Phase Transitions in the semi-infinite Ising model with a decaying field

We study the semi-infinite Ising model with an external field $h_i=\lambda \mathrm{\mid }{i}_d{\mathrm{\mid }}^{-\delta }$, $\lambda$ is the wall influence, and $\delta >0$. This external field decays as it gets further away from the wall. We are able to show that when $\delta >1$ and $\beta >{\beta }_c(d)$, there exists a critical value $0<{\lambda }_c:={\lambda }_c(\delta ,\beta )$ such that, for $\lambda <{\lambda }_c$ there is phase transition and for $\lambda >{\lambda }_c$ we have uniqueness of the Gibbs state. In addition, when $\delta <1$, we have only one Gibbs state for any positive $\beta$ and $\lambda$.