Opuscula Mathematica
Opuscula Math. 46, no. 5 (2026), 641-678

Solutions with prescribed mass for the Sobolev-critical Kirchhoff equation with trapping potential

Li Cai
Rong Pu
Abstract. We study prescribed-mass solutions of a nonautonomous Kirchhoff equation with a purely Sobolev-critical nonlinearity and a nonnegative trapping potential. The trapping potential breaks the scaling invariance of the autonomous problem, while the critical nonlinearity causes loss of compactness. We establish a global compactness alternative for constrained Palais-Smale sequences and combine it with sharp energy estimates adapted to the nonlocal Kirchhoff term. Under suitable structural assumptions and an explicit energy-separation condition, we obtain a positive normalized ground state as a local minimizer of the constrained energy. For sufficiently small Kirchhoff parameter \(b\gt 0\), we further prove the existence of a distinct positive normalized solution at a mountain-pass level.
Keywords: Kirchhoff equation, Sobolev-critical exponent, normalized ground state, mountain-pass solution.
Mathematics Subject Classification: 35J15, 35B33, 35A15.
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About the authors ▼
Li Cai (corresponding author)
https://orcid.org/0009-0007-1018-6282
School of Mathematics, Yangzhou University, Yangzhou 225002, P.R. China
Rong Pu
School of Mathematics, Yangzhou University, Yangzhou 225002, P.R. China
About the article ▼
Communicated by Vicenţiu D. Rădulescu.
Received: 2026-09-29.
Revised: 2026-10-11.
Accepted: 2026-10-12.
Published online: 2026-10-28.
Cite this article as ▼
Li Cai, Rong Pu, Solutions with prescribed mass for the Sobolev-critical Kirchhoff equation with trapping potential, Opuscula Math. 46, no. 5 (2026), 641-678, https://doi.org/10.7494/OpMath.202610121.
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