutions that are concentrated at local maximum points of $ K $ for the problem above.

References

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