On a class of generalized capillarity system involving fractional ψ#x02010;Hilfer derivative with p(·)-Laplacian operator

This research delves into a comprehensive investigation of a class of $\psi$-Hilfer generalized fractional nonlinear differential system originated from a capillarity phenomena with Dirichlet boundary conditions, focusing on issues of existence and multiplicity of nonnegative solutions. The nonlinearity of the problem, in general, does not satisfy the Ambrosetti–Rabinowitz type condition. We use minimization arguments of Nehari manifold together with variational approach to show the existence and multiplicity of positive solutions of our problem with respect to the parameter $\xi$ in appropriate fractional $\psi$-Hilfer spaces. Our main result is novel, and its investigation will enhance the scope of the literature on coupled systems of fractional $\psi$-Hilfer generalized capillary phenomena.