Normalized solutions to nonautonomous Kirchhoff equation

Xin Qiu; Zeng Qi Ou; Ying Lv; Xin Qiu; Zeng Qi Ou; Ying Lv

doi:10.3934/cam.2024022

Communications in Analysis and Mechanics

2024, Volume 16, Issue 3: 457-486. doi: 10.3934/cam.2024022

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Research article

Normalized solutions to nonautonomous Kirchhoff equation

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, P.R. China

Received: 31 October 2023 Revised: 11 April 2024 Accepted: 17 April 2024 Published: 03 July 2024
35A15, 35J60, 35J20

In this paper, we studied the existence of normalized solutions to the following Kirchhoff equation with a perturbation:
$ \left\{ \begin{aligned} &-\left(a+b\int _{\mathbb{R}^{N}}\left | \nabla u \right|^{2} dx\right)\Delta u+\lambda u = |u|^{p-2} u+h(x)\left |u\right |^{q-2}u, \quad \text{ in } \mathbb{R}^{N}, \\ &\int_{\mathbb{R}^{N}}\left|u\right|^{2}dx = c, \quad u \in H^{1}(\mathbb{R}^{N}), \end{aligned} \right. $
where $ 1\le N\le 3, a, b, c > 0, 1\leq q < 2 $, $ \lambda \in \mathbb{R} $. We treated three cases:
(i) When $ 2 < p < 2+\frac{4}{N}, h(x)\ge0 $, we obtained the existence of a global constraint minimizer.
(ii) When $ 2+\frac{8}{N} < p < 2^{*}, h(x)\ge0 $, we proved the existence of a mountain pass solution.
(iii) When $ 2+\frac{8}{N} < p < 2^{*}, h(x)\leq0 $, we established the existence of a bound state solution.
- nonautonomous Kirchhoff equations,
- normalized solutions,
- bound state solution,
- $ L^{2} $-critical exponent
Citation: Xin Qiu, Zeng Qi Ou, Ying Lv. Normalized solutions to nonautonomous Kirchhoff equation[J]. Communications in Analysis and Mechanics, 2024, 16(3): 457-486. doi: 10.3934/cam.2024022

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Abstract

In this paper, we studied the existence of normalized solutions to the following Kirchhoff equation with a perturbation:

$ \left\{ \begin{aligned} &-\left(a+b\int _{\mathbb{R}^{N}}\left | \nabla u \right|^{2} dx\right)\Delta u+\lambda u = |u|^{p-2} u+h(x)\left |u\right |^{q-2}u, \quad \text{ in } \mathbb{R}^{N}, \\ &\int_{\mathbb{R}^{N}}\left|u\right|^{2}dx = c, \quad u \in H^{1}(\mathbb{R}^{N}), \end{aligned} \right. $

where $ 1\le N\le 3, a, b, c > 0, 1\leq q < 2 $, $ \lambda \in \mathbb{R} $. We treated three cases:

(i) When $ 2 < p < 2+\frac{4}{N}, h(x)\ge0 $, we obtained the existence of a global constraint minimizer.

(ii) When $ 2+\frac{8}{N} < p < 2^{*}, h(x)\ge0 $, we proved the existence of a mountain pass solution.

(iii) When $ 2+\frac{8}{N} < p < 2^{*}, h(x)\leq0 $, we established the existence of a bound state solution.

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