Existence of solutions for a double phase problem with variable exponents

The purpose of this paper is to study a class of double-phase problems with a singular term and a superlinear parametric term on the right-hand side. Using the method of Nehari manifold combined with the fibering maps, we prove that for all small values of the parameter $\mu > 0,$ there exist at least two non-trivial positive solutions. Our results extend the previous works of Papageorgiou, Repov{\v{s}}, and Vetro %\cite{papageorgiou2020positive} and Liu, Dai, Papageorgiou, and Winkert, %\cite{liu2021existence}, from the case of Musielak-Orlicz Sobolev space, when exponents $p$ and $q$ are constant, to the case of Sobolev-Orlicz spaces with variable exponents in a complete manifold.