Infinitely many periodic solutions for the equation: $u_ {tt}-u_ {xx}\pm \vert u\vert ^ {p-1}u=f(x,t)$. II

Abstract:

Existence of forced vibrations of nonlinear wave equation: \[ \begin {array}{*{20}{c}} {{u_{tt}} - {u_{xx}} \pm |u{|^{p - 1}}u = f(x, t),} \hfill & {(x, t) \in (0, \pi ) \times {\mathbf {R}},} \hfill \\ {u(0, t) = u(\pi , t) = 0,} \hfill & {t \in {\mathbf {R}},} \hfill \\ {u(x, t + 2\pi ) = u(x, t),} \hfill & {(x, t) \in (0, \pi ) \times {\mathbf {R}},} \hfill \\ \end {array} \] is considered. For all $p \in (1, \infty )$ and $f(x, t) \in {L^{(p + 1)/p}}$, existence of infinitely many periodic solutions is proved. This improves the results of the author [29, 30]. We use variational methods to show the existence result. Minimax arguments and energy estimates for the corresponding functional play an essential role in the proof.