  We formulate analogues of the Birch and Swinnerton-Dyer conjecture for the
$p$-adic $L$-functions of Bertolini-Darmon-Prasanna attached to elliptic curves
$E/\mathbf{Q}$ at primes $p$ of good ordinary reduction. Using Iwasawa theory,
we then prove under mild hypotheses one of the inequalities predicted by the
rank part of our conjectures, as well as the predicted leading coefficient
formula up to a $p$-adic unit. Our conjectures are very closely related to
conjectures of Birch and Swinnerton-Dyer type formulated by Bertolini-Darmon in
1996 for certain Heegner distributions, and as application of our results we
also obtain the proof of an inequality in the rank part of their conjectures.
