  We deal with suitable nonlinear versions of Jauregui's isocapacitary mass in
3-manifolds with nonnegative scalar curvature and compact outermost minimal
boundary. These masses, which depend on a parameter $1<p\leq 2$, interpolate
between Jauregui's mass $p=2$ and Huisken's isoperimetric mass, as $p \to 1^+$.
We derive positive mass theorems for these masses under mild conditions at
infinity, and we show that these masses do coincide with the ADM mass when the
latter is defined. We finally work out a nonlinear potential theoretic proof of
the Penrose inequality in the optimal asymptotic regime.
