Ground state solution and optimal Gagliardo-Nirenberg constant for the p-Choquard functional

摘要

In the present paper we consider the minimization problem associated with the best constant CG_N of Gagliardo-Nirenberg interpolation inequality ∫_(R~3)∫_(R~3) (|ψ(x)|~P|ψ(y)~P|)/(|x - y|) dxdy ≤ C_(GN)‖{nalba}_ψ‖_(L~2)~a‖ψ‖_(L~p)~b, where a = (2p)/(6-p) and b = (2p(5-p))/(6-p). The corresponding optimal function is the unique solution to the following p-Choquard equation -Δψ + |ψ|~(p~(-2))ψ = I(|ψ|~P)|ψ|~(p-2)ψ, where I(·) is the classical Riesz potential.