The Nehari manifold for a fractional $(p(x,.),q(x,.))-$Laplacian elliptic system

Fractional elliptic system

Résumé

In this paper, we study the existence and multiplicity of weak solutions for a class of nonlocal elliptic systems involving fractional $(p(x,.),q(x,.))$-Laplacian operators under Neumann boundary conditions. The problem is formulated in fractional Sobolev spaces with variable exponents, where the interaction between nonlocality and space-dependent growth leads to an energy functional that is not lower bounded on the associated functional space. To overcome this difficulty, we use the Nehari manifold approach, which allows us to recover a natural constraint on which the functional becomes coercive and bounded from below. Our results extend recent advances in the field of fractional elliptic problems with variable exponents and provide new insights into nonlocal systems subject to Neumann boundary conditions.