In the paper, the first asymptotic approximations of the solutions for the displacements and the stress function for the four-frequency regime of forced nonlinear oscillations of a thin elastic shell with a positive Gaussian curve and finite deformations during oscillation were compiled. The four frequencies of the coercive force are either constant or slowly varying functions of time in the resonant range of the first four natural frequencies of undisturbed shell oscillation. Using the derived system of differential equations of the first approximation for the amplitudes and phases of the four frequency regimes of resonant oscillation, an analysis of the mutual influence of harmonics as a consequence of the nonlinearity of the system was performed.