We investigate theoretically the ground-state properties and breathing-mode collective excitations of three-dimensional dipolar Bose gases in anisotropic harmonic traps incorporating quantum fluctuations. Combining a Gaussian variational ansatz with a sum-rule analysis, we derive explicit analytical expressions for both axial and radial breathing-mode frequencies, which are validated by numerical solutions of the time-dependent extended Gross–Pitaevskii equation. Our theoretical predictions show great agreement with existing experimental data for ${}^{166}\mathrm{Er}$ and ${}^{162}\mathrm{Dy}$ gases. By constructing comprehensive phase diagrams across the parameter space of the $s$-wave scattering length, atom number, and trap aspect ratio, we reveal both discontinuous first-order phase transitions and smooth crossovers between the dilute Bose-Einstein condensate and dense quantum droplet phases. We confirm that the enhanced incompressibility induced by quantum fluctuations significantly elevates the breathing-mode frequencies in the droplet phase compared with conventional weakly interacting Bose gases. Furthermore, the system undergoes a phase transition and a crossover over the scattering length under the elongated and flattened confinements, characterized by discontinuous jumps and continuous crossovers in peak density and atomic cloud sizes, respectively. Our work offers a useful analytical framework to characterize collective excitations in dipolar quantum gases, providing quantitative insights for forthcoming ultracold atom experiments in lanthanide atoms and polar molecules.

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• Cold atoms & matter waves
• Dipolar gases
• Bose-Einstein condensates