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Research Article | Open Access

Normalized solutions to nonautonomous Kirchhoff equation

Xin Qiu, Zeng Qi Ou, Ying Lv(

)

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

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Abstract

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

${\begin{aligned} - (a + b \int_{R^{N}} {| \nabla u |}^{2} d x) Δ u + λ u = | u |^{p - 2} u + h (x) {| u |}^{q - 2} u, in R^{N}, \\ \int_{R^{N}} {| u |}^{2} d x = c, u \in H^{1} (R^{N}), \end{aligned}$

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

(i) When $2 < p < 2 + \frac{4}{N}, h (x) \geq 0$ , we obtained the existence of a global constraint minimizer.

(ii) When $2 + \frac{8}{N} < p < 2^{*}, h (x) \geq 0$ , we proved the existence of a mountain pass solution.

(iii) When $2 + \frac{8}{N} < p < 2^{*}, h (x) \leq 0$ , we established the existence of a bound state solution.

Keywords

nonautonomous Kirchhoff equations normalized solutions bound state solution L²-critical exponent

CLC number: 35A15, 35J60, 35J20

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Communications in Analysis and Mechanics

Volume 16 Issue 3,
September 2024

Pages 457-486

DOI: 10.3934/cam.2024022

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Cite this article:

Qiu X, Ou ZQ, Lv Y. Normalized solutions to nonautonomous Kirchhoff equation. Communications in Analysis and Mechanics, 2024, 16(3): 457-486. https://doi.org/10.3934/cam.2024022

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Citation

Received: 31 October 2023

Revised: 11 April 2024

Accepted: 17 April 2024

Published: 15 September 2024

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)