Normalized solutions for nonlinear Kirchhoff type equations in high dimensions

Lingzheng Kong; Haibo Chen; Lingzheng Kong; Haibo Chen

doi:10.3934/era.2022067

Electronic Research Archive

2022, Volume 30, Issue 4: 1282-1295. doi: 10.3934/era.2022067

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Research article

Normalized solutions for nonlinear Kirchhoff type equations in high dimensions

Lingzheng Kong ,
Haibo Chen ^,

School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha 410083, China

Received: 18 November 2021 Revised: 15 February 2022 Accepted: 02 March 2022 Published: 16 March 2022

We study the normalized solutions for nonlinear Kirchhoff equation with Sobolev critical exponent in high dimensions $ \mathbb{R}^N(N\geqslant4) $. In particular, in dimension $ N = 4 $, there is a special phenomenon for Kirchhoff equation that the mass critical exponent $ 2+\frac{8}{N} $ is equal to the energy critical exponent $ \frac{2N}{N-2} $, which leads to the fact that the equation no longer has a variational structure in dimensions $ N\geqslant 4 $ if we consider the mass supercritical case, and remains unsolved in the existing literature. In this paper, by using appropriate transform, we first get the equivalent system of Kirchhoff equation. With the equivalence result, we obtain the nonexistence, existence and multiplicity of normalized solutions by variational methods, Cardano's formulas and Pohožaev identity.
- Kirchhoff equation,
- normalized solutions,
- critical exponent,
- combined nonlinearities,
- pohožaev identity
Citation: Lingzheng Kong, Haibo Chen. Normalized solutions for nonlinear Kirchhoff type equations in high dimensions[J]. Electronic Research Archive, 2022, 30(4): 1282-1295. doi: 10.3934/era.2022067

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Abstract

We study the normalized solutions for nonlinear Kirchhoff equation with Sobolev critical exponent in high dimensions $ \mathbb{R}^N(N\geqslant4) $. In particular, in dimension $ N = 4 $, there is a special phenomenon for Kirchhoff equation that the mass critical exponent $ 2+\frac{8}{N} $ is equal to the energy critical exponent $ \frac{2N}{N-2} $, which leads to the fact that the equation no longer has a variational structure in dimensions $ N\geqslant 4 $ if we consider the mass supercritical case, and remains unsolved in the existing literature. In this paper, by using appropriate transform, we first get the equivalent system of Kirchhoff equation. With the equivalence result, we obtain the nonexistence, existence and multiplicity of normalized solutions by variational methods, Cardano's formulas and Pohožaev identity.

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