Multiple bounded variation solutions of a periodically perturbed sine-curvature equation

We prove the existence of at least two $T$-periodic solutions, not differing from each other by an integer multiple of $2\pi$, of the sine-curvature equation$$-\Big( u'/{ \sqrt{1+{u'}^2}}\Big)' = A \sin u + h(t).$$We assume that $A\in\RR$and $h\in L^1_{\rm loc}(\RR)$ is a $T$-periodic function such that $\int_0^T h \, dt=0$ and, e.g., $\|h\|_{L^\infty} < 4/T$. Our approach is variational and makes use of basic results of non-smooth critical point theory in the space of bounded variation functions.