Multiplicity and least energy sign-changing normalized solutions for a Kirchhoff equation on bounded domains
Abstract
In this paper, we study sign-changing normalized solutions of the Kirchhoff problem −(a+b∫Ω|∇ u|2 dx)Δu =|u|p−2u+μu &in Ω, u=0 &on ∂Ω, subject to the constraint ∫Ω|u|2 dx=c, where a,b>0, Ω⊂ℝN is a smooth bounded connected domain, 1≤ N≤3, c>0 is prescribed, and μ∈ℝ is a Lagrange multiplier. We assume 2<p<∞ for N=1,2 and 2<p≤6 for N=3. For every integer k≥1, we obtain at least k distinct pairs of sign-changing normalized solutions for sufficiently small c>0. We determine the small mass asymptotics of their energies, Lagrange multipliers, and rescaled profiles in terms of the distinct Dirichlet eigenvalues and their eigenspaces. When 2<p<2+8/N, we obtain infinitely many such pairs for every prescribed mass c>0, with energies tending to +∞. At p=2+8/N, this conclusion holds for every sufficiently small fixed mass. We also establish the existence of a least energy sign-changing normalized solution on general bounded domains for every prescribed mass when 2<p<2+8/N. For sufficiently small mass, such a solution exists on general bounded domains when p=2+8/N and on star-shaped domains when p>2+8/N. For any family of such least energy solutions, the rescaled profiles approach the L2-unit sphere of the second Dirichlet eigenspace in H01(Ω) as c→0+. The finite multiplicity and least energy results include the Sobolev critical case N=3, p=6. The proofs combine a localized symmetric minimax construction, energy comparisons with one-signed critical points, and compactness arguments that preserve the sign-changing property.
Related
- Multiplicity of normalized solutions for the L2-supercritical Kirchhoff equation with partial confinement
- Least energy sign-changing solutions for asymptotically cubic Kirchhoff equations on locally finite graphs
- Existence and multiplicity of solutions for mixed local and nonlocal Kirchhoff type problems
- Normalized solutions to the NLS equation in the ball for any prescribed mass
- Sufficient conditions for multiplicity and uniqueness of positive solutions of the Hénon equation
BibTeX
@misc{wang2026multiplicity,
title = {{Multiplicity and least energy sign-changing normalized solutions for a Kirchhoff equation on bounded domains}},
author = {Qun Wang and Xiaojun Chang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07199},
eprint = {2611.07199},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07199},
}
RIS
TY - UNPB
TI - Multiplicity and least energy sign-changing normalized solutions for a Kirchhoff equation on bounded domains
AU - Qun Wang
AU - Xiaojun Chang
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.07199
UR - https://arxiv.org/abs/2611.07199
AB - In this paper, we study sign-changing normalized solutions of the Kirchhoff problem −(a+b∫_Ω|∇ u|^2 dx)Δu =|u|^p−2u+μu &in Ω, u=0 &on ∂Ω, subject to the constraint ∫_Ω|u|^2 dx=c, where a,b>0, Ω⊂ℝ^N is a smooth bounded connected domain, 1≤ N≤3, c>0 is prescribed, and μ∈ℝ is a Lagrange multiplier. We assume 2<p<∞ for N=1,2 and 2<p≤6 for N=3. For every integer k≥1, we obtain at least k distinct pairs of sign-changing normalized solutions for sufficiently small c>0. We determine the small mass asymptotics of their energies, Lagrange multipliers, and rescaled profiles in terms of the distinct Dirichlet eigenvalues and their eigenspaces. When 2<p<2+8/N, we obtain infinitely many such pairs for every prescribed mass c>0, with energies tending to +∞. At p=2+8/N, this conclusion holds for every sufficiently small fixed mass. We also establish the existence of a least energy sign-changing normalized solution on general bounded domains for every prescribed mass when 2<p<2+8/N. For sufficiently small mass, such a solution exists on general bounded domains when p=2+8/N and on star-shaped domains when p>2+8/N. For any family of such least energy solutions, the rescaled profiles approach the L^2-unit sphere of the second Dirichlet eigenspace in H_0^1(Ω) as c→0^+. The finite multiplicity and least energy results include the Sobolev critical case N=3, p=6. The proofs combine a localized symmetric minimax construction, energy comparisons with one-signed critical points, and compactness arguments that preserve the sign-changing property.
ER -