A saturation phenomenon for a nonlinear nonlocal eigenvalue problem

Given 1 <= q <= 2 and alpha is an element of R, we study the properties of the solutions of the minimum problem lambda(alpha, q) = min {integral(1)(-1) vertical bar u'vertical bar(2) dx + alpha vertical bar integral(1)(-1) vertical bar u vertical bar(q-1) u dx vertical bar(2/q) /integral(1)(-1) vertical bar u vertical bar(2) dx , u is an element of H-0(1) (-1, 1), u not equivalent to 0). In particular, depending on alpha and q, we show that the minimizers have constant sign up to a critical value of alpha = alpha(q), and when alpha > alpha(q) the minimizers are odd.