On a Kondratiev problem

We consider the problem of minimizing, in the functional space W-p,0(2)(0,1), the functional f[y]=(integral(0)(1)\y ''\(p)dx)(1/p)/(integral(0)(1)\y'\(q)dx)(1/q), where p greater than or equal to 1, q > 0. This problem was posed by V.A. kondratiev in relation to the Lagrange problem. Some close problems were considered by Buslaev and Tikhomirov (see [1]). It is not difficult to prove the existence of the optimal solution. The main difficulty consists in the presence of two candidates for the optimal solutions, one symmetric, and another one non-symmetric. It is interesting to see that the optimal solution is really non-symmetric for some p, q.