Existence and multiplicity of solutions for singular differential equations with non-local boundary conditions*

Abstract

Abstract This paper focuses on the investigation of a class of singular differential systems with non-local boundary conditions. Its primary objective is to establish the variational structure of the non-local singular differential systems, and systematically explore multiple solutions through the application of the variational method. A significant achievement of this research is the demonstration that the energy functional satisfies the Cerami condition within an appropriate Sobolev's space. The derivation of key results is facilitated by the strategic application of the Minimization sequence and Fountain Theorem. Furthermore, even in cases where the nonlinear term fails to meet the Ambrosetti-Rabinowitz condition, this paper successfully applies the variational method to address the associated problems. The challenges addressed in this study include proving the convergence of the solution sequence, ensuring the continuous differentiability of the energy functional, and showcasing the embedding property of space. Distinguishing itself from existing research, this paper innovatively constructs a variational framework tailored specifically for singular differential systems with non-local boundary conditions. The utilization of variational methods enables a meticulous and systematic exploration into the solvability of these intricate problems.