Critical Point Approaches for the Existence of Multiple Solutions of a Mixed Boundary Value Problem for a Complete Sturm–Liouville Equation

In this paper, we present multiplicity results related to mixed boundary value problem for a complete Sturm–Liouville equation. Specifically, we utilize a consequence of Bonanno’s local minimum theorem to establish the existence of one solution under certain algebraic conditions on the nonlinear term. In addition, we demonstrate that two solutions can be obtained under algebraic conditions while adhering to the classical Ambrosetti–Rabinowitz (AR) condition for the nonlinear term. Moreover, by applying two critical point theorems, one by Averna and Bonanno and another by Bonanno, we ensure the existence of two or three solutions for our problem in a particular scenario. We also provide an illustrative example to highlight our results.