UNIQUENESS AND RADIAL SYMMETRY OF MINIMIZERS FOR A NONLOCAL VARIATIONAL PROBLEM

For -n < p < 0, 0 < q and K(x) = parallel to x parallel to(q)/q - parallel to x parallel to(p)/p, the existence of minimizers of E(u) = integral(Rn x Rn) K(x - y)u(x)u(y) dx dy under integral(Rn) u(x)dx = m > 0; 0 <= u(x) <= M, with given m and M, is proved in [3]. Moreover, except for translation, uniqueness and radial symmetry of the minimizer is proved for -n < p < 0 and q = 2. Here in the present paper, we show that, except for translation, uniqueness and radial symmetry of the minimizer hold for -n < p < 0 and 2 <= q <= 4. Applications are given.