Global Structure of Solution Sets for Two Classes of Fourth-Order Beam Equations with MEMS Applications

In this paper, we study the structure of the solution sets of the nonlinear fourth-order beam equation: [Formula: see text] subject to clamped boundary conditions [Formula: see text] and pinned boundary conditions [Formula: see text], where [Formula: see text] and [Formula: see text] denote parameters, and [Formula: see text] is a convex, increasing, and positive nonlinearity. We prove that all solutions lie on a single smooth solution curve containing exactly one turning point. This characterizes the global structure of the solution sets and determines the exact multiplicity of solutions as the parameters vary. We propose a method that excludes the possibility of homoclinic branches and proves the uniqueness of global branches, in the absence of global parametrization by the maximum norm. In the case of singular nonlinearities, we also provide an explicit expression for the endpoint of the global branch using Green’s function and the singular value. Our results directly apply to the governing equations of several specific MEMS/NEMS models.