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Photonic crystal fibers doped with silver nanoparticles exhibit a Kerr nonlinearity that can be positive or negative depending on the input wavelength and which vanishes at a specific wavelength. The existence of negative nonlinearity allows soliton formation even in the normal-dispersion region of the fiber, and the zero-nonlinearity wavelength (ZNW) acts as a barrier for the Raman-induced redshift of solitons. We adopted the variational principle to understand the role of the zero-nonlinearity point on Raman redshift and verified its prediction numerically for fundamental and higher-order solitons. We show how the simultaneous presence of a ZNW and a zero-dispersion wavelength affects soliton evolution inside such fibers and find a number of unique features such as the position and the spectral bandwidth of the dispersive wave that change with the location of the ZNW.