Stationary fully nonlinear mean-field games

In this paper we examine fully nonlinear mean-field games associated with a minimization problem. The variational setting is driven by functional depending on its argument through Hessian matrix. We work under fairly natural conditions and establish improved (sharp) regularity for the solutions in Sobolev spaces. Then, prove existence of minimizers problem to system. also investigate unidimensional example unveil new information explicit solutions. Our findings can be generalized larger class operators, yielding broader range examples.