  The molecular ion H$_3^+$ is a potentially powerful tracer of the ionospheres
and thermal structures of Jovian planets, but has never been detected in a
planetary mass object outside of the solar system. Models predict that H$_3^+$
emission driven by EUV flux and solar wind on hot Jupiters, or by powerful
aurorae on brown dwarfs, will be between $10^2$ and $10^5\times$ more intense
than that of Jupiter. If optimal conditions for the production of emission do
exist, the emission may be detectable by current ground-based instruments or in
the near future. We present the first search for H$_3^+$ line emission in brown
dwarfs with Keck/NIRSPEC $L^\prime$ high-resolution spectroscopy. Additionally,
we survey stars hosting giant planets at semi-major axes near $0.1-0.2$ au,
which models suggest may be the best planetary targets. No candidate H$_3^+$
emission is found. The limits we place on the emission of H$_3^+$ from brown
dwarfs indicates that auroral generation of H$_3^+$ in these environments
likely does not linearly scale from the processes found on Jupiter, plausibly
due to deeper atmospheric penetration by precipitating auroral electrons.
Detection of H$_3^+$ emission in brown dwarfs may be possible with the James
Webb Space Telescope (JWST), or future thirty-meter class telescopes.
