Critical points approaches to fourth-order elastic beam equations with variable coefficients having combined effects of concave and convex nonlinearities

This study investigates the theoretical and practical aspects of fourth-order elastic beam equations characterized by variable coefficients and influenced by combined nonlinearities. The equations under consideration arise naturally in the context of high-order differential problems encountered in material science, particularly in the study of phase transitions. The research focuses on establishing the existence and multiplicity of solutions for these equations, which involve both concave and convex nonlinearities. Using advanced mathematical techniques, such as variational methods and critical point theory, the study provides rigorous proofs for the existence of solutions under specific conditions. A central result is the application of Bonanno's local minimum theorem, which ensures the existence of at least one solution. Moreover, the research shows that by imposing additional algebraic conditions, particularly the classical Ambrosetti–Rabinowitz condition, the presence of two distinct solutions can be guaranteed. Beyond this, critical point theorems from Averna and Bonanno are employed to demonstrate scenarios where three solutions are possible.