On nonlinear boundary value problem corresponding to $N$-dimensional inverse spectral problem

We establish a relationship between an inverse optimization spectral problem for N-dimensional Schr"odinger equation $-\mathrm{\Delta }\psi +q\psi =\lambda \psi$ 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.