Ground states of a non‐local variational problem and Thomas–Fermi limit for the Choquard equation

Abstract We study non‐negative optimisers of a Gagliardo–Nirenberg‐type inequality which involves the non‐local Riesz energy with , , and . For , the equivalent problem has been studied in connection with the Keller–Segel diffusion–aggregation models in the past few decades. The general case considered here appears in the study of Thomas–Fermi limit regime for the Choquard equations with local repulsion. We establish, for the first time, optimal ranges of parameters for the validity of the above interpolation inequality, discuss the existence and qualitative properties of the non‐negative maximisers and in some special cases estimate the optimal constant. For , it is known that the maximisers are Hölder continuous and compactly supported on a ball. We show that for , the maximisers are smooth functions supported on , while for , the maximisers consist of a characteristic function of a ball and a non‐constant non‐increasing Hölder continuous function supported on the same ball. We use these qualitative properties of the maximisers to establish the validity of the Thomas–Fermi approximations for the Choquard equations with local repulsion. The results are verified numerically with extensive examples.