Existence of normalized solutions to Kirchhoff-Boussinesq equations in the subcritical and supercritical regime

In this article we study the existence of normalized solutions to the Kirchhoff-Boussinesq equation under the mass constraint $\|u\|_{2}=c$. In the $L^{2}$-subcritical regime, we apply Ekeland's variational principle and concentration compactness method to minimize the energy functional on the mass-constrained manifold. In the $L^{2}$-supercritical regime, we introduce a Pohozaev-constrained minimization approach, combined with scaling arguments to recover compactness. To handle the additional difficulties posed by $q$-Laplacian, we treat distinct ranges of $q$ separately.