Response solutions for the nonlinear Euler-Bernoulli beam equation with axial force and nonzero potential

This paper investigates the response solutions of the nonlinear Euler-Bernoulli beam equation with axial force and nonzero potential. The axial force modifies the structural stability of the beam and may induce buckling. Response solutions, which characterize synchronization and resonance phenomena, are driven by an external force and share its frequency. Without recourse to the Lyapunov-Schmidt reduction method, we employ a differential Nash-Moser iterative scheme to construct response solutions for frequencies and perturbation parameters belonging to a Cantor set of asymptotically full measure, under the assumption that the nonlinearity possesses only finite differentiability. The parameters associated with the axial force and the potential are both dense in R² and our solutions exhibit Sobolev regularity in time.