The nonlinear Penner external interaction is introduced and studied in the random matrix model of homo RNA. A numerical technique is developed to study the partition function, and a general formula is obtained for all lengths. The genus distribution function for the system is obtained, plotted, and compared with the genus distribution for the real RNA structures found from the protein databank. The genus distribution shows that the nonlinear interaction favors the formation of low genus structures and matches the result for real RNA structures. The distribution of structure with temperature suggests that nonlinear interaction is biased toward the planar structures. The variation of chemical potential with temperature and interaction strength indicates the presence of additional molecules in the system other than the magnesium ions and possibly represents a phase transition. The specific heat has a bump and its derivatives shows a double-peak behavior at a particular temperature. On analyzing the specific heat and derivatives for each genus separately, the planar structure (genus zero) is shown to contribute the most to the bump and double peak. This observation in the nonlinear model is similar to that observed in the unfolding experiments on RNA.