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Normalized Solutions for a Kirchhoff Equation with Potential in $\mathbb{R}^3$

Normalized Solutions for a Kirchhoff Equation with Potential in $\mathbb{R}^3$

Year:    2024

Author:    Yuan Xu, Yongyi Lan

Journal of Mathematical Study, Vol. 57 (2024), Iss. 4 : pp. 509–527

Abstract

In this paper, for given mass $c>0,$ we study the existence of normalized solutions to the following nonlinear Kirchhoff equation  $$\begin{cases} (a+b\int_{\mathbb{R}^3}[|\nabla u|^2+V(x)u^2]dx)[-\Delta u+V(x)u]=\lambda u+\mu|u|^{q-2}u+|u|^{p-2}u, \ \ \ {\rm in}\ \ \mathbb{R}^3, \\ \int_{\mathbb{R}^3}|u|^2dx=c^2, \end{cases}$$where $a>0, b>0, λ∈\mathbb{R},$ $5<q< p<6,$ $\mu>0$ and $V$ is a continuous non-positive function vanishing at infinity. Under some mild assumptions on $V,$ we prove the existence of a mountain pass normalized solution via the minimax principle.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jms.v57n4.24.08

Journal of Mathematical Study, Vol. 57 (2024), Iss. 4 : pp. 509–527

Published online:    2024-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    19

Keywords:    Kirchhoff equation normalized solutions minimax principle.

Author Details

Yuan Xu

Yongyi Lan