  Instabilities in a neutron star can generate Alfv\'en waves in its
magnetosphere. Propagation along the curved magnetic field lines strongly
shears the wave, boosting its electric current $j_{\rm A}$. We derive an
analytic expression for the evolution of the wave vector $\boldsymbol{k}$ and
the growth of $j_{\rm A}$. In the strongly sheared regime, $j_{\rm A}$ may
exceed the maximum current $j_{0}$ that can be supported by the background
$e^{\pm}$ plasma. We investigate these "charge-starved" waves, first using a
simplified two-fluid analytic model, then with first-principles kinetic
simulations. We find that the Alfv\'en wave continues to propagate successfully
even when $\kappa \equiv j_{\rm A}/j_{0} \gg 1$. It sustains $j_{\rm A}$ by
compressing and advecting the plasma along the magnetic field lines with
particle Lorentz factors $\sim \kappa^{1/2}$. The simulations show how plasma
instabilities lead to gradual dissipation of the wave energy, giving a
dissipation power $L_{\rm diss}\sim 10^{35}(\kappa/100)^{1/2}
(B_w/10^{11}\,{\rm G})\,\mathrm{erg/s}$, where $B_w$ is the wave amplitude. Our
results imply that dissipation due to charge starvation is not sufficient to
power observed fast radio bursts (FRBs), in contrast to recent proposals.
