On the Kirchhoff equation with prescribed mass and general nonlinearities

Abstract

In the present paper, we apply a global branch approach to study the existence, non-existence and multiplicity of positive normalized solutions $({\lambda }_c,u_c)\in \mathbb{R}\times H^1({\mathbb{R}}^N)$ to the following Kirchhoff problem $$-\left(a+b{\int }_{{\mathbb{R}}^N}|\nabla u{|}^{2}dx\right)\Delta u+\lambda u=g(u)in{\mathbb{R}}^N,N\ge 1$$ satisfying the normalization constraint ${\int }_{{\mathbb{R}}^N}{u}^2=c,$ which appears in free vibrations of elastic strings. The parameters $a,b>0$ are prescribed as well as the mass $c>0$. Due to the presence of the non-local term $b{\int }_{{\mathbb{R}}^N}|\nabla u{|}^{2}dx\Delta u$, such problems lack the mountain pass geometry in the higher dimension case $N\ge 5$. Our result seems to be the first attempt in this aspect.

Published in Discrete and Continuous Dynamical Systems - Series S, Vol. 16, No. 11, November 2023, pp. 3394-3409

References in corpus (3)

• Multiple normalized solutions for a Sobolev critical Schrödinger equation
• On global minimizers for a mass constrained problem
• Normalized ground states for Kirchhoff equations in ${\mathbb{R}}^3$ with a critical nonlinearity