A note on a generalized singular capillarity system with \(\Im \)-Hilfer fractional derivative

We consider a generalized singular capillarity problem driven by a fractional Hilfer derivative with respect to a function, with Dirichlet boundary conditions. The nonlinearity of the problem generally exhibits some singular characteristics and is characterized by a variable exponent function, which displays critical behavior at infinity. Using the combination of the Nehari manifold method with a variational approach on fractional spaces in the sense of Hilfer, we prove the existence and multiplicity of positive solutions to such a problem provided that the parameters that appear in the problem satisfy some appropriate conditions. Our main results are novel, and their investigation will enhance the scope of the literature on singular coupled systems involving generalized fractional derivatives.