  We establish a relationship between an inverse optimization spectral problem
for N-dimensional Schr\"odinger equation $ -\Delta \psi+q\psi=\lambda \psi $
and a solution of the nonlinear boundary value problem $-\Delta u+q_0 u=\lambda
u- u^{\gamma-1},~~u>0,~~
  u|_{\partial \Omega}=0$. Using this relationship, we find an exact solution
for the inverse optimization spectral problem, investigate its stability and
obtain new results on the existence and uniqueness of the solution for the
nonlinear boundary value problem.
