Solutions with prescribed mass for the Sobolev-critical Kirchhoff equation with trapping potential
Li Cai, Rong PuWe 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.