On some nonlocal variational problems

Abstract

We study uniqueness and non uniqueness of minimizers of functionals involving nonlocal quantities. We give also conditions which lead to a lack of minimizers and we show how minimization on an infinite dimensional space reduces here to a minimization on ℝ. Among other things, we prove that uniqueness of minimizers of functionals of the form ∫Ω a(∫Ω gu dx)|∇u|2 dx - 2 ∫Ω fu dx is ensured if a > 0 and 1/a is strictly concave in the sense that (1/a)″ < 0 on (0, ∞).