  Consider first passage percolation with identical and independent weight
distributions and first passage time ${\rm T}$. In this paper, we study the
upper tail large deviations $\mathbb{P}({\rm T}(0,nx)>n(\mu+\xi))$, for $\xi>0$
and $x\neq 0$ with a time constant $\mu$ and a dimension $d$, for weights that
satisfy a tail assumption $ \beta_1\exp{(-\alpha t^r)}\leq \mathbb
P(\tau_e>t)\leq \beta_2\exp{(-\alpha t^r)}.$ When $r\leq 1$ (this includes the
well-known Eden growth model), we show that the upper tail large deviation
decays as $\exp{(-(2d\xi +o(1))n)}$. When $1< r\leq d$, we find that the rate
function can be naturally described by a variational formula, called the
discrete p-Capacity, and we study its asymptotics. For $r<d$, we show that the
large deviation event ${\rm T}(0,nx)>n(\mu+\xi)$ is described by a localization
of high weights around the origin. The picture changes for $r\geq d$ where the
configuration is not anymore localized.
