A Maximization Problem Involving a Fractional Laplace Type Operator

Abstract

Fractional Laplacian is an important nonlocal operator which has many applications in different kinds of differential equations. Recently, optimization problems involving the fractional Laplacian have been studied a lot by many authors. However, most of these papers are focusing on the optimization problems related to the first eigenvalue of the equation. Optimization problems related to the energy functional of the equation have not been investigated well enough. In this paper, we are going to study a maximization problem related to the energy functional of an equation involving a fractional Laplace type operator. Firstly, by using suitable variational framework in a fractional Sobolev space, we can show that a fractional equation has a solution which is in fact the global minimum of the corresponding energy functional. Moreover, by using reduction to absurdity we can obtain the uniqueness of the solution of the fractional equation. Then, we focus on a maximization problem related to the equation which takes the energy functional as the objective functional. Finally, by carefully analysing the properties of an arbitrarily choosen minimizing sequence and the tools of the rearrangement theory, we can prove that the maximization problem is solvable.