Open AccessNormalized solutions for nonlinear Kirchhoff type equations in high dimensions
Author(s) -
Lingzheng Kong,
Haibo Chen
Publication year - 2022
Publication title -
electronic research archive
Language(s) - English
Resource type - Journals
ISSN - 2688-1594
DOI - 10.3934/era.2022067
Subject(s) - mathematics , exponent , sobolev space , critical exponent , nonlinear system , mathematical analysis , equivalence (formal languages) , multiplicity (mathematics) , dimension (graph theory) , mathematical physics , pure mathematics , physics , geometry , quantum mechanics , scaling , philosophy , linguistics
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.