  In this paper, we obtain bounds for the Mordell-Weil ranks over cyclotomic
extensions of a wide range of abelian varieties defined over a number field $F$
whose primes above $p$ are totally ramified over $F/\mathbb{Q}$. We assume that
the abelian varieties may have good non-ordinary reduction at those primes. Our
work is a generalization of \cite{Kim}, in which the second author generalized
Perrin-Riou's Iwasawa theory for elliptic curves over $\mathbb{Q}$ with
supersingular reduction (\cite{Perrin-Riou}) to elliptic curves defined over
the above-mentioned number field $F$. On top of non-ordinary reduction and the
ramification of the field $F$, we deal with the additional difficulty that the
dimensions of the abelian varieties can be any number bigger than 1 which
causes a variety of issues. As a result, we obtain bounds for the ranks over
cyclotomic extensions $\mathbb{Q}(\mu_{p^{\max(M,N)+n}})$ of the Jacobian
varieties of {\it ramified} hyperelliptic curves $y^{2p^M}=x^{3p^N}+ax^{p^N}+b$
among others.
