Non-spherical minimizers in the generalized liquid drop model for Yukawa and truncated Coulomb potentials

We investigate generalized liquid drop models with screened Riesz-type interactions, focusing in particular on truncated Coulomb and Yukawa potentials in three dimensions. While the classical Gamow model with Coulomb interaction is conjectured to admit only spherical minimizers below a critical mass and no minimizer above, we show that this conjecture fails if the interaction is screened. In the case of truncated Coulomb and Yukawa potentials, we establish the existence of non-spherical minimizers for some values of the screening parameter. This gives the first evidence of such minimizers in the class of repulsive, radial, and radially nonincreasing kernels in three dimensions. Our approach relies on a comparison of the energy-per-mass ratios of balls and cylinders, in contrast with recent two-dimensional results obtained via $\mathrm{\Gamma }$-convergence. We further show that in the unscreened Riesz case, the conjecture remains consistent, though delicate. Indeed we observe that the energy-per-mass ratios of balls and of cylinders are surprisingly close.