A nonlinear eigenvalue problem arising in a nanostructured quantum dot

In this paper we investigate a minimization problem related to the principal eigenvalue of the \(s\)-wave Schr\"{o}dinger operator. The operator depends nonlinearly on the eigenparameter. We prove the existence of a solution for the optimization problem and the uniqueness will be addressed when the domain is a ball. The optimized solution can be applied to design new electronic and photonic devices based on the quantum dots.