Multiplicity of normalized solutions for the $L^2$-supercritical Kirchhoff equation with partial confinement

Abstract

We study the existence of positive normalized solutions for the nonlocal Kirchhoff equation with partial harmonic confinement \[-\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\,dx\right)Δu +(x_1^2+x_2^2)u+λu=|u|^{p-2}u,\] subject to the mass constraint $\int_{\mathbb{R}^{3}}|u|^2\,dx=μ$, where $a,b>0$ are constants, $\frac{14}{3} 0$ is a prescribed mass, and $λ\in\mathbb{R}$ is a Lagrange multiplier associated with the mass constraint. For every sufficiently small $μ>0$, we prove the existence of two positive normalized solutions. One of these solutions is a local minimizer, while the other is obtained at a mountain-pass level. The construction of the local minimizer relies on a combination of profile decomposition and a new nonlocal merging technique, yielding compactness up to translations in the unconfined $x_3$-direction. The mountain-pass solution is obtained by considering positive solutions defined on expanding balls. A uniform bound on the Morse index allows us to obtain a spectral estimate for the limiting Lagrange multiplier. Together with an energy comparison, this spectral estimate yields strong convergence in the energy space and recovers both the prescribed mass and the Kirchhoff coefficient.