| Abstract: |
| We study nonnegative optimizers of a Gagliardo-Nirenberg type inequality that interpolates between the nonlocal Riesz energy and two $L^q$-norms. A particular case of the equivalent problem has been studied in connection with the Keller-Segel diffusion-aggregation models over the past few decades. The more general case considered in our work arises in the study of the Thomas-Fermi limit regime for the Choquard equations with local repulsion. We establish, for the first time, the optimal ranges of parameters for the validity of the above interpolation inequality, discuss the existence and qualitative properties of the nonnegative maximisers, and in some special cases estimate the optimal constant. In special cases it is known that the maximisers are H\older continuous and compactly supported in a ball. We show that, depending on the regime, maximisers may also be smooth functions supported on the entire space, or discontinuous functions consisting of the characteristic function of a ball and a nonconstant, nonincreasing H\older continuous function supported on the same ball. We use these qualitative properties of the maximisers to justify the validity of the Thomas-Fermi approximations for the Choquard equations with local repulsion. The results are verified numerically through extensive examples. |
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