  In the present paper, we study the $p$-adic $L$-functions and the (strict)
Selmer groups over $\mathbb{Q}_{\infty}$, the cyclotomic
$\mathbb{Z}_p$-extension of $\mathbb{Q}$, of the $p$-adic weight one cusp forms
$f$, obtained via the $p$-stabilization of weight one Eisenstein series, under
the assumption that a certain Eisenstein component of the $p$-ordinary
universal cuspidal Hecke algebra is Gorenstein. As an application, we compute
the Iwasawa invariants of ordinary modular forms of weight $k\geq 2$ with the
same residual Galois representations as the one of $f$, which in our setting,
is reducible and non-$p$-distinguished. Combining this with a result of Kato
\cite[Theorem~17.4.2]{kato04}, we prove the Iwasawa main conjecture for these
forms. Also, we give numerical examples that satisfy the Gorenstein hypothesis.
  The crucial point on the analytic counter part is that under the Gorenstein
hypothesis, we are able to define, following Greenberg--Vatsal, the $p$-adic
$L$-functions of $p$-adic weight one forms $f$ as an element in the
one-dimensional Iwasawa algebra by using Mazur--Kitagawa two-variable $p$-adic
$L$-function and then, to compute them explicitly via local explicit
reciprocity law. On the algebraic counter part, we compute the (strict) Selmer
groups of $f$ over $\mathbb{Q}_{\infty}$ via the knowledge of the Galois
representations of $f$ studied in \cite{BDP}.
