p-Laplacian Fractional Differential Equations With Singular Terms and Infinite Point Boundary Conditions

Fractional differential equations (FDEs) are of great significance for describing complex physical processes with memory and genetic characteristics. The introduction of the p-Laplacian operator can expand its applicability in nonlinear problems. To explore the uniqueness multiplicity, and variety of solutions to p-Laplacian differential equations with singular terms, the influences of fractional order and P-Laplace exponent on the solution morphology are analyzed. Based on the Riemann–Liouville theory, the boundary value problem is transformed into an equivalent integral equation through integral transformation and the Green's function is constructed. The existence theory of the solution is established, and then the influence of parameters is verified and analyzed through numerical examples. The results show that when the p-Laplacian index is 1.8, 2.5, and 3.2, the peak solutions are 0.7768, 0.6393, and 0.5229, respectively. The results show that when a specific integral inequality is satisfied reducing the p-Laplacian exponent increases the amplitude of the solution. Under appropriate nonlinear conditions, the equation has three positive solutions (PS) with large amplitude differences. This research provides methodological references and theoretical foundations for the analysis of related nonlinear FDEs.