We establish a relationship between an inverse optimization spectral problem for Ndimensional Schrödinger equation −∆ψ+qψ = λψ and a solution of the nonlinear boundary value problem −∆u + q 0 u = λu − u γ−1 , u > 0, u| ∂Ω = 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.

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We consider an inverse optimization spectral problem for the Sturm-Liouville operator L[q]u := −u ′′ + q(x)u subject to the separated boundary conditions. In the main result, we prove that this problem is related to the existence of solutions of the boundary value problems for the nonlinear equations of the form −u ′′ + q0(x)u = λu + σu 3 with σ = 1 or σ = −1.The key outcome of this relationship is a generalized Sturm's nodal theorem for the nonlinear boundary value problems. *

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