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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, underthe assumption that a certain Eisenstein component of the $p$-ordinaryuniversal cuspidal Hecke algebra is Gorenstein. As an application, we computethe Iwasawa invariants of ordinary modular forms of weight $k\geq 2$ with thesame 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 theseforms. Also, we give numerical examples that satisfy the Gorenstein hypothesis.  The crucial point on the analytic counter part is that under the Gorensteinhypothesis, we are able to define, following Greenberg--Vatsal, the $p$-adic$L$-functions of $p$-adic weight one forms $f$ as an element in theone-dimensional Iwasawa algebra by using Mazur--Kitagawa two-variable $p$-adic$L$-function and then, to compute them explicitly via local explicitreciprocity law. On the algebraic counter part, we compute the (strict) Selmergroups of $f$ over $\mathbb{Q}_{\infty}$ via the knowledge of the Galoisrepresentations of $f$ studied in \cite{BDP}.

On Iwasawa invariants of modular forms with reducible and non-$p$-distinguished residual Galois representations