arrow
Volume 40, Issue 2
The Existence and Multiplicity of Normalized Solutions for Kirchhoff Equations in Defocusing Case

Lin Xu

Anal. Theory Appl., 40 (2024), pp. 191-207.

Published online: 2024-07

Export citation
  • Abstract

In this paper, we study the existence of solutions for Kirchhoff equation

1ata.JPG

with mass constraint condition

2ata.JPG

where $a$, $b$, $c>0$, $\mu\in \mathbb{R}$ and $2<q<p<6$. The $\lambda \in \mathbb{R}$ appears as a Lagrange multiplier. For the range of $p$ and $q$, the Sobolev critical exponent $6$ and mass critical exponent $\frac{14}{3}$ are involved which corresponding energy functional is unbounded from below on $S_{c}$. We consider the defocusing case, i.e. $\mu<0$ when $(p, q)$ belongs to a certain domain in $\mathbb{R}^{2}$. We prove the existence and multiplicity of normalized solutions by using constraint minimization, concentration compactness principle and Minimax methods. We partially extend the results that have been studied.


  • AMS Subject Headings

35B08, 35J47, 35P30, 35Q55

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{ATA-40-191, author = {Xu , Lin}, title = {The Existence and Multiplicity of Normalized Solutions for Kirchhoff Equations in Defocusing Case}, journal = {Analysis in Theory and Applications}, year = {2024}, volume = {40}, number = {2}, pages = {191--207}, abstract = {

In this paper, we study the existence of solutions for Kirchhoff equation

1ata.JPG

with mass constraint condition

2ata.JPG

where $a$, $b$, $c>0$, $\mu\in \mathbb{R}$ and $2<q<p<6$. The $\lambda \in \mathbb{R}$ appears as a Lagrange multiplier. For the range of $p$ and $q$, the Sobolev critical exponent $6$ and mass critical exponent $\frac{14}{3}$ are involved which corresponding energy functional is unbounded from below on $S_{c}$. We consider the defocusing case, i.e. $\mu<0$ when $(p, q)$ belongs to a certain domain in $\mathbb{R}^{2}$. We prove the existence and multiplicity of normalized solutions by using constraint minimization, concentration compactness principle and Minimax methods. We partially extend the results that have been studied.


}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2023-0027}, url = {http://global-sci.org/intro/article_detail/ata/23234.html} }
TY - JOUR T1 - The Existence and Multiplicity of Normalized Solutions for Kirchhoff Equations in Defocusing Case AU - Xu , Lin JO - Analysis in Theory and Applications VL - 2 SP - 191 EP - 207 PY - 2024 DA - 2024/07 SN - 40 DO - http://doi.org/10.4208/ata.OA-2023-0027 UR - https://global-sci.org/intro/article_detail/ata/23234.html KW - Normalized solutions, Kirchhoff-type equation, mixed nonlinearity. AB -

In this paper, we study the existence of solutions for Kirchhoff equation

1ata.JPG

with mass constraint condition

2ata.JPG

where $a$, $b$, $c>0$, $\mu\in \mathbb{R}$ and $2<q<p<6$. The $\lambda \in \mathbb{R}$ appears as a Lagrange multiplier. For the range of $p$ and $q$, the Sobolev critical exponent $6$ and mass critical exponent $\frac{14}{3}$ are involved which corresponding energy functional is unbounded from below on $S_{c}$. We consider the defocusing case, i.e. $\mu<0$ when $(p, q)$ belongs to a certain domain in $\mathbb{R}^{2}$. We prove the existence and multiplicity of normalized solutions by using constraint minimization, concentration compactness principle and Minimax methods. We partially extend the results that have been studied.


Xu , Lin. (2024). The Existence and Multiplicity of Normalized Solutions for Kirchhoff Equations in Defocusing Case. Analysis in Theory and Applications. 40 (2). 191-207. doi:10.4208/ata.OA-2023-0027
Copy to clipboard
The citation has been copied to your clipboard