  A new stabilization phenomenon induced by degenerate diffusion is discovered
in the context of pinned planar $p$-elasticae. It was known that in the
non-degenerate regime $p\in(1,2]$, including the classical case of Euler's
elastica, there are no local minimizers other than unique global minimizers.
Here we prove that, in stark contrast, in the degenerate regime
$p\in(2,\infty)$ there emerge uncountably many local minimizers with diverging
energy.
