Critical points approaches for multiple solutions of a quasilinear periodic boundary value problem

Optimization problems are omnipresent in the mathematical modeling of real world systems and cover a very extensive range of applications becoming apparent in all branches of Economics, Finance, Materials Science, Astronomy, Physics, Structural and Molecular Biology, Engineering, Computer Science, and Medicine. In this paper, we aim to delve deeper into the multiplicity findings concerning a specific class of quasilinear periodic boundary value problems. In fact, as an optimization problem, we look for the critical points of the energy functional related to the problem. Utilizing a corollary derived from Bonanno’s local minimum theorem, we investigate the existence of a one solution under certain algebraic conditions on the nonlinear term. Additionally, we explore conditions that lead to the existence of two solutions, incorporating the classical Ambrosetti-Rabinowitz (AR) condition alongside algebraic criteria. Moreover, by employing two critical point theorems one by Averna and Bonanno, and another by Bonanno, we establish the existence of two and three solutions in a particular scenario. To illustrate our findings, we provide an example.