Weighted EGZ Constant for p-groups of rank 2

Let $G$ be a finite abelian group of exponent $n$, written additively, and let $A$ be a subset of $mathbb{Z}$. The constant $s_A(G)$ is defined as the smallest integer $ell$ such that any sequence over $G$ of length at least $ell$ has an $A$-weighted zero-sum of length $n$ and $eta_A(G)$ defined as the smallest integer $ell$ such that any sequence over $G$ of length at least $ell$ has an $A$-weighted zero-sum of length at most $n$. Here we prove that, for $alpha geq beta$, and $A=left{xinmathbb{N}; : ; 1 le a le p^{alpha} ; mbox{ and }; gcd(a, p) = 1right }$, we have $s_{A}(mathbb{Z}_{p^{alpha}}oplus mathbb{Z}_{p^beta}) = eta_A(mathbb{Z}_{p^{alpha}}oplus mathbb{Z}_{p^beta}) + p^{alpha}-1 = p^{alpha} + alpha +beta$ and classify all the extremal $A$-weighted zero-sum free sequences.